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7252019 Maths Formula Class10

httpslidepdfcomreaderfullmaths-formula-class10 149

983107983148983137983155983155 983089983088 983117983137983156983144991257983155 983110983151983154983149983157983148983137 983107983106983123983109 983107983148983137983155983155 10 983117983137983156983144991257983155 983123983157983149983149983137983154983161

983124983144983145983155 983152983140983142 983148983145983155983156 983137983148983148 983156983144983141 983107983148983137983155983155 10 983107983106983123983109 983149983137983156983144991257983155 983142983151983154983149983157983148983137 983145983150 983137 983139983151983150983139983145983155983141

983149983137983150983150983141983154 983156983151 983144983141983148983152 983156983144983141 983155983156983157983140983141983150983156983155 983145983150 983154983141983158983145983155983145983151983150 983137983150983140 983141983160983137983149983145983150983137983156983145983151983150 983137983155

983152983141983154 983118983107983109983122983124 983155983161983148983148983137983138983157983155

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983151983150983148983161

1

983122983141983137983148 983118983157983149983138983141983154983155

Sno Type of Numbers Description

1 Natural Numbers N = 12345helliphelliphellipIt is the counting numbers

2 Whole number W= 012345helliphellipIt is the counting numbers + zero

3 Integers Z=hellip-7-6-5-4-3-2-10123456hellip

4 Positive integers Z+= 12345helliphellip

5 Negative integers Z-=hellip-7-6-5-4-3-2-1

6 Rational Number A number is called rational if it can be expressed

in the form pq where p and q are integers ( qgt

0)

Example frac12 43 57 1 etc

7 Irrational Number A number is called rational if it cannot be

expressed in the form pq where p and q are

integers ( qgt 0)

Example radic 3 radic 2 radic 5 etc

8 Real Numbers All rational and all irrational number makes the

collection of real number It is denoted by the

letter R

Sno Terms Descriptions

1 Euclidrsquos Division

Lemma

For a and b any two positive integer we can always

find unique integer q and r such that

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983151983150983148983161

2

a=bq + r 0 le r le b

If r =0 then b is divisor of a

2 HCF (Highest

common factor)

HCF of two positive integers can be find using the

Euclidrsquos Division Lemma algorithm

We know that for any two integers a b we can write

following expression

a=bq + r 0 le r le b

If r=0 then

HCF( a b) =b

If rne0 then

HCF ( a b) = HCF ( br)

Again expressing the integer br in Euclidrsquos Division

Lemma we get

b=pr + r 1

HCF ( br)=HCF ( rr 1)

Similarly successive Euclid lsquos division can be written

until we get the remainder zero the divisor at that

point is called the HCF of the a and b

3 HCF ( ab) =1 Then a and b are co primes

4 Fundamental

Theorem of

Arithmetic

Composite number = Product of primes

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983151983150983148983161

983091

5 HCF and LCM byprime factorization

method

HCF = Product of the smallest power of eachcommon factor in the numbers

LCM = Product of the greatest power of each prime

factor involved in the number

6 Important Formula 983112983107983110 (983137983138) 983128 983116983107983117 (983137983138) 983101983137 983128 983138

7 Important concept

for rational Number

Terminating decimal expression can be written in

the form

p2n

5m

Polynomial expressions

A polynomial expression S(x) in one variable x is an algebraic expression in x term

as

⋯ 991270 991270 991270 Where anan-1hellipaa0 are constant and real numbers and an is not equal to zero

Some Important point to Note

Sno Points

1 an an-1 an-2 hellipa1a0 are called the coefficients for xnxn-1 hellipx1x0

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983151983150983148983161

983092

2 n is called the degree of the polynomial

3 when an an-1 an-2 hellipa1a0 all are zero it is called zero polynomial

4 A constant polynomial is the polynomial with zero degree it is a constant

value polynomial

5 A polynomial of one item is called monomial two items binomial and three

items as trinomial

6 A polynomial of one degree is called linear polynomial two degree as

quadratic polynomial and degree three as cubic polynomial

Important concepts on Polynomial

Concept Description

Zerorsquos or roots

of the

polynomial

It is a solution to the polynomial equation S(x)=0 ie a number

a is said to be a zero of a polynomial if S(a) = 0

If we draw the graph of S(x) =0 the values where the curve

cuts the X-axis are called Zeroes of the polynomial

Remainder

Theoremrsquos

If p(x) is an polynomial of degree greater than or equal to 1

and p(x) is divided by the expression (x-a)then the remainder

will be p(a)

FactorrsquosTheoremrsquos

If x-a is a factor of polynomial p(x) then p(a)=0 or if p(a)=0x-a is the factor the polynomial p(x)

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983151983150983148983161

983093

Geometric Meaning of the Zeroes of the polynomial

Letrsquos us assume

y= p(x) where p(x) is the polynomial of any form

Now we can plot the equation y=p(x) on the Cartesian plane by taking various values of x and y obtained

by putting the values The plot or graph obtained can be of any shapes

The zeroes of the polynomial are the points where the graph meet x axis in the Cartesian plane If the

graph does not meet x axis then the polynomial does not have any zerorsquos

Let us take some useful polynomial and shapes obtained on the Cartesian plane

Sno y=p(x) Graph obtained Name of the

graph

Name of the

equation

1 y=ax+b where a and b can be

any values (ane0)

Example y=2x+3

Straight line

It intersect the x-

axis at ( -ba 0)

Example ( -320)

Linear polynomial

2 y=ax2+bx+c

where

b2-4ac gt 0 and ane0 and agt 0

Example

y=x2-7x+12

Parabola

It intersect the x-

axis at two points

Example

(30) and (40)

Quadratic

polynomial

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983151983150983148983161

983094

3 y=ax2+bx+c

where

b2-4ac gt 0 and ane0 and a lt 0

Example

y=-x2+2x+8

Parabola

It intersect the x-

axis at two points

Example (-20)

and (40)

Quadratic

polynomial

4 y=ax +bx+c

where

b2-4ac = 0 and ane0 a gt 0

Example

y=(x-2)2

Parabola

It intersect the x-

axis at one points

Quadratic

polynomial

5 y=ax2+bx+c

where

b

2

-4ac lt 0 and ane0 a gt 0

Example

y=x2-2x+6

Parabola

It does not

intersect the x-axis

It has no zerorsquos

Quadratic

polynomial

6 y=ax +bx+c

where

b2-4ac lt 0 and ane0 a lt 0

Example

y=-x2-2x-6

Parabola

It does not

intersect the x-axis

It has no zerorsquos

Quadratic

polynomial

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983124983144983145983155 983149983137983156983141983154983145983137983148 983145983155 983139983154983141983137983156983141983140 983138983161 983144983156983156983152983098983152983144983161983155983145983139983155983139983137983156983137983148983161983155983156983139983151983149 983137983150983140 983145983155 983142983151983154 983161983151983157983154 983152983141983154983155983151983150983137983148 983137983150983140 983150983151983150983085983139983151983149983149983141983154983139983145983137983148 983157983155983141

983151983150983148983161

983095

7 y=ax3 +bx

2+cx+d

where ane0

It can be of any

shape

It will cut the x-axis

at the most 3 times

Cubic Polynomial

8 ⋯ 991270 991270 991270

Where an ne0

It can be of any

shape

It will cut the x-axis

at the most n times

Polynomial of n

degree

Relation between coefficient and zeroes of the Polynomial

Sno Type of Polynomial General form Zerorsquos Relationship between Zerorsquos

and coefficients

1 Linear polynomial ax+b ane0 1

2 Quadratic ax +bx+c ane0 2

3 Cubic ax +bx +cx+d ane0 3

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983151983150983148983161

983096

Formation of polynomial when the zeroes are given

Type ofpolynomial

Zerorsquos Polynomial Formed

Linear k=a (x-a)

Quadratic k1=a and

k2=b

(x-a)(x-b)

Or

x2-( a+b)x +ab

Or

x2

-( Sum of the zerorsquos)x +product of the zerorsquos

Cubic k1=a k2=b

and k3=c

(x-a)(x-b)(x-c)

Division algorithm for Polynomial

Letlsquos p(x) and q(x) are any two polynomial with q(x) ne0 then we can find polynomial s(x) and r(x) suchthat

P(x)=s(x) q(x) + r(x)

Where r(x) can be zero or degree of r(x) lt degree of g(x)

983108983145983158983145983140983141983150983140 983101983121983157983151983156983145983141983150983156 983128 983108983145983158983145983155983151983154 + 983122983141983149983137983145983150983140983141983154

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983124983144983145983155 983149983137983156983141983154983145983137983148 983145983155 983139983154983141983137983156983141983140 983138983161 983144983156983156983152983098983152983144983161983155983145983139983155983139983137983156983137983148983161983155983156983139983151983149 983137983150983140 983145983155 983142983151983154 983161983151983157983154 983152983141983154983155983151983150983137983148 983137983150983140 983150983151983150983085983139983151983149983149983141983154983139983145983137983148 983157983155983141

983151983150983148983161

983097

COORDINATE GEOMETRYSno Points

1 We require two perpendicular axes to locate a point in the plane One of

them is horizontal and other is Vertical

2 The plane is called Cartesian plane and axis are called the coordinates axis

3 The horizontal axis is called x-axis and Vertical axis is called Y-axis

4 The point of intersection of axis is called origin

5 The distance of a point from y axis is called x ndashcoordinate or abscissa and

the distance of the point from x ndashaxis is called y ndash coordinate or Ordinate

6 The distance of a point from y axis is called x ndashcoordinate or abscissa and

the distance of the point from x ndashaxis is called y ndash coordinate or Ordinate

7 The Origin has zero distance from both x-axis and y-axis so that its

abscissa and ordinate both are zero So the coordinate of the origin is (0

0)

8 A point on the x ndashaxis has zero distance from x-axis so coordinate of any

point on the x-axis will be (x 0)

9 A point on the y ndashaxis has zero distance from y-axis so coordinate of any

point on the y-axis will be (0 y)

10 The axes divide the Cartesian plane in to four parts These Four parts arecalled the quadrants

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983151983150983148983161

10

The coordinates of the points in the four quadrants will have sign according to thebelow table

Quadrant x-coordinate y-coordinate

Ist Quadrant + +

IInd quadrant - +

IIIrd quadrant - -

IVth quadrant + -

Sno Terms Descriptions

1 Distance formula Distance between the points AB is given by

Distance of Point A from Origin

2 Section Formula A point P(xy) which divide the line segment AB in

the ratio m1 and m2 is given by

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983151983150983148983161

11

The midpoint P is given by

3 Area of Triangle Area of triangle ABC of coordinates A(x 1y 1 )

B(x 2 y 2 ) and C(x 3y 3 )

12 For point AB and C to be collinear The value of A

should be zero

LINEAR EQUATIONS IN TWO

VARIABLESAn equation of the form ax + by + c = 0 where a b and c are real numbers such

that a andb are not both zero is called a linear equation in two variablesImportant points to Note

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983151983150983148983161

12

Sno Points

1 A linear equation in two variable has infinite solutions

2 The graph of every linear equation in two variable is a straight line

3 x = 0 is the equation of the y-axis and y = 0 is the equation of the x-axis

4 The graph x=a is a line parallel to y -axis

5 The graph y=b is a line parallel to x -axis

6 An equation of the type y = mx represents a line passing through theorigin

7 Every point on the graph of a linear equation in two variables is a solution

of the linear

equation Moreover every solution of the linear equation is a point on the

graph

Sno Type of equation Mathematical

representation

Solutions

1 Linear equation in one Variable ax+b=0 ane0

a and b are real

number

One solution

2 Linear equation in two Variable ax+by+c=0 ane0 and

bne0

a b and c are real

number

Infinite solution

possible

3 Linear equation in three Variable ax+by+cz+d=0 ane0

bne0 and cne0

Infinite solution

possible

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983151983150983148983161

1983091

a b c d are realnumber

Simultaneous pair of Linear equation

A pair of Linear equation in two variables

a1x+b1y+c1=0

a2x +b2y+c2=0

Graphically it is represented by two straight lines on Cartesian plane

Simultaneous pair of

Linear equation

Condition Graphical

representation

Algebraic

interpretation

a1x+b1y+c1=0

a2x +b2y+c2=0

Example

x-4y+14=0

3x+2y-14=0

Intersecting lines The

intersecting point

coordinate is the only

solution

One unique solution

only

a1x+b1y+c1=0

a2x +b2y+c2=0

Coincident lines Theany coordinate on the

line is the solution

Infinite solution

7252019 Maths Formula Class10

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983124983144983145983155 983149983137983156983141983154983145983137983148 983145983155 983139983154983141983137983156983141983140 983138983161 983144983156983156983152983098983152983144983161983155983145983139983155983139983137983156983137983148983161983155983156983139983151983149 983137983150983140 983145983155 983142983151983154 983161983151983157983154 983152983141983154983155983151983150983137983148 983137983150983140 983150983151983150983085983139983151983149983149983141983154983139983145983137983148 983157983155983141

983151983150983148983161

1983092

Example

2x+4y=16

3x+6y=24

a1x+b1y+c1=0

a2x +b2y+c2=0

Example

2x+4y=6

4x+8y=18

Parallel Lines No solution

The graphical solution can be obtained by drawing the lines on the Cartesian plane

Algebraic Solution of system of Linear equation

Sno Type of method Working of method

1 Method of elimination by

substitution

1) Suppose the equation are

a1x+b1y+c1=0

a2x +b2y+c2=0

2) Find the value of variable of either x or y in other

variable term in first equation

3) Substitute the value of that variable in second

equation

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4) Now this is a linear equation in one variable Findthe value of the variable

5) Substitute this value in first equation and get the

second variable

2 Method of elimination by

equating the coefficients

1) Suppose the equation are

a1x+b1y+c1=0

a2x +b2y+c2=0

2) Find the LCM of a1 and a2 Let it k

3) Multiple the first equation by the value ka1

4) Multiple the first equation by the value ka2

4) Subtract the equation obtained This way one

variable will be eliminated and we can solve to get the

value of variable y

5) Substitute this value in first equation and get the

second variable

3 Cross Multiplication method 1) Suppose the equation are

a1x+b1y+c1=0

a2x +b2y+c2=0

2) This can be written as

3) This can be written as

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4) Value of x and y can be find using the

x =gt first and last expression

y=gt second and last expression

Quadratic EquationsSno Terms Descriptions

1 Quadratic Polynomial P(x) = ax +bx+c where ane0

2 Quadratic equation ax +bx+c =0 where ane0

3 Solution or root of the

Quadratic equation

A real number α is called the root or solution of the

quadratic equation if

aα2 +bα+c=0

4 zeroes of the

polynomial p(x)

The root of the quadratic equation are called zeroes

5 Maximum roots of

quadratic equations

We know from chapter two that a polynomial of

degree can have max two zeroes So a quadratic

equation can have maximum two roots

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6 Condition for realroots A quadratic equation has real roots if b

2

- 4ac gt 0

How to Solve Quadratic equation

Sno Method Working

1 factorization This method we factorize the equation by splitting themiddle term b

In ax2+bx+c=0

Example

6x2-x-2=0

1) First we need to multiple the coefficient a and cIn this

case =6X-2=-12

2) Splitting the middle term so that multiplication is 12

and difference is the coefficient b

6x2 +3x-4x-2=0

3x( 2x+1) -2(2x+1)=0

(3x-2) (2x+1)=0

3) Roots of the equation can be find equating the factorsto zero

3x-2=0 =gt x=32

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2x+1=0 =gt x=-12

2 Square method In this method we create square on LHS and RHS and

then find the value

ax2 +bx+c=0

1) x2 +(ba) x+(ca)=0

2) ( x+b2a)2 ndash(b2a)2 +(ca)=0

3) ( x+b2a)2=(b2-4ac)4a2

4) radic

Example

x2 +4x-5=0

1) (x+2)2 -4-5=0

2) (x+2)2=9

3) Roots of the equation can be find using square root on

both the sides

x+2 =-3 =gt x=-5

x+2=3=gt x=1

3 Quadratic

method

For quadratic equation

ax2 +bx+c=0

roots are given by radic radic

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For b2 -4ac gt 0 Quadratic equation has two real roots of

different value

For b2-4ac =0 quadratic equation has one real root

For b2-4ac lt 0 no real roots for quadratic equation

Nature of roots of Quadratic equation

Sno Condition Nature of roots

1 b -4ac gt 0 Two distinct real roots

2 b -4ac =0 One real root

3 b

2

-4ac lt 0 No real roots

Triangles

Sno Terms Descriptions

1 Congruence Two Geometric figure are said to be congruence ifthey are exactly same size and shapeSymbol used is cong Two angles are congruent if they are equalTwo circle are congruent if they have equal radiiTwo squares are congruent if the sides are equal

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2 Triangle Congruencebull

Two triangles are congruent if three sidesand three angles of one triangle is

congruent to the corresponding sides and

angles of the other

bull Corresponding sides are equal

AB=DE BC=EF AC=DF

bull Corresponding angles are equalang angang angang ang

bull We write this as

cong

bull The above six equalities are between the

corresponding parts of the two congruent

triangles In short form this is called

CPCT

bull We should keep the letters in correct order

on both sides

3 Inequalities in

Triangles

1) In a triangle angle opposite to longer side is

larger

2) In a triangle side opposite to larger angle is

larger

3) The sum of any two sides of the triangle is

greater than the third side

983105

983107983106

983108

983109 983110

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In triangle ABC

AB +BC gt AC

Different Criterion for Congruence of the triangles

N Criterion Description Figures and

expression

1 Side angle

Side (SAS)

congruence

bull Two triangles are congruent if the

two sides and included angles of

one triangle is equal to the two

sides and included angle

bull It is an axiom as it cannot be

proved so it is an accepted truth

bull ASS and SSA type two triangles

may not be congruent alwaysIf following

condition

983105

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983107983106

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22

AB=DE BC=EFang ang

Then

cong

2 Angle side

angle (ASA)

congruence

bull Two triangles are congruent if the

two angles and included side of

one triangle is equal to the

corresponding angles and side

bull It is a theorem and can be proved

If following

condition

BC=EF

ang angang ang

Then

cong

3 Angle angle

side( AAS)

congruence

bull Two triangles are congruent if the

any two pair of angles and any

side of one triangle is equal to the

corresponding angles and side

bull It is a theorem and can be proved

983110983109

983108

983107983106

983105

983110983109

983108

983107983106

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If followingcondition

BC=EF

ang angang ang

Then

cong

4 Side-Side-Side

(SSS)congruence

bull Two triangles are congruent if the

three sides of one triangle is equalto the three sides of the another

If following

condition

BC=EFAB=DEDF

=AC

Then

cong

5 Right angle ndash

hypotenuse-

side(RHS)

bull Two right triangles are congruent if

the hypotenuse and a side of the

one triangle are equal to

983110983109

983108

983107983106

983105

983105 983108

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congruence corresponding hypotenuse and sideof the another

If following

condition

AC=DFBC=EF

Then

cong

Some Important points on Triangles

Terms Description

Orthocenter Point of intersection of the three altitudeof the triangle

Equilateral triangle whose all sides are equal and allangles are equal to 600

Median A line Segment joining the corner of thetriangle to the midpoint of the oppositeside of the triangle

Altitude A line Segment from the corner of thetriangle and perpendicular to theopposite side of the triangle

Isosceles A triangle whose two sides are equalCentroid Point of intersection of the three median

of the triangle is called the centroid of

the triangleIn center All the angle bisector of the trianglepasses through same point

Circumcenter The perpendicular bisector of the sidesof the triangles passes through samepoint

983106 983107 983109 983110

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Scalene triangle Triangle having no equal angles and noequal sidesRight Triangle Right triangle has one angle equal to 900 Obtuse Triangle One angle is obtuse angle while other

two are acute anglesAcute Triangle All the angles are acute

Similarity of Triangles

Sno Points

1 Two figures having the same shape but not necessarily the same size are called similarfigures

2 All the congruent figures are similar but the converse is not true

3 If a line is drawn parallel to one side of a triangle to intersect the other two sides in

distinct points then the other two sides are divided in the same ratio

4 If a line divides any two sides of a triangle in the same ratio then the line is parallel tothe third side

Different Criterion for Similarity of the triangles

N Criterion Description Expression

1 Angle Angle

angle(AAA)

similarity

bull Two triangles are similar if

corresponding angle are equal

If following

condition

ang ang

ang ang

ang ang

Then

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Then 983166

2 Angle angle

(AA) similarity

bull Two triangles are similar if the two

corresponding angles are equal as

by angle property third angle will

be also equal

If following

condition

ang ang

ang ang

Then

ang ang

Then

Then

983166

3 Side side

side(SSS)

Similarity

Two triangles are similar if the

sides of one triangle is

proportional to the sides of other

triangle

If following

condition

Then

ang ang

ang ang

ang ang

Then

cong

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4 Side-Angle-Side (SAS)

similarity

bull

Two triangles are similar if the oneangle of a triangle is equal to one

angle of other triangles and sides

including that angle is proportional

If followingcondition

And ang ang

Then

cong

Area of Similar triangles

If the two triangle ABC and DEF are similar

cong

Then

Pythagoras Theorem

Sno Points

1 If a perpendicular is drawn from the vertex of the right angle of a right triangle to thehypotenuse then the triangles on both sides of the perpendicular are similar to the

whole triangle and also to each other

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2 In a right triangle the square of the hypotenuse is equal to the sum of the squares of the

other two sides (Pythagoras Theorem)

(hyp)2 =(base)

2 + (perp)

2

3 If in a triangle square of one side is equal to the sum of the squares of the other twosides then the angle opposite the first side is a right angle

Arithmetic Progression

Sno Terms Descriptions

1 ArithmeticProgression

An arithmetic progression is a sequence ofnumbers such that the difference of any two

successive members is a constant

Examples

1) 1591317hellip

2) 12345hellip

2 common difference of

the AP

the difference between any successive members is a

constant and it is called the common difference of AP

1) If a1 a2a3a4a5 are the terms in AP then

D=a2 -a1 =a3 - a2 =a4 ndash a3=a5 ndasha4

2) We can represent the general form of AP inthe form

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aa+da+2da+3da+4d

Where a is first term and d is the commondifference

3 nth term of Arithmetic Progression

n term = a + (n - 1)d

4 Sum of nth item in ArithmeticProgression

Sn =(n2)[a + (n-1)d]

Or

Sn =(n2)[t1+ tn]

Trigonometry

Sno Terms Descriptions

1 What is

Trigonometry

Trigonometry from Greek trigotildenon triangle and

metron measure) is a branch of mathematics

that studies relationships involving lengths and

angles of triangles The field emerged during the

3rd century BC from applications of geometry to

astronomical studies

Trigonometry is most simply associated with

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planar right angle triangles (each of which is atwo-dimensional triangle with one angle equal to

90 degrees) The applicability to non-right-angle

triangles exists but since any non-right-angle

triangle (on a flat plane) can be bisected to

create two right-angle triangles most problems

can be reduced to calculations on right-angle

triangles Thus the majority of applications relate

to right-angle triangles

2 Trigonometric

Ratiorsquos

In a right angle triangle ABC where B=90deg

We can define following term for angle A

Base Side adjacent to angle

Perpendicular Side Opposite of angleHypotenuse Side opposite to right angle

We can define the trigonometric ratios for

angle A as

sin A= PerpendicularHypotenuse =BCAC

cosec A= HypotenusePerpendicular

=ACBC

cos A= BaseHypotenuse =ABAC

sec A= HypotenuseBase=ACAB

tan A= PerpendicularBase =BCAB

cot A= BasePerpendicular=ABBC

Notice that each ratio in the right-hand column is

the inverse or the reciprocal of the ratio in the

left-hand column

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3 Reciprocal offunctions The reciprocal of sin A is cosec A and vice-versa

The reciprocal of cos A is sec A

And the reciprocal of tan A is cot A

These are valid for acute angles

We can define tan A = sin Acos A

And Cot A =cos A Sin A

4 Value of of sin and

cos

Is always less 1

5 Trigonometric ration

from another angle

We can define the trigonometric ratios for angle Cas

sin C= PerpendicularHypotenuse =ABACcosec C= HypotenusePerpendicular =ACABBcos C= BaseHypotenuse =BCACsec C= HypotenuseBase=ACBCtan A= PerpendicularBase =ABBCcot A= BasePerpendicular=BCAB

6 Trigonometric ratios

of complimentary

Sin (90-A) =cos(A)

Cos(90-A) = sin A

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angles Tan(90-A) =cot A

Sec(90-A)= cosec A

Cosec (90-A) =sec A

Cot(90- A) =tan A7 Trigonometric

identities

Sin2 A + cos2 A =0

1 + tan2 A =sec2 A

1 + cot2 A =cosec2 A

Trigonometric Ratios of Common angles

We can find the values of trigonometric ratiorsquos various angle

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Area of Circles

Sno Points

1 A circle is a collection of all the points in a plane which are equidistant

from a fixed point in the plane

2 Equal chords of a circle (or of congruent circles) subtend equal angles at

the center

3 If the angles subtended by two chords of a circle (or of congruent circles)

at the center (corresponding center) are equal the chords are equal

4 The perpendicular from the center of a circle to a chord bisects the chord

5 The line drawn through the center of a circle to bisect a chord is

perpendicular to the chord

6 There is one and only one circle passing through three non-collinear points

7 Equal chords of a circle (or of congruent circles) are equidistant from the

center (or corresponding centers)

8 Chords equidistant from the center (or corresponding centers) of a circle

(or of congruent circles) are equal

9 If two arcs of a circle are congruent then their corresponding chords are

equal and conversely if two chords of a circle are equal then their

corresponding arcs (minor major) are congruent

10 Congruent arcs of a circle subtend equal angles at the center

11 The angle subtended by an arc at the center is double the angle subtended

by it at any point on the remaining part of the circle

12 Angles in the same segment of a circle are equal

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13 Angle in a semicircle is a right angle

14 If a line segment joining two points subtends equal angles at two other

points lying on the same side of the line containing the line segment the

four points lie on a circle

15 The sum of either pair of opposite angles of a cyclic quadrilateral is 180deg

16 If the sum of a pair of opposite angles of a quadrilateral is 180deg then the

quadrilateral is cyclic

Sno Terms Descriptions

1 Circumference of a circle 2 π r

2 Area of circle π r 2

3 Length of the arc of

the sector of angle

Length of the arc of the sector of angle θ 2

4 Area of the sector of

angle

Area of the sector of angle θ

5 Area of segment of a

circle

Area of the corresponding sector ndash Area of the

corresponding triangle

Surface Area and Volume

Sno Term Description

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1 Mensuration It is branch of mathematics which is concernedabout the measurement of length area and

Volume of plane and Solid figure

2 Perimeter a)The perimeter of plane figure is defined as thelength of the boundaryb)It units is same as that of length ie m cmkm

3 Area a)The area of the plane figure is the surfaceenclosed by its boundaryb) It unit is square of length unit ie m2 km2

4 Volume Volume is the measure of the amount of spaceinside of a solid figure like a cube ball cylinderor pyramid Its units are always cubic that isthe number of little element cubes that fit insidethe figure

Volume Unit conversion

1 cm3 1mL 1000 mm3

1 Litre 1000ml 1000 cm3

1 m3 106 cm3 1000 L

1 dm3 1000 cm3 1 L

Surface Area and Volume of Cube and Cuboid

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Type Measurement

Surface Area of Cuboid of Length L

Breadth B and Height H

2(LB + BH + LH)

Lateral surface area of the cuboids 2( L + B ) H

Diagonal of the cuboids Volume of a cuboids LBH

Length of all 12 edges of the cuboids 4 (L+B+H)

Surface Area of Cube of side L 6L2

Lateral surface area of the cube 4L2

Diagonal of the cube radic3 Volume of a cube L3

Surface Area and Volume of Right circular cylinder

Radius The radius (r) of the circular base is called the radius of the

cylinder

Height The length of the axis of the cylinder is called the height (h) of the

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cylinder

Lateral

Surface

The curved surface joining the two base of a right circular cylinder is

called Lateral Surface

Type Measurement

Curved or lateral Surface Area of

cylinder

2πrh

Total surface area of cylinder 2πr (h+r)

Volume of Cylinder π r2h

Surface Area and Volume of Right circular cone

Radius The radius (r) of the circular base is called the radius of the

cone

Height The length of the line segment joining the vertex to the center of

base is called the height (h) of the cone

Slant

Height

The length of the segment joining the vertex to any point on the

circular edge of the base is called the slant height (L) of the cone

Lateral

surface

Area

The curved surface joining the base and uppermost point of a right

circular cone is called Lateral Surface

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Type Measurement

Curved or lateral Surface Area of

cone

πrL

Total surface area of cone πr (L+r)

Volume of Cone

Surface Area and Volume of sphere and hemisphere

Sphere A sphere can also be considered as a solid obtained on

rotating a circle About its diameter

Hemisphere A plane through the centre of the sphere divides the sphere into two

equal parts each of which is called a hemisphere

radius The radius of the circle by which it is formed

SphericalShell

The difference of two solid concentric spheres is called a sphericalshell

Lateral

Surface

Area for

Total surface area of the sphere

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Sphere

Lateral

Surface

area of

Hemisphere

It is the curved surface area leaving the circular base

Type Measurement

Surface area of Sphere 4πr2

Volume of Sphere 43 Curved Surface area of hemisphere 2πr2

Total Surface area of hemisphere 3πr2

Volume of hemisphere 23 Volume of the spherical shell whose

outer and inner radii and lsquoRrsquo and lsquorrsquorespectively

43

How the Surface area and Volume are determined

Area of Circle The circumference of a circle is 2πr

This is the definition of π (pi) Divide

the circle into many triangular

segments The area of the triangles

is 12 times the sum of their bases

2πr (the circumference) times their

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height r

12 2 Surface Area of cylinder This can be imagined as unwrapping the

surface into a rectangle

Surface area of cone This can be achieved by divide the

surface of the cone into its triangles or

the surface of the cone into many thin

triangles The area of the triangles is 12

times the sum of their bases p times

their height

12 2

Surface Area and Volume of frustum of cone

h = vertical height of the frustum

l = slant height of the frustum

r 1 and r 2 are radii of the two bases (ends) of the frustum

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Type Measurement

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Statistics

Sno Term Description

1 Statistics Statistics is a broad mathematical disciplinewhich studies ways to collect summarize and

draw conclusions from data

2 Data A systematic record of facts or different values of

a quantity is called data

Data is of two types - Primary data and Secondary

data

Primary Data The data collected by a researcher

with a specific purpose in mind is called primarydata

Secondary Data The data gathered from a

source where it already exists is called secondary

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data

3 Features of data bull Statistics deals with collectionpresentation analysis and interpretation ofnumerical data

bull Arranging data in an order to study theirsalient features is called presentation ofdata

bull Data arranged in ascending or descendingorder is called arrayed data or an array

bull Range of the data is the differencebetween the maximum and the minimum

values of the observationsbull Table that shows the frequency of different

values in the given data is called afrequency distribution table

bull A frequency distribution table that showsthe frequency of each individual value in thegiven data is called an ungrouped frequencydistribution table

bull A table that shows the frequency of groupsof values in the given data is called agrouped frequency distribution table

bull

The groupings used to group the values ingiven data are called classes or class-intervals The number of values that eachclass contains is called the class size orclass width The lower value in a class iscalled the lower class limit The higher valuein a class is called the upper class limit

bull Class mark of a class is the mid value ofthe two limits of that class

bull A frequency distribution in which the upperlimit of one class differs from the lower limitof the succeeding class is called an

Inclusive or discontinuous FrequencyDistribution

bull A frequency distribution in which the upperlimit of one class coincides from the lowerlimit of the succeeding class is called anexclusive or continuous Frequency

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Distribution bull

4 Bar graph A bar graph is a pictorial representation of data inwhich rectangular bars of uniform width are drawnwith equal spacing between them on one axisusually the x axis The value of the variable isshown on the other axis that is the y axis

5 Histogram A histogram is a set of adjacent rectangles whoseareas are proportional to the frequencies of agiven continuous frequency distribution

6 Mean The mean value of a variable is defined as thesum of all the values of the variable divided by thenumber of values

4 sum

7 Median The median of a set of data values is the middle

value of the data set when it has been arranged inascending order That is from the smallest valueto the highest value

Median is calculated as

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12 1 Where n is the number of values in the data

If the number of values in the data set is eventhen the median is the average of the two middlevalues

8 Mode Mode of a statistical data is the value of thatvariable which has the maximum frequency

Sno Term Description

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Mean is given by

sum sum

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In these distribution it is assumed that frequency of each classinterval is centered around its mid-point ie class marks

2

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983137983081 983108983145983154983141983139983156 983149983141983156983144983151983140

sum

sum

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sum sum

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Where

a=gt Assumed mean

di =gt xi ndasha

983139983081 983123983156983141983152 983140983141983158983145983137983156983145983151983150 983117983141983156983144983151983140

sum sum

Where

a=gt Assumed mean

ui =gt (xi ndasha)h

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Modal class The class interval having highest frequency is

called the modal class and Mode is obtained using the modal

class

2

983127983144983141983154983141

l = lower limit of the modal classh = size of the class interval (assuming all class sizes tobe equal)f 1 = frequency of the modal classf 0 = frequency of the class preceding the modal classf 2 = frequency of the class succeeding the modal class

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983143983154983151983157983152983141983140 983140983137983156983137

983142983154983141983153983157983141983150983139983161 983156983137983138983148983141

For the given data we need to have class interval frequency

distribution and cumulative frequency distribution

Median is calculated as

7252019 Maths Formula Class10

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983124983144983145983155 983149983137983156983141983154983145983137983148 983145983155 983139983154983141983137983156983141983140 983138983161 983144983156983156983152983098983152983144983161983155983145983139983155983139983137983156983137983148983161983155983156983139983151983149 983137983150983140 983145983155 983142983151983154 983161983151983157983154 983152983141983154983155983151983150983137983148 983137983150983140 983150983151983150983085983139983151983149983149983141983154983139983145983137983148 983157983155983141

983151983150983148983161

983092983094

2

983127983144983141983154983141

983148 983101 983148983151983159983141983154 983148983145983149983145983156 983151983142 983149983141983140983145983137983150 983139983148983137983155983155

983150 983101 983150983157983149983138983141983154 983151983142 983151983138983155983141983154983158983137983156983145983151983150983155

983139983142 983101 983139983157983149983157983148983137983156983145983158983141 983142983154983141983153983157983141983150983139983161 983151983142 983139983148983137983155983155 983152983154983141983139983141983140983145983150983143 983156983144983141 983149983141983140983145983137983150 983139983148983137983155983155

983142 983101 983142983154983141983153983157983141983150983139983161 983151983142 983149983141983140983145983137983150 983139983148983137983155983155

983144 983101 983139983148983137983155983155 983155983145983162983141 (983137983155983155983157983149983145983150983143 983139983148983137983155983155 983155983145983162983141 983156983151 983138983141 983141983153983157983137983148)

5983109983149983152983145983154983145983139983137983148

983110983151983154983149983157983148983137 983138983141983156983159983141983141983150

983117983151983140983141983084 983117983141983137983150 983137983150983140

983117983141983140983145983137983150

3 Median=Mode +2 Mean

ProbabilitySn

o

Term Description

1 Empirical

probability

It is a probability of event which is calculated based on

experiments

Example

A coin is tossed 1000 times we get 499 times head and

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983151983150983148983161

983092983095

501 times tail

So empirical or experimental probability of getting head is

calculated as

4991000 499

Empirical probability depends on experiment and

different will get different values based on the

experiment

2 Important point

about events

If the event A B C covers the entire possible outcome inthe experiment Then

P (A) +P (B) +P(C)=1

3 impossible

event

The probability of an event (U) which is impossible tooccur is 0 Such an event is called an impossible event

P (U)=0

4 Sure or certain

event

The probability of an event (X) which is sure (or certain)to occur is 1 Such an event is called a sure event or acertain event

P(X) =1

5 Probability of

any event

Probability of any event can be as

0 1

Sn

o

Term Description

7252019 Maths Formula Class10

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983092983096

1

983124983144983141983151983154983141983156983145983139983137983148

983120983154983151983138983137983138983145983148983145983156983161

The theoretical probability or the classical probability of the event isdefined as

2 Elementary

events

983105983150 983141983158983141983150983156 983144983137983158983145983150983143 983151983150983148983161 983151983150983141 983151983157983156983139983151983149983141 983151983142 983156983144983141 983141983160983152983141983154983145983149983141983150983156 983145983155 983139983137983148983148983141983140 983137983150

983141983148983141983149983141983150983156983137983154983161 983141983158983141983150983156

991260983124983144983141 983155983157983149 983151983142 983156983144983141 983152983154983151983138983137983138983145983148983145983156983145983141983155 983151983142 983137983148983148 983156983144983141 983141983148983141983149983141983150983156983137983154983161 983141983158983141983150983156983155 983151983142983137983150 983141983160983152983141983154983145983149983141983150983156 983145983155 1991261

983113983141 983113983142 983159983141 983156983144983154983141983141 983141983148983141983149983141983150983156983137983154983161 983141983158983141983150983156 983105983106983107 983145983150 983156983144983141 983141983160983152983141983154983145983149983141983150983156 983156983144983141983150

983120(983105)+983120(983106) +983120(983107)9831011

3 Complementar

y events

983124983144983141 983141983158983141983150983156 Ᾱ 983154983141983152983154983141983155983141983150983156983145983150983143 991256983150983151983156 983105991257 983145983155 983139983137983148983148983141983140 983156983144983141 983139983151983149983152983148983141983149983141983150983156 983151983142 983156983144983141 983141983158983141983150983156

983105 983127983141 983137983148983155983151 983155983137983161 983156983144983137983156 Ᾱ 983137983150983140 983105 983137983154983141 983139983151983149983152983148983141983149983141983150983156983137983154983161 983141983158983141983150983156983155 983105983148983155983151

983120(983105) +983120(Ᾱ)9831011

4 Sure or certain

event

The probability of an event (X) which is sure (or certain)

to occur is 1 Such an event is called a sure event or acertain event

P(X) =1

5 Probability of

any event

Probability of any event can be as

0 1

Page 2: Maths Formula Class10

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983151983150983148983161

1

983122983141983137983148 983118983157983149983138983141983154983155

Sno Type of Numbers Description

1 Natural Numbers N = 12345helliphelliphellipIt is the counting numbers

2 Whole number W= 012345helliphellipIt is the counting numbers + zero

3 Integers Z=hellip-7-6-5-4-3-2-10123456hellip

4 Positive integers Z+= 12345helliphellip

5 Negative integers Z-=hellip-7-6-5-4-3-2-1

6 Rational Number A number is called rational if it can be expressed

in the form pq where p and q are integers ( qgt

0)

Example frac12 43 57 1 etc

7 Irrational Number A number is called rational if it cannot be

expressed in the form pq where p and q are

integers ( qgt 0)

Example radic 3 radic 2 radic 5 etc

8 Real Numbers All rational and all irrational number makes the

collection of real number It is denoted by the

letter R

Sno Terms Descriptions

1 Euclidrsquos Division

Lemma

For a and b any two positive integer we can always

find unique integer q and r such that

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983151983150983148983161

2

a=bq + r 0 le r le b

If r =0 then b is divisor of a

2 HCF (Highest

common factor)

HCF of two positive integers can be find using the

Euclidrsquos Division Lemma algorithm

We know that for any two integers a b we can write

following expression

a=bq + r 0 le r le b

If r=0 then

HCF( a b) =b

If rne0 then

HCF ( a b) = HCF ( br)

Again expressing the integer br in Euclidrsquos Division

Lemma we get

b=pr + r 1

HCF ( br)=HCF ( rr 1)

Similarly successive Euclid lsquos division can be written

until we get the remainder zero the divisor at that

point is called the HCF of the a and b

3 HCF ( ab) =1 Then a and b are co primes

4 Fundamental

Theorem of

Arithmetic

Composite number = Product of primes

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983151983150983148983161

983091

5 HCF and LCM byprime factorization

method

HCF = Product of the smallest power of eachcommon factor in the numbers

LCM = Product of the greatest power of each prime

factor involved in the number

6 Important Formula 983112983107983110 (983137983138) 983128 983116983107983117 (983137983138) 983101983137 983128 983138

7 Important concept

for rational Number

Terminating decimal expression can be written in

the form

p2n

5m

Polynomial expressions

A polynomial expression S(x) in one variable x is an algebraic expression in x term

as

⋯ 991270 991270 991270 Where anan-1hellipaa0 are constant and real numbers and an is not equal to zero

Some Important point to Note

Sno Points

1 an an-1 an-2 hellipa1a0 are called the coefficients for xnxn-1 hellipx1x0

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983151983150983148983161

983092

2 n is called the degree of the polynomial

3 when an an-1 an-2 hellipa1a0 all are zero it is called zero polynomial

4 A constant polynomial is the polynomial with zero degree it is a constant

value polynomial

5 A polynomial of one item is called monomial two items binomial and three

items as trinomial

6 A polynomial of one degree is called linear polynomial two degree as

quadratic polynomial and degree three as cubic polynomial

Important concepts on Polynomial

Concept Description

Zerorsquos or roots

of the

polynomial

It is a solution to the polynomial equation S(x)=0 ie a number

a is said to be a zero of a polynomial if S(a) = 0

If we draw the graph of S(x) =0 the values where the curve

cuts the X-axis are called Zeroes of the polynomial

Remainder

Theoremrsquos

If p(x) is an polynomial of degree greater than or equal to 1

and p(x) is divided by the expression (x-a)then the remainder

will be p(a)

FactorrsquosTheoremrsquos

If x-a is a factor of polynomial p(x) then p(a)=0 or if p(a)=0x-a is the factor the polynomial p(x)

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983151983150983148983161

983093

Geometric Meaning of the Zeroes of the polynomial

Letrsquos us assume

y= p(x) where p(x) is the polynomial of any form

Now we can plot the equation y=p(x) on the Cartesian plane by taking various values of x and y obtained

by putting the values The plot or graph obtained can be of any shapes

The zeroes of the polynomial are the points where the graph meet x axis in the Cartesian plane If the

graph does not meet x axis then the polynomial does not have any zerorsquos

Let us take some useful polynomial and shapes obtained on the Cartesian plane

Sno y=p(x) Graph obtained Name of the

graph

Name of the

equation

1 y=ax+b where a and b can be

any values (ane0)

Example y=2x+3

Straight line

It intersect the x-

axis at ( -ba 0)

Example ( -320)

Linear polynomial

2 y=ax2+bx+c

where

b2-4ac gt 0 and ane0 and agt 0

Example

y=x2-7x+12

Parabola

It intersect the x-

axis at two points

Example

(30) and (40)

Quadratic

polynomial

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983151983150983148983161

983094

3 y=ax2+bx+c

where

b2-4ac gt 0 and ane0 and a lt 0

Example

y=-x2+2x+8

Parabola

It intersect the x-

axis at two points

Example (-20)

and (40)

Quadratic

polynomial

4 y=ax +bx+c

where

b2-4ac = 0 and ane0 a gt 0

Example

y=(x-2)2

Parabola

It intersect the x-

axis at one points

Quadratic

polynomial

5 y=ax2+bx+c

where

b

2

-4ac lt 0 and ane0 a gt 0

Example

y=x2-2x+6

Parabola

It does not

intersect the x-axis

It has no zerorsquos

Quadratic

polynomial

6 y=ax +bx+c

where

b2-4ac lt 0 and ane0 a lt 0

Example

y=-x2-2x-6

Parabola

It does not

intersect the x-axis

It has no zerorsquos

Quadratic

polynomial

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983124983144983145983155 983149983137983156983141983154983145983137983148 983145983155 983139983154983141983137983156983141983140 983138983161 983144983156983156983152983098983152983144983161983155983145983139983155983139983137983156983137983148983161983155983156983139983151983149 983137983150983140 983145983155 983142983151983154 983161983151983157983154 983152983141983154983155983151983150983137983148 983137983150983140 983150983151983150983085983139983151983149983149983141983154983139983145983137983148 983157983155983141

983151983150983148983161

983095

7 y=ax3 +bx

2+cx+d

where ane0

It can be of any

shape

It will cut the x-axis

at the most 3 times

Cubic Polynomial

8 ⋯ 991270 991270 991270

Where an ne0

It can be of any

shape

It will cut the x-axis

at the most n times

Polynomial of n

degree

Relation between coefficient and zeroes of the Polynomial

Sno Type of Polynomial General form Zerorsquos Relationship between Zerorsquos

and coefficients

1 Linear polynomial ax+b ane0 1

2 Quadratic ax +bx+c ane0 2

3 Cubic ax +bx +cx+d ane0 3

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983151983150983148983161

983096

Formation of polynomial when the zeroes are given

Type ofpolynomial

Zerorsquos Polynomial Formed

Linear k=a (x-a)

Quadratic k1=a and

k2=b

(x-a)(x-b)

Or

x2-( a+b)x +ab

Or

x2

-( Sum of the zerorsquos)x +product of the zerorsquos

Cubic k1=a k2=b

and k3=c

(x-a)(x-b)(x-c)

Division algorithm for Polynomial

Letlsquos p(x) and q(x) are any two polynomial with q(x) ne0 then we can find polynomial s(x) and r(x) suchthat

P(x)=s(x) q(x) + r(x)

Where r(x) can be zero or degree of r(x) lt degree of g(x)

983108983145983158983145983140983141983150983140 983101983121983157983151983156983145983141983150983156 983128 983108983145983158983145983155983151983154 + 983122983141983149983137983145983150983140983141983154

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983151983150983148983161

983097

COORDINATE GEOMETRYSno Points

1 We require two perpendicular axes to locate a point in the plane One of

them is horizontal and other is Vertical

2 The plane is called Cartesian plane and axis are called the coordinates axis

3 The horizontal axis is called x-axis and Vertical axis is called Y-axis

4 The point of intersection of axis is called origin

5 The distance of a point from y axis is called x ndashcoordinate or abscissa and

the distance of the point from x ndashaxis is called y ndash coordinate or Ordinate

6 The distance of a point from y axis is called x ndashcoordinate or abscissa and

the distance of the point from x ndashaxis is called y ndash coordinate or Ordinate

7 The Origin has zero distance from both x-axis and y-axis so that its

abscissa and ordinate both are zero So the coordinate of the origin is (0

0)

8 A point on the x ndashaxis has zero distance from x-axis so coordinate of any

point on the x-axis will be (x 0)

9 A point on the y ndashaxis has zero distance from y-axis so coordinate of any

point on the y-axis will be (0 y)

10 The axes divide the Cartesian plane in to four parts These Four parts arecalled the quadrants

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983151983150983148983161

10

The coordinates of the points in the four quadrants will have sign according to thebelow table

Quadrant x-coordinate y-coordinate

Ist Quadrant + +

IInd quadrant - +

IIIrd quadrant - -

IVth quadrant + -

Sno Terms Descriptions

1 Distance formula Distance between the points AB is given by

Distance of Point A from Origin

2 Section Formula A point P(xy) which divide the line segment AB in

the ratio m1 and m2 is given by

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11

The midpoint P is given by

3 Area of Triangle Area of triangle ABC of coordinates A(x 1y 1 )

B(x 2 y 2 ) and C(x 3y 3 )

12 For point AB and C to be collinear The value of A

should be zero

LINEAR EQUATIONS IN TWO

VARIABLESAn equation of the form ax + by + c = 0 where a b and c are real numbers such

that a andb are not both zero is called a linear equation in two variablesImportant points to Note

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Sno Points

1 A linear equation in two variable has infinite solutions

2 The graph of every linear equation in two variable is a straight line

3 x = 0 is the equation of the y-axis and y = 0 is the equation of the x-axis

4 The graph x=a is a line parallel to y -axis

5 The graph y=b is a line parallel to x -axis

6 An equation of the type y = mx represents a line passing through theorigin

7 Every point on the graph of a linear equation in two variables is a solution

of the linear

equation Moreover every solution of the linear equation is a point on the

graph

Sno Type of equation Mathematical

representation

Solutions

1 Linear equation in one Variable ax+b=0 ane0

a and b are real

number

One solution

2 Linear equation in two Variable ax+by+c=0 ane0 and

bne0

a b and c are real

number

Infinite solution

possible

3 Linear equation in three Variable ax+by+cz+d=0 ane0

bne0 and cne0

Infinite solution

possible

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a b c d are realnumber

Simultaneous pair of Linear equation

A pair of Linear equation in two variables

a1x+b1y+c1=0

a2x +b2y+c2=0

Graphically it is represented by two straight lines on Cartesian plane

Simultaneous pair of

Linear equation

Condition Graphical

representation

Algebraic

interpretation

a1x+b1y+c1=0

a2x +b2y+c2=0

Example

x-4y+14=0

3x+2y-14=0

Intersecting lines The

intersecting point

coordinate is the only

solution

One unique solution

only

a1x+b1y+c1=0

a2x +b2y+c2=0

Coincident lines Theany coordinate on the

line is the solution

Infinite solution

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Example

2x+4y=16

3x+6y=24

a1x+b1y+c1=0

a2x +b2y+c2=0

Example

2x+4y=6

4x+8y=18

Parallel Lines No solution

The graphical solution can be obtained by drawing the lines on the Cartesian plane

Algebraic Solution of system of Linear equation

Sno Type of method Working of method

1 Method of elimination by

substitution

1) Suppose the equation are

a1x+b1y+c1=0

a2x +b2y+c2=0

2) Find the value of variable of either x or y in other

variable term in first equation

3) Substitute the value of that variable in second

equation

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4) Now this is a linear equation in one variable Findthe value of the variable

5) Substitute this value in first equation and get the

second variable

2 Method of elimination by

equating the coefficients

1) Suppose the equation are

a1x+b1y+c1=0

a2x +b2y+c2=0

2) Find the LCM of a1 and a2 Let it k

3) Multiple the first equation by the value ka1

4) Multiple the first equation by the value ka2

4) Subtract the equation obtained This way one

variable will be eliminated and we can solve to get the

value of variable y

5) Substitute this value in first equation and get the

second variable

3 Cross Multiplication method 1) Suppose the equation are

a1x+b1y+c1=0

a2x +b2y+c2=0

2) This can be written as

3) This can be written as

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4) Value of x and y can be find using the

x =gt first and last expression

y=gt second and last expression

Quadratic EquationsSno Terms Descriptions

1 Quadratic Polynomial P(x) = ax +bx+c where ane0

2 Quadratic equation ax +bx+c =0 where ane0

3 Solution or root of the

Quadratic equation

A real number α is called the root or solution of the

quadratic equation if

aα2 +bα+c=0

4 zeroes of the

polynomial p(x)

The root of the quadratic equation are called zeroes

5 Maximum roots of

quadratic equations

We know from chapter two that a polynomial of

degree can have max two zeroes So a quadratic

equation can have maximum two roots

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6 Condition for realroots A quadratic equation has real roots if b

2

- 4ac gt 0

How to Solve Quadratic equation

Sno Method Working

1 factorization This method we factorize the equation by splitting themiddle term b

In ax2+bx+c=0

Example

6x2-x-2=0

1) First we need to multiple the coefficient a and cIn this

case =6X-2=-12

2) Splitting the middle term so that multiplication is 12

and difference is the coefficient b

6x2 +3x-4x-2=0

3x( 2x+1) -2(2x+1)=0

(3x-2) (2x+1)=0

3) Roots of the equation can be find equating the factorsto zero

3x-2=0 =gt x=32

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2x+1=0 =gt x=-12

2 Square method In this method we create square on LHS and RHS and

then find the value

ax2 +bx+c=0

1) x2 +(ba) x+(ca)=0

2) ( x+b2a)2 ndash(b2a)2 +(ca)=0

3) ( x+b2a)2=(b2-4ac)4a2

4) radic

Example

x2 +4x-5=0

1) (x+2)2 -4-5=0

2) (x+2)2=9

3) Roots of the equation can be find using square root on

both the sides

x+2 =-3 =gt x=-5

x+2=3=gt x=1

3 Quadratic

method

For quadratic equation

ax2 +bx+c=0

roots are given by radic radic

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For b2 -4ac gt 0 Quadratic equation has two real roots of

different value

For b2-4ac =0 quadratic equation has one real root

For b2-4ac lt 0 no real roots for quadratic equation

Nature of roots of Quadratic equation

Sno Condition Nature of roots

1 b -4ac gt 0 Two distinct real roots

2 b -4ac =0 One real root

3 b

2

-4ac lt 0 No real roots

Triangles

Sno Terms Descriptions

1 Congruence Two Geometric figure are said to be congruence ifthey are exactly same size and shapeSymbol used is cong Two angles are congruent if they are equalTwo circle are congruent if they have equal radiiTwo squares are congruent if the sides are equal

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20

2 Triangle Congruencebull

Two triangles are congruent if three sidesand three angles of one triangle is

congruent to the corresponding sides and

angles of the other

bull Corresponding sides are equal

AB=DE BC=EF AC=DF

bull Corresponding angles are equalang angang angang ang

bull We write this as

cong

bull The above six equalities are between the

corresponding parts of the two congruent

triangles In short form this is called

CPCT

bull We should keep the letters in correct order

on both sides

3 Inequalities in

Triangles

1) In a triangle angle opposite to longer side is

larger

2) In a triangle side opposite to larger angle is

larger

3) The sum of any two sides of the triangle is

greater than the third side

983105

983107983106

983108

983109 983110

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In triangle ABC

AB +BC gt AC

Different Criterion for Congruence of the triangles

N Criterion Description Figures and

expression

1 Side angle

Side (SAS)

congruence

bull Two triangles are congruent if the

two sides and included angles of

one triangle is equal to the two

sides and included angle

bull It is an axiom as it cannot be

proved so it is an accepted truth

bull ASS and SSA type two triangles

may not be congruent alwaysIf following

condition

983105

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983108

983107983106

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AB=DE BC=EFang ang

Then

cong

2 Angle side

angle (ASA)

congruence

bull Two triangles are congruent if the

two angles and included side of

one triangle is equal to the

corresponding angles and side

bull It is a theorem and can be proved

If following

condition

BC=EF

ang angang ang

Then

cong

3 Angle angle

side( AAS)

congruence

bull Two triangles are congruent if the

any two pair of angles and any

side of one triangle is equal to the

corresponding angles and side

bull It is a theorem and can be proved

983110983109

983108

983107983106

983105

983110983109

983108

983107983106

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If followingcondition

BC=EF

ang angang ang

Then

cong

4 Side-Side-Side

(SSS)congruence

bull Two triangles are congruent if the

three sides of one triangle is equalto the three sides of the another

If following

condition

BC=EFAB=DEDF

=AC

Then

cong

5 Right angle ndash

hypotenuse-

side(RHS)

bull Two right triangles are congruent if

the hypotenuse and a side of the

one triangle are equal to

983110983109

983108

983107983106

983105

983105 983108

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congruence corresponding hypotenuse and sideof the another

If following

condition

AC=DFBC=EF

Then

cong

Some Important points on Triangles

Terms Description

Orthocenter Point of intersection of the three altitudeof the triangle

Equilateral triangle whose all sides are equal and allangles are equal to 600

Median A line Segment joining the corner of thetriangle to the midpoint of the oppositeside of the triangle

Altitude A line Segment from the corner of thetriangle and perpendicular to theopposite side of the triangle

Isosceles A triangle whose two sides are equalCentroid Point of intersection of the three median

of the triangle is called the centroid of

the triangleIn center All the angle bisector of the trianglepasses through same point

Circumcenter The perpendicular bisector of the sidesof the triangles passes through samepoint

983106 983107 983109 983110

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Scalene triangle Triangle having no equal angles and noequal sidesRight Triangle Right triangle has one angle equal to 900 Obtuse Triangle One angle is obtuse angle while other

two are acute anglesAcute Triangle All the angles are acute

Similarity of Triangles

Sno Points

1 Two figures having the same shape but not necessarily the same size are called similarfigures

2 All the congruent figures are similar but the converse is not true

3 If a line is drawn parallel to one side of a triangle to intersect the other two sides in

distinct points then the other two sides are divided in the same ratio

4 If a line divides any two sides of a triangle in the same ratio then the line is parallel tothe third side

Different Criterion for Similarity of the triangles

N Criterion Description Expression

1 Angle Angle

angle(AAA)

similarity

bull Two triangles are similar if

corresponding angle are equal

If following

condition

ang ang

ang ang

ang ang

Then

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Then 983166

2 Angle angle

(AA) similarity

bull Two triangles are similar if the two

corresponding angles are equal as

by angle property third angle will

be also equal

If following

condition

ang ang

ang ang

Then

ang ang

Then

Then

983166

3 Side side

side(SSS)

Similarity

Two triangles are similar if the

sides of one triangle is

proportional to the sides of other

triangle

If following

condition

Then

ang ang

ang ang

ang ang

Then

cong

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4 Side-Angle-Side (SAS)

similarity

bull

Two triangles are similar if the oneangle of a triangle is equal to one

angle of other triangles and sides

including that angle is proportional

If followingcondition

And ang ang

Then

cong

Area of Similar triangles

If the two triangle ABC and DEF are similar

cong

Then

Pythagoras Theorem

Sno Points

1 If a perpendicular is drawn from the vertex of the right angle of a right triangle to thehypotenuse then the triangles on both sides of the perpendicular are similar to the

whole triangle and also to each other

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2 In a right triangle the square of the hypotenuse is equal to the sum of the squares of the

other two sides (Pythagoras Theorem)

(hyp)2 =(base)

2 + (perp)

2

3 If in a triangle square of one side is equal to the sum of the squares of the other twosides then the angle opposite the first side is a right angle

Arithmetic Progression

Sno Terms Descriptions

1 ArithmeticProgression

An arithmetic progression is a sequence ofnumbers such that the difference of any two

successive members is a constant

Examples

1) 1591317hellip

2) 12345hellip

2 common difference of

the AP

the difference between any successive members is a

constant and it is called the common difference of AP

1) If a1 a2a3a4a5 are the terms in AP then

D=a2 -a1 =a3 - a2 =a4 ndash a3=a5 ndasha4

2) We can represent the general form of AP inthe form

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2983097

aa+da+2da+3da+4d

Where a is first term and d is the commondifference

3 nth term of Arithmetic Progression

n term = a + (n - 1)d

4 Sum of nth item in ArithmeticProgression

Sn =(n2)[a + (n-1)d]

Or

Sn =(n2)[t1+ tn]

Trigonometry

Sno Terms Descriptions

1 What is

Trigonometry

Trigonometry from Greek trigotildenon triangle and

metron measure) is a branch of mathematics

that studies relationships involving lengths and

angles of triangles The field emerged during the

3rd century BC from applications of geometry to

astronomical studies

Trigonometry is most simply associated with

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planar right angle triangles (each of which is atwo-dimensional triangle with one angle equal to

90 degrees) The applicability to non-right-angle

triangles exists but since any non-right-angle

triangle (on a flat plane) can be bisected to

create two right-angle triangles most problems

can be reduced to calculations on right-angle

triangles Thus the majority of applications relate

to right-angle triangles

2 Trigonometric

Ratiorsquos

In a right angle triangle ABC where B=90deg

We can define following term for angle A

Base Side adjacent to angle

Perpendicular Side Opposite of angleHypotenuse Side opposite to right angle

We can define the trigonometric ratios for

angle A as

sin A= PerpendicularHypotenuse =BCAC

cosec A= HypotenusePerpendicular

=ACBC

cos A= BaseHypotenuse =ABAC

sec A= HypotenuseBase=ACAB

tan A= PerpendicularBase =BCAB

cot A= BasePerpendicular=ABBC

Notice that each ratio in the right-hand column is

the inverse or the reciprocal of the ratio in the

left-hand column

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3 Reciprocal offunctions The reciprocal of sin A is cosec A and vice-versa

The reciprocal of cos A is sec A

And the reciprocal of tan A is cot A

These are valid for acute angles

We can define tan A = sin Acos A

And Cot A =cos A Sin A

4 Value of of sin and

cos

Is always less 1

5 Trigonometric ration

from another angle

We can define the trigonometric ratios for angle Cas

sin C= PerpendicularHypotenuse =ABACcosec C= HypotenusePerpendicular =ACABBcos C= BaseHypotenuse =BCACsec C= HypotenuseBase=ACBCtan A= PerpendicularBase =ABBCcot A= BasePerpendicular=BCAB

6 Trigonometric ratios

of complimentary

Sin (90-A) =cos(A)

Cos(90-A) = sin A

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angles Tan(90-A) =cot A

Sec(90-A)= cosec A

Cosec (90-A) =sec A

Cot(90- A) =tan A7 Trigonometric

identities

Sin2 A + cos2 A =0

1 + tan2 A =sec2 A

1 + cot2 A =cosec2 A

Trigonometric Ratios of Common angles

We can find the values of trigonometric ratiorsquos various angle

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Area of Circles

Sno Points

1 A circle is a collection of all the points in a plane which are equidistant

from a fixed point in the plane

2 Equal chords of a circle (or of congruent circles) subtend equal angles at

the center

3 If the angles subtended by two chords of a circle (or of congruent circles)

at the center (corresponding center) are equal the chords are equal

4 The perpendicular from the center of a circle to a chord bisects the chord

5 The line drawn through the center of a circle to bisect a chord is

perpendicular to the chord

6 There is one and only one circle passing through three non-collinear points

7 Equal chords of a circle (or of congruent circles) are equidistant from the

center (or corresponding centers)

8 Chords equidistant from the center (or corresponding centers) of a circle

(or of congruent circles) are equal

9 If two arcs of a circle are congruent then their corresponding chords are

equal and conversely if two chords of a circle are equal then their

corresponding arcs (minor major) are congruent

10 Congruent arcs of a circle subtend equal angles at the center

11 The angle subtended by an arc at the center is double the angle subtended

by it at any point on the remaining part of the circle

12 Angles in the same segment of a circle are equal

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13 Angle in a semicircle is a right angle

14 If a line segment joining two points subtends equal angles at two other

points lying on the same side of the line containing the line segment the

four points lie on a circle

15 The sum of either pair of opposite angles of a cyclic quadrilateral is 180deg

16 If the sum of a pair of opposite angles of a quadrilateral is 180deg then the

quadrilateral is cyclic

Sno Terms Descriptions

1 Circumference of a circle 2 π r

2 Area of circle π r 2

3 Length of the arc of

the sector of angle

Length of the arc of the sector of angle θ 2

4 Area of the sector of

angle

Area of the sector of angle θ

5 Area of segment of a

circle

Area of the corresponding sector ndash Area of the

corresponding triangle

Surface Area and Volume

Sno Term Description

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1 Mensuration It is branch of mathematics which is concernedabout the measurement of length area and

Volume of plane and Solid figure

2 Perimeter a)The perimeter of plane figure is defined as thelength of the boundaryb)It units is same as that of length ie m cmkm

3 Area a)The area of the plane figure is the surfaceenclosed by its boundaryb) It unit is square of length unit ie m2 km2

4 Volume Volume is the measure of the amount of spaceinside of a solid figure like a cube ball cylinderor pyramid Its units are always cubic that isthe number of little element cubes that fit insidethe figure

Volume Unit conversion

1 cm3 1mL 1000 mm3

1 Litre 1000ml 1000 cm3

1 m3 106 cm3 1000 L

1 dm3 1000 cm3 1 L

Surface Area and Volume of Cube and Cuboid

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Type Measurement

Surface Area of Cuboid of Length L

Breadth B and Height H

2(LB + BH + LH)

Lateral surface area of the cuboids 2( L + B ) H

Diagonal of the cuboids Volume of a cuboids LBH

Length of all 12 edges of the cuboids 4 (L+B+H)

Surface Area of Cube of side L 6L2

Lateral surface area of the cube 4L2

Diagonal of the cube radic3 Volume of a cube L3

Surface Area and Volume of Right circular cylinder

Radius The radius (r) of the circular base is called the radius of the

cylinder

Height The length of the axis of the cylinder is called the height (h) of the

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cylinder

Lateral

Surface

The curved surface joining the two base of a right circular cylinder is

called Lateral Surface

Type Measurement

Curved or lateral Surface Area of

cylinder

2πrh

Total surface area of cylinder 2πr (h+r)

Volume of Cylinder π r2h

Surface Area and Volume of Right circular cone

Radius The radius (r) of the circular base is called the radius of the

cone

Height The length of the line segment joining the vertex to the center of

base is called the height (h) of the cone

Slant

Height

The length of the segment joining the vertex to any point on the

circular edge of the base is called the slant height (L) of the cone

Lateral

surface

Area

The curved surface joining the base and uppermost point of a right

circular cone is called Lateral Surface

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Type Measurement

Curved or lateral Surface Area of

cone

πrL

Total surface area of cone πr (L+r)

Volume of Cone

Surface Area and Volume of sphere and hemisphere

Sphere A sphere can also be considered as a solid obtained on

rotating a circle About its diameter

Hemisphere A plane through the centre of the sphere divides the sphere into two

equal parts each of which is called a hemisphere

radius The radius of the circle by which it is formed

SphericalShell

The difference of two solid concentric spheres is called a sphericalshell

Lateral

Surface

Area for

Total surface area of the sphere

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Sphere

Lateral

Surface

area of

Hemisphere

It is the curved surface area leaving the circular base

Type Measurement

Surface area of Sphere 4πr2

Volume of Sphere 43 Curved Surface area of hemisphere 2πr2

Total Surface area of hemisphere 3πr2

Volume of hemisphere 23 Volume of the spherical shell whose

outer and inner radii and lsquoRrsquo and lsquorrsquorespectively

43

How the Surface area and Volume are determined

Area of Circle The circumference of a circle is 2πr

This is the definition of π (pi) Divide

the circle into many triangular

segments The area of the triangles

is 12 times the sum of their bases

2πr (the circumference) times their

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height r

12 2 Surface Area of cylinder This can be imagined as unwrapping the

surface into a rectangle

Surface area of cone This can be achieved by divide the

surface of the cone into its triangles or

the surface of the cone into many thin

triangles The area of the triangles is 12

times the sum of their bases p times

their height

12 2

Surface Area and Volume of frustum of cone

h = vertical height of the frustum

l = slant height of the frustum

r 1 and r 2 are radii of the two bases (ends) of the frustum

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Type Measurement

983126983151983148983157983149983141 983151983142 983137 983142983154983157983155983156983157983149 983151983142 983137 983139983151983150983141 983123983148983137983150983156 983144983141983145983143983144983156 983151983142 983142983154983157983155983156983157983149 983151983142 983137 983139983151983150983141 983107983157983154983158983141983140 983155983157983154983142983137983139983141 983137983154983141983137 983151983142 983137 983142983154983157983155983156983157983149 983151983142 983137 983139983151983150983141 983124983151983156983137983148 983155983157983154983142983137983139983141 983137983154983141983137 983151983142 983142983154983157983155983156983157983149 983151983142 983137 983139983151983150983141

Statistics

Sno Term Description

1 Statistics Statistics is a broad mathematical disciplinewhich studies ways to collect summarize and

draw conclusions from data

2 Data A systematic record of facts or different values of

a quantity is called data

Data is of two types - Primary data and Secondary

data

Primary Data The data collected by a researcher

with a specific purpose in mind is called primarydata

Secondary Data The data gathered from a

source where it already exists is called secondary

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data

3 Features of data bull Statistics deals with collectionpresentation analysis and interpretation ofnumerical data

bull Arranging data in an order to study theirsalient features is called presentation ofdata

bull Data arranged in ascending or descendingorder is called arrayed data or an array

bull Range of the data is the differencebetween the maximum and the minimum

values of the observationsbull Table that shows the frequency of different

values in the given data is called afrequency distribution table

bull A frequency distribution table that showsthe frequency of each individual value in thegiven data is called an ungrouped frequencydistribution table

bull A table that shows the frequency of groupsof values in the given data is called agrouped frequency distribution table

bull

The groupings used to group the values ingiven data are called classes or class-intervals The number of values that eachclass contains is called the class size orclass width The lower value in a class iscalled the lower class limit The higher valuein a class is called the upper class limit

bull Class mark of a class is the mid value ofthe two limits of that class

bull A frequency distribution in which the upperlimit of one class differs from the lower limitof the succeeding class is called an

Inclusive or discontinuous FrequencyDistribution

bull A frequency distribution in which the upperlimit of one class coincides from the lowerlimit of the succeeding class is called anexclusive or continuous Frequency

7252019 Maths Formula Class10

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983124983144983145983155 983149983137983156983141983154983145983137983148 983145983155 983139983154983141983137983156983141983140 983138983161 983144983156983156983152983098983152983144983161983155983145983139983155983139983137983156983137983148983161983155983156983139983151983149 983137983150983140 983145983155 983142983151983154 983161983151983157983154 983152983141983154983155983151983150983137983148 983137983150983140 983150983151983150983085983139983151983149983149983141983154983139983145983137983148 983157983155983141

983151983150983148983161

983092983091

Distribution bull

4 Bar graph A bar graph is a pictorial representation of data inwhich rectangular bars of uniform width are drawnwith equal spacing between them on one axisusually the x axis The value of the variable isshown on the other axis that is the y axis

5 Histogram A histogram is a set of adjacent rectangles whoseareas are proportional to the frequencies of agiven continuous frequency distribution

6 Mean The mean value of a variable is defined as thesum of all the values of the variable divided by thenumber of values

4 sum

7 Median The median of a set of data values is the middle

value of the data set when it has been arranged inascending order That is from the smallest valueto the highest value

Median is calculated as

7252019 Maths Formula Class10

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983124983144983145983155 983149983137983156983141983154983145983137983148 983145983155 983139983154983141983137983156983141983140 983138983161 983144983156983156983152983098983152983144983161983155983145983139983155983139983137983156983137983148983161983155983156983139983151983149 983137983150983140 983145983155 983142983151983154 983161983151983157983154 983152983141983154983155983151983150983137983148 983137983150983140 983150983151983150983085983139983151983149983149983141983154983139983145983137983148 983157983155983141

983151983150983148983161

983092983092

12 1 Where n is the number of values in the data

If the number of values in the data set is eventhen the median is the average of the two middlevalues

8 Mode Mode of a statistical data is the value of thatvariable which has the maximum frequency

Sno Term Description

1983117983141983137983150 983142983151983154

983125983150983143983154983151983157983152

983110983154983141983153983157983141983150983139983161 983156983137983138983148983141

Mean is given by

sum sum

2983117983141983137983150 983142983151983154 983143983154983151983157983152

983110983154983141983153983157983141983150983139983161 983156983137983138983148983141

In these distribution it is assumed that frequency of each classinterval is centered around its mid-point ie class marks

2

983117983141983137983150 983139983137983150 983138983141 983139983137983148983139983157983148983137983156983141983140 983157983155983145983150983143 983156983144983154983141983141 983149983141983156983144983151983140

983137983081 983108983145983154983141983139983156 983149983141983156983144983151983140

sum

sum

983138983081 983105983155983155983157983149983141983140 983149983141983137983150 983149983141983156983144983151983140

sum sum

7252019 Maths Formula Class10

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983124983144983145983155 983149983137983156983141983154983145983137983148 983145983155 983139983154983141983137983156983141983140 983138983161 983144983156983156983152983098983152983144983161983155983145983139983155983139983137983156983137983148983161983155983156983139983151983149 983137983150983140 983145983155 983142983151983154 983161983151983157983154 983152983141983154983155983151983150983137983148 983137983150983140 983150983151983150983085983139983151983149983149983141983154983139983145983137983148 983157983155983141

983151983150983148983161

983092983093

Where

a=gt Assumed mean

di =gt xi ndasha

983139983081 983123983156983141983152 983140983141983158983145983137983156983145983151983150 983117983141983156983144983151983140

sum sum

Where

a=gt Assumed mean

ui =gt (xi ndasha)h

3983117983151983140983141 983142983151983154

983143983154983151983157983152983141983140

983142983154983141983153983157983141983150983139983161 983156983137983138983148983141

Modal class The class interval having highest frequency is

called the modal class and Mode is obtained using the modal

class

2

983127983144983141983154983141

l = lower limit of the modal classh = size of the class interval (assuming all class sizes tobe equal)f 1 = frequency of the modal classf 0 = frequency of the class preceding the modal classf 2 = frequency of the class succeeding the modal class

4983117983141983140983145983137983150 983151983142 983137

983143983154983151983157983152983141983140 983140983137983156983137

983142983154983141983153983157983141983150983139983161 983156983137983138983148983141

For the given data we need to have class interval frequency

distribution and cumulative frequency distribution

Median is calculated as

7252019 Maths Formula Class10

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983124983144983145983155 983149983137983156983141983154983145983137983148 983145983155 983139983154983141983137983156983141983140 983138983161 983144983156983156983152983098983152983144983161983155983145983139983155983139983137983156983137983148983161983155983156983139983151983149 983137983150983140 983145983155 983142983151983154 983161983151983157983154 983152983141983154983155983151983150983137983148 983137983150983140 983150983151983150983085983139983151983149983149983141983154983139983145983137983148 983157983155983141

983151983150983148983161

983092983094

2

983127983144983141983154983141

983148 983101 983148983151983159983141983154 983148983145983149983145983156 983151983142 983149983141983140983145983137983150 983139983148983137983155983155

983150 983101 983150983157983149983138983141983154 983151983142 983151983138983155983141983154983158983137983156983145983151983150983155

983139983142 983101 983139983157983149983157983148983137983156983145983158983141 983142983154983141983153983157983141983150983139983161 983151983142 983139983148983137983155983155 983152983154983141983139983141983140983145983150983143 983156983144983141 983149983141983140983145983137983150 983139983148983137983155983155

983142 983101 983142983154983141983153983157983141983150983139983161 983151983142 983149983141983140983145983137983150 983139983148983137983155983155

983144 983101 983139983148983137983155983155 983155983145983162983141 (983137983155983155983157983149983145983150983143 983139983148983137983155983155 983155983145983162983141 983156983151 983138983141 983141983153983157983137983148)

5983109983149983152983145983154983145983139983137983148

983110983151983154983149983157983148983137 983138983141983156983159983141983141983150

983117983151983140983141983084 983117983141983137983150 983137983150983140

983117983141983140983145983137983150

3 Median=Mode +2 Mean

ProbabilitySn

o

Term Description

1 Empirical

probability

It is a probability of event which is calculated based on

experiments

Example

A coin is tossed 1000 times we get 499 times head and

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983124983144983145983155 983149983137983156983141983154983145983137983148 983145983155 983139983154983141983137983156983141983140 983138983161 983144983156983156983152983098983152983144983161983155983145983139983155983139983137983156983137983148983161983155983156983139983151983149 983137983150983140 983145983155 983142983151983154 983161983151983157983154 983152983141983154983155983151983150983137983148 983137983150983140 983150983151983150983085983139983151983149983149983141983154983139983145983137983148 983157983155983141

983151983150983148983161

983092983095

501 times tail

So empirical or experimental probability of getting head is

calculated as

4991000 499

Empirical probability depends on experiment and

different will get different values based on the

experiment

2 Important point

about events

If the event A B C covers the entire possible outcome inthe experiment Then

P (A) +P (B) +P(C)=1

3 impossible

event

The probability of an event (U) which is impossible tooccur is 0 Such an event is called an impossible event

P (U)=0

4 Sure or certain

event

The probability of an event (X) which is sure (or certain)to occur is 1 Such an event is called a sure event or acertain event

P(X) =1

5 Probability of

any event

Probability of any event can be as

0 1

Sn

o

Term Description

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983092983096

1

983124983144983141983151983154983141983156983145983139983137983148

983120983154983151983138983137983138983145983148983145983156983161

The theoretical probability or the classical probability of the event isdefined as

2 Elementary

events

983105983150 983141983158983141983150983156 983144983137983158983145983150983143 983151983150983148983161 983151983150983141 983151983157983156983139983151983149983141 983151983142 983156983144983141 983141983160983152983141983154983145983149983141983150983156 983145983155 983139983137983148983148983141983140 983137983150

983141983148983141983149983141983150983156983137983154983161 983141983158983141983150983156

991260983124983144983141 983155983157983149 983151983142 983156983144983141 983152983154983151983138983137983138983145983148983145983156983145983141983155 983151983142 983137983148983148 983156983144983141 983141983148983141983149983141983150983156983137983154983161 983141983158983141983150983156983155 983151983142983137983150 983141983160983152983141983154983145983149983141983150983156 983145983155 1991261

983113983141 983113983142 983159983141 983156983144983154983141983141 983141983148983141983149983141983150983156983137983154983161 983141983158983141983150983156 983105983106983107 983145983150 983156983144983141 983141983160983152983141983154983145983149983141983150983156 983156983144983141983150

983120(983105)+983120(983106) +983120(983107)9831011

3 Complementar

y events

983124983144983141 983141983158983141983150983156 Ᾱ 983154983141983152983154983141983155983141983150983156983145983150983143 991256983150983151983156 983105991257 983145983155 983139983137983148983148983141983140 983156983144983141 983139983151983149983152983148983141983149983141983150983156 983151983142 983156983144983141 983141983158983141983150983156

983105 983127983141 983137983148983155983151 983155983137983161 983156983144983137983156 Ᾱ 983137983150983140 983105 983137983154983141 983139983151983149983152983148983141983149983141983150983156983137983154983161 983141983158983141983150983156983155 983105983148983155983151

983120(983105) +983120(Ᾱ)9831011

4 Sure or certain

event

The probability of an event (X) which is sure (or certain)

to occur is 1 Such an event is called a sure event or acertain event

P(X) =1

5 Probability of

any event

Probability of any event can be as

0 1

Page 3: Maths Formula Class10

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983124983144983145983155 983149983137983156983141983154983145983137983148 983145983155 983139983154983141983137983156983141983140 983138983161 983144983156983156983152983098983152983144983161983155983145983139983155983139983137983156983137983148983161983155983156983139983151983149 983137983150983140 983145983155 983142983151983154 983161983151983157983154 983152983141983154983155983151983150983137983148 983137983150983140 983150983151983150983085983139983151983149983149983141983154983139983145983137983148 983157983155983141

983151983150983148983161

2

a=bq + r 0 le r le b

If r =0 then b is divisor of a

2 HCF (Highest

common factor)

HCF of two positive integers can be find using the

Euclidrsquos Division Lemma algorithm

We know that for any two integers a b we can write

following expression

a=bq + r 0 le r le b

If r=0 then

HCF( a b) =b

If rne0 then

HCF ( a b) = HCF ( br)

Again expressing the integer br in Euclidrsquos Division

Lemma we get

b=pr + r 1

HCF ( br)=HCF ( rr 1)

Similarly successive Euclid lsquos division can be written

until we get the remainder zero the divisor at that

point is called the HCF of the a and b

3 HCF ( ab) =1 Then a and b are co primes

4 Fundamental

Theorem of

Arithmetic

Composite number = Product of primes

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983151983150983148983161

983091

5 HCF and LCM byprime factorization

method

HCF = Product of the smallest power of eachcommon factor in the numbers

LCM = Product of the greatest power of each prime

factor involved in the number

6 Important Formula 983112983107983110 (983137983138) 983128 983116983107983117 (983137983138) 983101983137 983128 983138

7 Important concept

for rational Number

Terminating decimal expression can be written in

the form

p2n

5m

Polynomial expressions

A polynomial expression S(x) in one variable x is an algebraic expression in x term

as

⋯ 991270 991270 991270 Where anan-1hellipaa0 are constant and real numbers and an is not equal to zero

Some Important point to Note

Sno Points

1 an an-1 an-2 hellipa1a0 are called the coefficients for xnxn-1 hellipx1x0

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983151983150983148983161

983092

2 n is called the degree of the polynomial

3 when an an-1 an-2 hellipa1a0 all are zero it is called zero polynomial

4 A constant polynomial is the polynomial with zero degree it is a constant

value polynomial

5 A polynomial of one item is called monomial two items binomial and three

items as trinomial

6 A polynomial of one degree is called linear polynomial two degree as

quadratic polynomial and degree three as cubic polynomial

Important concepts on Polynomial

Concept Description

Zerorsquos or roots

of the

polynomial

It is a solution to the polynomial equation S(x)=0 ie a number

a is said to be a zero of a polynomial if S(a) = 0

If we draw the graph of S(x) =0 the values where the curve

cuts the X-axis are called Zeroes of the polynomial

Remainder

Theoremrsquos

If p(x) is an polynomial of degree greater than or equal to 1

and p(x) is divided by the expression (x-a)then the remainder

will be p(a)

FactorrsquosTheoremrsquos

If x-a is a factor of polynomial p(x) then p(a)=0 or if p(a)=0x-a is the factor the polynomial p(x)

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983151983150983148983161

983093

Geometric Meaning of the Zeroes of the polynomial

Letrsquos us assume

y= p(x) where p(x) is the polynomial of any form

Now we can plot the equation y=p(x) on the Cartesian plane by taking various values of x and y obtained

by putting the values The plot or graph obtained can be of any shapes

The zeroes of the polynomial are the points where the graph meet x axis in the Cartesian plane If the

graph does not meet x axis then the polynomial does not have any zerorsquos

Let us take some useful polynomial and shapes obtained on the Cartesian plane

Sno y=p(x) Graph obtained Name of the

graph

Name of the

equation

1 y=ax+b where a and b can be

any values (ane0)

Example y=2x+3

Straight line

It intersect the x-

axis at ( -ba 0)

Example ( -320)

Linear polynomial

2 y=ax2+bx+c

where

b2-4ac gt 0 and ane0 and agt 0

Example

y=x2-7x+12

Parabola

It intersect the x-

axis at two points

Example

(30) and (40)

Quadratic

polynomial

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983124983144983145983155 983149983137983156983141983154983145983137983148 983145983155 983139983154983141983137983156983141983140 983138983161 983144983156983156983152983098983152983144983161983155983145983139983155983139983137983156983137983148983161983155983156983139983151983149 983137983150983140 983145983155 983142983151983154 983161983151983157983154 983152983141983154983155983151983150983137983148 983137983150983140 983150983151983150983085983139983151983149983149983141983154983139983145983137983148 983157983155983141

983151983150983148983161

983094

3 y=ax2+bx+c

where

b2-4ac gt 0 and ane0 and a lt 0

Example

y=-x2+2x+8

Parabola

It intersect the x-

axis at two points

Example (-20)

and (40)

Quadratic

polynomial

4 y=ax +bx+c

where

b2-4ac = 0 and ane0 a gt 0

Example

y=(x-2)2

Parabola

It intersect the x-

axis at one points

Quadratic

polynomial

5 y=ax2+bx+c

where

b

2

-4ac lt 0 and ane0 a gt 0

Example

y=x2-2x+6

Parabola

It does not

intersect the x-axis

It has no zerorsquos

Quadratic

polynomial

6 y=ax +bx+c

where

b2-4ac lt 0 and ane0 a lt 0

Example

y=-x2-2x-6

Parabola

It does not

intersect the x-axis

It has no zerorsquos

Quadratic

polynomial

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983124983144983145983155 983149983137983156983141983154983145983137983148 983145983155 983139983154983141983137983156983141983140 983138983161 983144983156983156983152983098983152983144983161983155983145983139983155983139983137983156983137983148983161983155983156983139983151983149 983137983150983140 983145983155 983142983151983154 983161983151983157983154 983152983141983154983155983151983150983137983148 983137983150983140 983150983151983150983085983139983151983149983149983141983154983139983145983137983148 983157983155983141

983151983150983148983161

983095

7 y=ax3 +bx

2+cx+d

where ane0

It can be of any

shape

It will cut the x-axis

at the most 3 times

Cubic Polynomial

8 ⋯ 991270 991270 991270

Where an ne0

It can be of any

shape

It will cut the x-axis

at the most n times

Polynomial of n

degree

Relation between coefficient and zeroes of the Polynomial

Sno Type of Polynomial General form Zerorsquos Relationship between Zerorsquos

and coefficients

1 Linear polynomial ax+b ane0 1

2 Quadratic ax +bx+c ane0 2

3 Cubic ax +bx +cx+d ane0 3

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Formation of polynomial when the zeroes are given

Type ofpolynomial

Zerorsquos Polynomial Formed

Linear k=a (x-a)

Quadratic k1=a and

k2=b

(x-a)(x-b)

Or

x2-( a+b)x +ab

Or

x2

-( Sum of the zerorsquos)x +product of the zerorsquos

Cubic k1=a k2=b

and k3=c

(x-a)(x-b)(x-c)

Division algorithm for Polynomial

Letlsquos p(x) and q(x) are any two polynomial with q(x) ne0 then we can find polynomial s(x) and r(x) suchthat

P(x)=s(x) q(x) + r(x)

Where r(x) can be zero or degree of r(x) lt degree of g(x)

983108983145983158983145983140983141983150983140 983101983121983157983151983156983145983141983150983156 983128 983108983145983158983145983155983151983154 + 983122983141983149983137983145983150983140983141983154

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COORDINATE GEOMETRYSno Points

1 We require two perpendicular axes to locate a point in the plane One of

them is horizontal and other is Vertical

2 The plane is called Cartesian plane and axis are called the coordinates axis

3 The horizontal axis is called x-axis and Vertical axis is called Y-axis

4 The point of intersection of axis is called origin

5 The distance of a point from y axis is called x ndashcoordinate or abscissa and

the distance of the point from x ndashaxis is called y ndash coordinate or Ordinate

6 The distance of a point from y axis is called x ndashcoordinate or abscissa and

the distance of the point from x ndashaxis is called y ndash coordinate or Ordinate

7 The Origin has zero distance from both x-axis and y-axis so that its

abscissa and ordinate both are zero So the coordinate of the origin is (0

0)

8 A point on the x ndashaxis has zero distance from x-axis so coordinate of any

point on the x-axis will be (x 0)

9 A point on the y ndashaxis has zero distance from y-axis so coordinate of any

point on the y-axis will be (0 y)

10 The axes divide the Cartesian plane in to four parts These Four parts arecalled the quadrants

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10

The coordinates of the points in the four quadrants will have sign according to thebelow table

Quadrant x-coordinate y-coordinate

Ist Quadrant + +

IInd quadrant - +

IIIrd quadrant - -

IVth quadrant + -

Sno Terms Descriptions

1 Distance formula Distance between the points AB is given by

Distance of Point A from Origin

2 Section Formula A point P(xy) which divide the line segment AB in

the ratio m1 and m2 is given by

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11

The midpoint P is given by

3 Area of Triangle Area of triangle ABC of coordinates A(x 1y 1 )

B(x 2 y 2 ) and C(x 3y 3 )

12 For point AB and C to be collinear The value of A

should be zero

LINEAR EQUATIONS IN TWO

VARIABLESAn equation of the form ax + by + c = 0 where a b and c are real numbers such

that a andb are not both zero is called a linear equation in two variablesImportant points to Note

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Sno Points

1 A linear equation in two variable has infinite solutions

2 The graph of every linear equation in two variable is a straight line

3 x = 0 is the equation of the y-axis and y = 0 is the equation of the x-axis

4 The graph x=a is a line parallel to y -axis

5 The graph y=b is a line parallel to x -axis

6 An equation of the type y = mx represents a line passing through theorigin

7 Every point on the graph of a linear equation in two variables is a solution

of the linear

equation Moreover every solution of the linear equation is a point on the

graph

Sno Type of equation Mathematical

representation

Solutions

1 Linear equation in one Variable ax+b=0 ane0

a and b are real

number

One solution

2 Linear equation in two Variable ax+by+c=0 ane0 and

bne0

a b and c are real

number

Infinite solution

possible

3 Linear equation in three Variable ax+by+cz+d=0 ane0

bne0 and cne0

Infinite solution

possible

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a b c d are realnumber

Simultaneous pair of Linear equation

A pair of Linear equation in two variables

a1x+b1y+c1=0

a2x +b2y+c2=0

Graphically it is represented by two straight lines on Cartesian plane

Simultaneous pair of

Linear equation

Condition Graphical

representation

Algebraic

interpretation

a1x+b1y+c1=0

a2x +b2y+c2=0

Example

x-4y+14=0

3x+2y-14=0

Intersecting lines The

intersecting point

coordinate is the only

solution

One unique solution

only

a1x+b1y+c1=0

a2x +b2y+c2=0

Coincident lines Theany coordinate on the

line is the solution

Infinite solution

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Example

2x+4y=16

3x+6y=24

a1x+b1y+c1=0

a2x +b2y+c2=0

Example

2x+4y=6

4x+8y=18

Parallel Lines No solution

The graphical solution can be obtained by drawing the lines on the Cartesian plane

Algebraic Solution of system of Linear equation

Sno Type of method Working of method

1 Method of elimination by

substitution

1) Suppose the equation are

a1x+b1y+c1=0

a2x +b2y+c2=0

2) Find the value of variable of either x or y in other

variable term in first equation

3) Substitute the value of that variable in second

equation

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4) Now this is a linear equation in one variable Findthe value of the variable

5) Substitute this value in first equation and get the

second variable

2 Method of elimination by

equating the coefficients

1) Suppose the equation are

a1x+b1y+c1=0

a2x +b2y+c2=0

2) Find the LCM of a1 and a2 Let it k

3) Multiple the first equation by the value ka1

4) Multiple the first equation by the value ka2

4) Subtract the equation obtained This way one

variable will be eliminated and we can solve to get the

value of variable y

5) Substitute this value in first equation and get the

second variable

3 Cross Multiplication method 1) Suppose the equation are

a1x+b1y+c1=0

a2x +b2y+c2=0

2) This can be written as

3) This can be written as

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4) Value of x and y can be find using the

x =gt first and last expression

y=gt second and last expression

Quadratic EquationsSno Terms Descriptions

1 Quadratic Polynomial P(x) = ax +bx+c where ane0

2 Quadratic equation ax +bx+c =0 where ane0

3 Solution or root of the

Quadratic equation

A real number α is called the root or solution of the

quadratic equation if

aα2 +bα+c=0

4 zeroes of the

polynomial p(x)

The root of the quadratic equation are called zeroes

5 Maximum roots of

quadratic equations

We know from chapter two that a polynomial of

degree can have max two zeroes So a quadratic

equation can have maximum two roots

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6 Condition for realroots A quadratic equation has real roots if b

2

- 4ac gt 0

How to Solve Quadratic equation

Sno Method Working

1 factorization This method we factorize the equation by splitting themiddle term b

In ax2+bx+c=0

Example

6x2-x-2=0

1) First we need to multiple the coefficient a and cIn this

case =6X-2=-12

2) Splitting the middle term so that multiplication is 12

and difference is the coefficient b

6x2 +3x-4x-2=0

3x( 2x+1) -2(2x+1)=0

(3x-2) (2x+1)=0

3) Roots of the equation can be find equating the factorsto zero

3x-2=0 =gt x=32

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2x+1=0 =gt x=-12

2 Square method In this method we create square on LHS and RHS and

then find the value

ax2 +bx+c=0

1) x2 +(ba) x+(ca)=0

2) ( x+b2a)2 ndash(b2a)2 +(ca)=0

3) ( x+b2a)2=(b2-4ac)4a2

4) radic

Example

x2 +4x-5=0

1) (x+2)2 -4-5=0

2) (x+2)2=9

3) Roots of the equation can be find using square root on

both the sides

x+2 =-3 =gt x=-5

x+2=3=gt x=1

3 Quadratic

method

For quadratic equation

ax2 +bx+c=0

roots are given by radic radic

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For b2 -4ac gt 0 Quadratic equation has two real roots of

different value

For b2-4ac =0 quadratic equation has one real root

For b2-4ac lt 0 no real roots for quadratic equation

Nature of roots of Quadratic equation

Sno Condition Nature of roots

1 b -4ac gt 0 Two distinct real roots

2 b -4ac =0 One real root

3 b

2

-4ac lt 0 No real roots

Triangles

Sno Terms Descriptions

1 Congruence Two Geometric figure are said to be congruence ifthey are exactly same size and shapeSymbol used is cong Two angles are congruent if they are equalTwo circle are congruent if they have equal radiiTwo squares are congruent if the sides are equal

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20

2 Triangle Congruencebull

Two triangles are congruent if three sidesand three angles of one triangle is

congruent to the corresponding sides and

angles of the other

bull Corresponding sides are equal

AB=DE BC=EF AC=DF

bull Corresponding angles are equalang angang angang ang

bull We write this as

cong

bull The above six equalities are between the

corresponding parts of the two congruent

triangles In short form this is called

CPCT

bull We should keep the letters in correct order

on both sides

3 Inequalities in

Triangles

1) In a triangle angle opposite to longer side is

larger

2) In a triangle side opposite to larger angle is

larger

3) The sum of any two sides of the triangle is

greater than the third side

983105

983107983106

983108

983109 983110

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In triangle ABC

AB +BC gt AC

Different Criterion for Congruence of the triangles

N Criterion Description Figures and

expression

1 Side angle

Side (SAS)

congruence

bull Two triangles are congruent if the

two sides and included angles of

one triangle is equal to the two

sides and included angle

bull It is an axiom as it cannot be

proved so it is an accepted truth

bull ASS and SSA type two triangles

may not be congruent alwaysIf following

condition

983105

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983107983106

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AB=DE BC=EFang ang

Then

cong

2 Angle side

angle (ASA)

congruence

bull Two triangles are congruent if the

two angles and included side of

one triangle is equal to the

corresponding angles and side

bull It is a theorem and can be proved

If following

condition

BC=EF

ang angang ang

Then

cong

3 Angle angle

side( AAS)

congruence

bull Two triangles are congruent if the

any two pair of angles and any

side of one triangle is equal to the

corresponding angles and side

bull It is a theorem and can be proved

983110983109

983108

983107983106

983105

983110983109

983108

983107983106

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If followingcondition

BC=EF

ang angang ang

Then

cong

4 Side-Side-Side

(SSS)congruence

bull Two triangles are congruent if the

three sides of one triangle is equalto the three sides of the another

If following

condition

BC=EFAB=DEDF

=AC

Then

cong

5 Right angle ndash

hypotenuse-

side(RHS)

bull Two right triangles are congruent if

the hypotenuse and a side of the

one triangle are equal to

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983151983150983148983161

2983092

congruence corresponding hypotenuse and sideof the another

If following

condition

AC=DFBC=EF

Then

cong

Some Important points on Triangles

Terms Description

Orthocenter Point of intersection of the three altitudeof the triangle

Equilateral triangle whose all sides are equal and allangles are equal to 600

Median A line Segment joining the corner of thetriangle to the midpoint of the oppositeside of the triangle

Altitude A line Segment from the corner of thetriangle and perpendicular to theopposite side of the triangle

Isosceles A triangle whose two sides are equalCentroid Point of intersection of the three median

of the triangle is called the centroid of

the triangleIn center All the angle bisector of the trianglepasses through same point

Circumcenter The perpendicular bisector of the sidesof the triangles passes through samepoint

983106 983107 983109 983110

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983151983150983148983161

2983093

Scalene triangle Triangle having no equal angles and noequal sidesRight Triangle Right triangle has one angle equal to 900 Obtuse Triangle One angle is obtuse angle while other

two are acute anglesAcute Triangle All the angles are acute

Similarity of Triangles

Sno Points

1 Two figures having the same shape but not necessarily the same size are called similarfigures

2 All the congruent figures are similar but the converse is not true

3 If a line is drawn parallel to one side of a triangle to intersect the other two sides in

distinct points then the other two sides are divided in the same ratio

4 If a line divides any two sides of a triangle in the same ratio then the line is parallel tothe third side

Different Criterion for Similarity of the triangles

N Criterion Description Expression

1 Angle Angle

angle(AAA)

similarity

bull Two triangles are similar if

corresponding angle are equal

If following

condition

ang ang

ang ang

ang ang

Then

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983151983150983148983161

2983094

Then 983166

2 Angle angle

(AA) similarity

bull Two triangles are similar if the two

corresponding angles are equal as

by angle property third angle will

be also equal

If following

condition

ang ang

ang ang

Then

ang ang

Then

Then

983166

3 Side side

side(SSS)

Similarity

Two triangles are similar if the

sides of one triangle is

proportional to the sides of other

triangle

If following

condition

Then

ang ang

ang ang

ang ang

Then

cong

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983151983150983148983161

2983095

4 Side-Angle-Side (SAS)

similarity

bull

Two triangles are similar if the oneangle of a triangle is equal to one

angle of other triangles and sides

including that angle is proportional

If followingcondition

And ang ang

Then

cong

Area of Similar triangles

If the two triangle ABC and DEF are similar

cong

Then

Pythagoras Theorem

Sno Points

1 If a perpendicular is drawn from the vertex of the right angle of a right triangle to thehypotenuse then the triangles on both sides of the perpendicular are similar to the

whole triangle and also to each other

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2983096

2 In a right triangle the square of the hypotenuse is equal to the sum of the squares of the

other two sides (Pythagoras Theorem)

(hyp)2 =(base)

2 + (perp)

2

3 If in a triangle square of one side is equal to the sum of the squares of the other twosides then the angle opposite the first side is a right angle

Arithmetic Progression

Sno Terms Descriptions

1 ArithmeticProgression

An arithmetic progression is a sequence ofnumbers such that the difference of any two

successive members is a constant

Examples

1) 1591317hellip

2) 12345hellip

2 common difference of

the AP

the difference between any successive members is a

constant and it is called the common difference of AP

1) If a1 a2a3a4a5 are the terms in AP then

D=a2 -a1 =a3 - a2 =a4 ndash a3=a5 ndasha4

2) We can represent the general form of AP inthe form

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983151983150983148983161

2983097

aa+da+2da+3da+4d

Where a is first term and d is the commondifference

3 nth term of Arithmetic Progression

n term = a + (n - 1)d

4 Sum of nth item in ArithmeticProgression

Sn =(n2)[a + (n-1)d]

Or

Sn =(n2)[t1+ tn]

Trigonometry

Sno Terms Descriptions

1 What is

Trigonometry

Trigonometry from Greek trigotildenon triangle and

metron measure) is a branch of mathematics

that studies relationships involving lengths and

angles of triangles The field emerged during the

3rd century BC from applications of geometry to

astronomical studies

Trigonometry is most simply associated with

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9830910

planar right angle triangles (each of which is atwo-dimensional triangle with one angle equal to

90 degrees) The applicability to non-right-angle

triangles exists but since any non-right-angle

triangle (on a flat plane) can be bisected to

create two right-angle triangles most problems

can be reduced to calculations on right-angle

triangles Thus the majority of applications relate

to right-angle triangles

2 Trigonometric

Ratiorsquos

In a right angle triangle ABC where B=90deg

We can define following term for angle A

Base Side adjacent to angle

Perpendicular Side Opposite of angleHypotenuse Side opposite to right angle

We can define the trigonometric ratios for

angle A as

sin A= PerpendicularHypotenuse =BCAC

cosec A= HypotenusePerpendicular

=ACBC

cos A= BaseHypotenuse =ABAC

sec A= HypotenuseBase=ACAB

tan A= PerpendicularBase =BCAB

cot A= BasePerpendicular=ABBC

Notice that each ratio in the right-hand column is

the inverse or the reciprocal of the ratio in the

left-hand column

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9830911

3 Reciprocal offunctions The reciprocal of sin A is cosec A and vice-versa

The reciprocal of cos A is sec A

And the reciprocal of tan A is cot A

These are valid for acute angles

We can define tan A = sin Acos A

And Cot A =cos A Sin A

4 Value of of sin and

cos

Is always less 1

5 Trigonometric ration

from another angle

We can define the trigonometric ratios for angle Cas

sin C= PerpendicularHypotenuse =ABACcosec C= HypotenusePerpendicular =ACABBcos C= BaseHypotenuse =BCACsec C= HypotenuseBase=ACBCtan A= PerpendicularBase =ABBCcot A= BasePerpendicular=BCAB

6 Trigonometric ratios

of complimentary

Sin (90-A) =cos(A)

Cos(90-A) = sin A

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983151983150983148983161

9830912

angles Tan(90-A) =cot A

Sec(90-A)= cosec A

Cosec (90-A) =sec A

Cot(90- A) =tan A7 Trigonometric

identities

Sin2 A + cos2 A =0

1 + tan2 A =sec2 A

1 + cot2 A =cosec2 A

Trigonometric Ratios of Common angles

We can find the values of trigonometric ratiorsquos various angle

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Area of Circles

Sno Points

1 A circle is a collection of all the points in a plane which are equidistant

from a fixed point in the plane

2 Equal chords of a circle (or of congruent circles) subtend equal angles at

the center

3 If the angles subtended by two chords of a circle (or of congruent circles)

at the center (corresponding center) are equal the chords are equal

4 The perpendicular from the center of a circle to a chord bisects the chord

5 The line drawn through the center of a circle to bisect a chord is

perpendicular to the chord

6 There is one and only one circle passing through three non-collinear points

7 Equal chords of a circle (or of congruent circles) are equidistant from the

center (or corresponding centers)

8 Chords equidistant from the center (or corresponding centers) of a circle

(or of congruent circles) are equal

9 If two arcs of a circle are congruent then their corresponding chords are

equal and conversely if two chords of a circle are equal then their

corresponding arcs (minor major) are congruent

10 Congruent arcs of a circle subtend equal angles at the center

11 The angle subtended by an arc at the center is double the angle subtended

by it at any point on the remaining part of the circle

12 Angles in the same segment of a circle are equal

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983091983092

13 Angle in a semicircle is a right angle

14 If a line segment joining two points subtends equal angles at two other

points lying on the same side of the line containing the line segment the

four points lie on a circle

15 The sum of either pair of opposite angles of a cyclic quadrilateral is 180deg

16 If the sum of a pair of opposite angles of a quadrilateral is 180deg then the

quadrilateral is cyclic

Sno Terms Descriptions

1 Circumference of a circle 2 π r

2 Area of circle π r 2

3 Length of the arc of

the sector of angle

Length of the arc of the sector of angle θ 2

4 Area of the sector of

angle

Area of the sector of angle θ

5 Area of segment of a

circle

Area of the corresponding sector ndash Area of the

corresponding triangle

Surface Area and Volume

Sno Term Description

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983091983093

1 Mensuration It is branch of mathematics which is concernedabout the measurement of length area and

Volume of plane and Solid figure

2 Perimeter a)The perimeter of plane figure is defined as thelength of the boundaryb)It units is same as that of length ie m cmkm

3 Area a)The area of the plane figure is the surfaceenclosed by its boundaryb) It unit is square of length unit ie m2 km2

4 Volume Volume is the measure of the amount of spaceinside of a solid figure like a cube ball cylinderor pyramid Its units are always cubic that isthe number of little element cubes that fit insidethe figure

Volume Unit conversion

1 cm3 1mL 1000 mm3

1 Litre 1000ml 1000 cm3

1 m3 106 cm3 1000 L

1 dm3 1000 cm3 1 L

Surface Area and Volume of Cube and Cuboid

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Type Measurement

Surface Area of Cuboid of Length L

Breadth B and Height H

2(LB + BH + LH)

Lateral surface area of the cuboids 2( L + B ) H

Diagonal of the cuboids Volume of a cuboids LBH

Length of all 12 edges of the cuboids 4 (L+B+H)

Surface Area of Cube of side L 6L2

Lateral surface area of the cube 4L2

Diagonal of the cube radic3 Volume of a cube L3

Surface Area and Volume of Right circular cylinder

Radius The radius (r) of the circular base is called the radius of the

cylinder

Height The length of the axis of the cylinder is called the height (h) of the

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983091983095

cylinder

Lateral

Surface

The curved surface joining the two base of a right circular cylinder is

called Lateral Surface

Type Measurement

Curved or lateral Surface Area of

cylinder

2πrh

Total surface area of cylinder 2πr (h+r)

Volume of Cylinder π r2h

Surface Area and Volume of Right circular cone

Radius The radius (r) of the circular base is called the radius of the

cone

Height The length of the line segment joining the vertex to the center of

base is called the height (h) of the cone

Slant

Height

The length of the segment joining the vertex to any point on the

circular edge of the base is called the slant height (L) of the cone

Lateral

surface

Area

The curved surface joining the base and uppermost point of a right

circular cone is called Lateral Surface

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983091983096

Type Measurement

Curved or lateral Surface Area of

cone

πrL

Total surface area of cone πr (L+r)

Volume of Cone

Surface Area and Volume of sphere and hemisphere

Sphere A sphere can also be considered as a solid obtained on

rotating a circle About its diameter

Hemisphere A plane through the centre of the sphere divides the sphere into two

equal parts each of which is called a hemisphere

radius The radius of the circle by which it is formed

SphericalShell

The difference of two solid concentric spheres is called a sphericalshell

Lateral

Surface

Area for

Total surface area of the sphere

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983091983097

Sphere

Lateral

Surface

area of

Hemisphere

It is the curved surface area leaving the circular base

Type Measurement

Surface area of Sphere 4πr2

Volume of Sphere 43 Curved Surface area of hemisphere 2πr2

Total Surface area of hemisphere 3πr2

Volume of hemisphere 23 Volume of the spherical shell whose

outer and inner radii and lsquoRrsquo and lsquorrsquorespectively

43

How the Surface area and Volume are determined

Area of Circle The circumference of a circle is 2πr

This is the definition of π (pi) Divide

the circle into many triangular

segments The area of the triangles

is 12 times the sum of their bases

2πr (the circumference) times their

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983151983150983148983161

9830920

height r

12 2 Surface Area of cylinder This can be imagined as unwrapping the

surface into a rectangle

Surface area of cone This can be achieved by divide the

surface of the cone into its triangles or

the surface of the cone into many thin

triangles The area of the triangles is 12

times the sum of their bases p times

their height

12 2

Surface Area and Volume of frustum of cone

h = vertical height of the frustum

l = slant height of the frustum

r 1 and r 2 are radii of the two bases (ends) of the frustum

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983151983150983148983161

9830921

Type Measurement

983126983151983148983157983149983141 983151983142 983137 983142983154983157983155983156983157983149 983151983142 983137 983139983151983150983141 983123983148983137983150983156 983144983141983145983143983144983156 983151983142 983142983154983157983155983156983157983149 983151983142 983137 983139983151983150983141 983107983157983154983158983141983140 983155983157983154983142983137983139983141 983137983154983141983137 983151983142 983137 983142983154983157983155983156983157983149 983151983142 983137 983139983151983150983141 983124983151983156983137983148 983155983157983154983142983137983139983141 983137983154983141983137 983151983142 983142983154983157983155983156983157983149 983151983142 983137 983139983151983150983141

Statistics

Sno Term Description

1 Statistics Statistics is a broad mathematical disciplinewhich studies ways to collect summarize and

draw conclusions from data

2 Data A systematic record of facts or different values of

a quantity is called data

Data is of two types - Primary data and Secondary

data

Primary Data The data collected by a researcher

with a specific purpose in mind is called primarydata

Secondary Data The data gathered from a

source where it already exists is called secondary

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983151983150983148983161

9830922

data

3 Features of data bull Statistics deals with collectionpresentation analysis and interpretation ofnumerical data

bull Arranging data in an order to study theirsalient features is called presentation ofdata

bull Data arranged in ascending or descendingorder is called arrayed data or an array

bull Range of the data is the differencebetween the maximum and the minimum

values of the observationsbull Table that shows the frequency of different

values in the given data is called afrequency distribution table

bull A frequency distribution table that showsthe frequency of each individual value in thegiven data is called an ungrouped frequencydistribution table

bull A table that shows the frequency of groupsof values in the given data is called agrouped frequency distribution table

bull

The groupings used to group the values ingiven data are called classes or class-intervals The number of values that eachclass contains is called the class size orclass width The lower value in a class iscalled the lower class limit The higher valuein a class is called the upper class limit

bull Class mark of a class is the mid value ofthe two limits of that class

bull A frequency distribution in which the upperlimit of one class differs from the lower limitof the succeeding class is called an

Inclusive or discontinuous FrequencyDistribution

bull A frequency distribution in which the upperlimit of one class coincides from the lowerlimit of the succeeding class is called anexclusive or continuous Frequency

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983124983144983145983155 983149983137983156983141983154983145983137983148 983145983155 983139983154983141983137983156983141983140 983138983161 983144983156983156983152983098983152983144983161983155983145983139983155983139983137983156983137983148983161983155983156983139983151983149 983137983150983140 983145983155 983142983151983154 983161983151983157983154 983152983141983154983155983151983150983137983148 983137983150983140 983150983151983150983085983139983151983149983149983141983154983139983145983137983148 983157983155983141

983151983150983148983161

983092983091

Distribution bull

4 Bar graph A bar graph is a pictorial representation of data inwhich rectangular bars of uniform width are drawnwith equal spacing between them on one axisusually the x axis The value of the variable isshown on the other axis that is the y axis

5 Histogram A histogram is a set of adjacent rectangles whoseareas are proportional to the frequencies of agiven continuous frequency distribution

6 Mean The mean value of a variable is defined as thesum of all the values of the variable divided by thenumber of values

4 sum

7 Median The median of a set of data values is the middle

value of the data set when it has been arranged inascending order That is from the smallest valueto the highest value

Median is calculated as

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983124983144983145983155 983149983137983156983141983154983145983137983148 983145983155 983139983154983141983137983156983141983140 983138983161 983144983156983156983152983098983152983144983161983155983145983139983155983139983137983156983137983148983161983155983156983139983151983149 983137983150983140 983145983155 983142983151983154 983161983151983157983154 983152983141983154983155983151983150983137983148 983137983150983140 983150983151983150983085983139983151983149983149983141983154983139983145983137983148 983157983155983141

983151983150983148983161

983092983092

12 1 Where n is the number of values in the data

If the number of values in the data set is eventhen the median is the average of the two middlevalues

8 Mode Mode of a statistical data is the value of thatvariable which has the maximum frequency

Sno Term Description

1983117983141983137983150 983142983151983154

983125983150983143983154983151983157983152

983110983154983141983153983157983141983150983139983161 983156983137983138983148983141

Mean is given by

sum sum

2983117983141983137983150 983142983151983154 983143983154983151983157983152

983110983154983141983153983157983141983150983139983161 983156983137983138983148983141

In these distribution it is assumed that frequency of each classinterval is centered around its mid-point ie class marks

2

983117983141983137983150 983139983137983150 983138983141 983139983137983148983139983157983148983137983156983141983140 983157983155983145983150983143 983156983144983154983141983141 983149983141983156983144983151983140

983137983081 983108983145983154983141983139983156 983149983141983156983144983151983140

sum

sum

983138983081 983105983155983155983157983149983141983140 983149983141983137983150 983149983141983156983144983151983140

sum sum

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983124983144983145983155 983149983137983156983141983154983145983137983148 983145983155 983139983154983141983137983156983141983140 983138983161 983144983156983156983152983098983152983144983161983155983145983139983155983139983137983156983137983148983161983155983156983139983151983149 983137983150983140 983145983155 983142983151983154 983161983151983157983154 983152983141983154983155983151983150983137983148 983137983150983140 983150983151983150983085983139983151983149983149983141983154983139983145983137983148 983157983155983141

983151983150983148983161

983092983093

Where

a=gt Assumed mean

di =gt xi ndasha

983139983081 983123983156983141983152 983140983141983158983145983137983156983145983151983150 983117983141983156983144983151983140

sum sum

Where

a=gt Assumed mean

ui =gt (xi ndasha)h

3983117983151983140983141 983142983151983154

983143983154983151983157983152983141983140

983142983154983141983153983157983141983150983139983161 983156983137983138983148983141

Modal class The class interval having highest frequency is

called the modal class and Mode is obtained using the modal

class

2

983127983144983141983154983141

l = lower limit of the modal classh = size of the class interval (assuming all class sizes tobe equal)f 1 = frequency of the modal classf 0 = frequency of the class preceding the modal classf 2 = frequency of the class succeeding the modal class

4983117983141983140983145983137983150 983151983142 983137

983143983154983151983157983152983141983140 983140983137983156983137

983142983154983141983153983157983141983150983139983161 983156983137983138983148983141

For the given data we need to have class interval frequency

distribution and cumulative frequency distribution

Median is calculated as

7252019 Maths Formula Class10

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983124983144983145983155 983149983137983156983141983154983145983137983148 983145983155 983139983154983141983137983156983141983140 983138983161 983144983156983156983152983098983152983144983161983155983145983139983155983139983137983156983137983148983161983155983156983139983151983149 983137983150983140 983145983155 983142983151983154 983161983151983157983154 983152983141983154983155983151983150983137983148 983137983150983140 983150983151983150983085983139983151983149983149983141983154983139983145983137983148 983157983155983141

983151983150983148983161

983092983094

2

983127983144983141983154983141

983148 983101 983148983151983159983141983154 983148983145983149983145983156 983151983142 983149983141983140983145983137983150 983139983148983137983155983155

983150 983101 983150983157983149983138983141983154 983151983142 983151983138983155983141983154983158983137983156983145983151983150983155

983139983142 983101 983139983157983149983157983148983137983156983145983158983141 983142983154983141983153983157983141983150983139983161 983151983142 983139983148983137983155983155 983152983154983141983139983141983140983145983150983143 983156983144983141 983149983141983140983145983137983150 983139983148983137983155983155

983142 983101 983142983154983141983153983157983141983150983139983161 983151983142 983149983141983140983145983137983150 983139983148983137983155983155

983144 983101 983139983148983137983155983155 983155983145983162983141 (983137983155983155983157983149983145983150983143 983139983148983137983155983155 983155983145983162983141 983156983151 983138983141 983141983153983157983137983148)

5983109983149983152983145983154983145983139983137983148

983110983151983154983149983157983148983137 983138983141983156983159983141983141983150

983117983151983140983141983084 983117983141983137983150 983137983150983140

983117983141983140983145983137983150

3 Median=Mode +2 Mean

ProbabilitySn

o

Term Description

1 Empirical

probability

It is a probability of event which is calculated based on

experiments

Example

A coin is tossed 1000 times we get 499 times head and

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983151983150983148983161

983092983095

501 times tail

So empirical or experimental probability of getting head is

calculated as

4991000 499

Empirical probability depends on experiment and

different will get different values based on the

experiment

2 Important point

about events

If the event A B C covers the entire possible outcome inthe experiment Then

P (A) +P (B) +P(C)=1

3 impossible

event

The probability of an event (U) which is impossible tooccur is 0 Such an event is called an impossible event

P (U)=0

4 Sure or certain

event

The probability of an event (X) which is sure (or certain)to occur is 1 Such an event is called a sure event or acertain event

P(X) =1

5 Probability of

any event

Probability of any event can be as

0 1

Sn

o

Term Description

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983092983096

1

983124983144983141983151983154983141983156983145983139983137983148

983120983154983151983138983137983138983145983148983145983156983161

The theoretical probability or the classical probability of the event isdefined as

2 Elementary

events

983105983150 983141983158983141983150983156 983144983137983158983145983150983143 983151983150983148983161 983151983150983141 983151983157983156983139983151983149983141 983151983142 983156983144983141 983141983160983152983141983154983145983149983141983150983156 983145983155 983139983137983148983148983141983140 983137983150

983141983148983141983149983141983150983156983137983154983161 983141983158983141983150983156

991260983124983144983141 983155983157983149 983151983142 983156983144983141 983152983154983151983138983137983138983145983148983145983156983145983141983155 983151983142 983137983148983148 983156983144983141 983141983148983141983149983141983150983156983137983154983161 983141983158983141983150983156983155 983151983142983137983150 983141983160983152983141983154983145983149983141983150983156 983145983155 1991261

983113983141 983113983142 983159983141 983156983144983154983141983141 983141983148983141983149983141983150983156983137983154983161 983141983158983141983150983156 983105983106983107 983145983150 983156983144983141 983141983160983152983141983154983145983149983141983150983156 983156983144983141983150

983120(983105)+983120(983106) +983120(983107)9831011

3 Complementar

y events

983124983144983141 983141983158983141983150983156 Ᾱ 983154983141983152983154983141983155983141983150983156983145983150983143 991256983150983151983156 983105991257 983145983155 983139983137983148983148983141983140 983156983144983141 983139983151983149983152983148983141983149983141983150983156 983151983142 983156983144983141 983141983158983141983150983156

983105 983127983141 983137983148983155983151 983155983137983161 983156983144983137983156 Ᾱ 983137983150983140 983105 983137983154983141 983139983151983149983152983148983141983149983141983150983156983137983154983161 983141983158983141983150983156983155 983105983148983155983151

983120(983105) +983120(Ᾱ)9831011

4 Sure or certain

event

The probability of an event (X) which is sure (or certain)

to occur is 1 Such an event is called a sure event or acertain event

P(X) =1

5 Probability of

any event

Probability of any event can be as

0 1

Page 4: Maths Formula Class10

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983151983150983148983161

983091

5 HCF and LCM byprime factorization

method

HCF = Product of the smallest power of eachcommon factor in the numbers

LCM = Product of the greatest power of each prime

factor involved in the number

6 Important Formula 983112983107983110 (983137983138) 983128 983116983107983117 (983137983138) 983101983137 983128 983138

7 Important concept

for rational Number

Terminating decimal expression can be written in

the form

p2n

5m

Polynomial expressions

A polynomial expression S(x) in one variable x is an algebraic expression in x term

as

⋯ 991270 991270 991270 Where anan-1hellipaa0 are constant and real numbers and an is not equal to zero

Some Important point to Note

Sno Points

1 an an-1 an-2 hellipa1a0 are called the coefficients for xnxn-1 hellipx1x0

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983151983150983148983161

983092

2 n is called the degree of the polynomial

3 when an an-1 an-2 hellipa1a0 all are zero it is called zero polynomial

4 A constant polynomial is the polynomial with zero degree it is a constant

value polynomial

5 A polynomial of one item is called monomial two items binomial and three

items as trinomial

6 A polynomial of one degree is called linear polynomial two degree as

quadratic polynomial and degree three as cubic polynomial

Important concepts on Polynomial

Concept Description

Zerorsquos or roots

of the

polynomial

It is a solution to the polynomial equation S(x)=0 ie a number

a is said to be a zero of a polynomial if S(a) = 0

If we draw the graph of S(x) =0 the values where the curve

cuts the X-axis are called Zeroes of the polynomial

Remainder

Theoremrsquos

If p(x) is an polynomial of degree greater than or equal to 1

and p(x) is divided by the expression (x-a)then the remainder

will be p(a)

FactorrsquosTheoremrsquos

If x-a is a factor of polynomial p(x) then p(a)=0 or if p(a)=0x-a is the factor the polynomial p(x)

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983151983150983148983161

983093

Geometric Meaning of the Zeroes of the polynomial

Letrsquos us assume

y= p(x) where p(x) is the polynomial of any form

Now we can plot the equation y=p(x) on the Cartesian plane by taking various values of x and y obtained

by putting the values The plot or graph obtained can be of any shapes

The zeroes of the polynomial are the points where the graph meet x axis in the Cartesian plane If the

graph does not meet x axis then the polynomial does not have any zerorsquos

Let us take some useful polynomial and shapes obtained on the Cartesian plane

Sno y=p(x) Graph obtained Name of the

graph

Name of the

equation

1 y=ax+b where a and b can be

any values (ane0)

Example y=2x+3

Straight line

It intersect the x-

axis at ( -ba 0)

Example ( -320)

Linear polynomial

2 y=ax2+bx+c

where

b2-4ac gt 0 and ane0 and agt 0

Example

y=x2-7x+12

Parabola

It intersect the x-

axis at two points

Example

(30) and (40)

Quadratic

polynomial

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3 y=ax2+bx+c

where

b2-4ac gt 0 and ane0 and a lt 0

Example

y=-x2+2x+8

Parabola

It intersect the x-

axis at two points

Example (-20)

and (40)

Quadratic

polynomial

4 y=ax +bx+c

where

b2-4ac = 0 and ane0 a gt 0

Example

y=(x-2)2

Parabola

It intersect the x-

axis at one points

Quadratic

polynomial

5 y=ax2+bx+c

where

b

2

-4ac lt 0 and ane0 a gt 0

Example

y=x2-2x+6

Parabola

It does not

intersect the x-axis

It has no zerorsquos

Quadratic

polynomial

6 y=ax +bx+c

where

b2-4ac lt 0 and ane0 a lt 0

Example

y=-x2-2x-6

Parabola

It does not

intersect the x-axis

It has no zerorsquos

Quadratic

polynomial

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7 y=ax3 +bx

2+cx+d

where ane0

It can be of any

shape

It will cut the x-axis

at the most 3 times

Cubic Polynomial

8 ⋯ 991270 991270 991270

Where an ne0

It can be of any

shape

It will cut the x-axis

at the most n times

Polynomial of n

degree

Relation between coefficient and zeroes of the Polynomial

Sno Type of Polynomial General form Zerorsquos Relationship between Zerorsquos

and coefficients

1 Linear polynomial ax+b ane0 1

2 Quadratic ax +bx+c ane0 2

3 Cubic ax +bx +cx+d ane0 3

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Formation of polynomial when the zeroes are given

Type ofpolynomial

Zerorsquos Polynomial Formed

Linear k=a (x-a)

Quadratic k1=a and

k2=b

(x-a)(x-b)

Or

x2-( a+b)x +ab

Or

x2

-( Sum of the zerorsquos)x +product of the zerorsquos

Cubic k1=a k2=b

and k3=c

(x-a)(x-b)(x-c)

Division algorithm for Polynomial

Letlsquos p(x) and q(x) are any two polynomial with q(x) ne0 then we can find polynomial s(x) and r(x) suchthat

P(x)=s(x) q(x) + r(x)

Where r(x) can be zero or degree of r(x) lt degree of g(x)

983108983145983158983145983140983141983150983140 983101983121983157983151983156983145983141983150983156 983128 983108983145983158983145983155983151983154 + 983122983141983149983137983145983150983140983141983154

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COORDINATE GEOMETRYSno Points

1 We require two perpendicular axes to locate a point in the plane One of

them is horizontal and other is Vertical

2 The plane is called Cartesian plane and axis are called the coordinates axis

3 The horizontal axis is called x-axis and Vertical axis is called Y-axis

4 The point of intersection of axis is called origin

5 The distance of a point from y axis is called x ndashcoordinate or abscissa and

the distance of the point from x ndashaxis is called y ndash coordinate or Ordinate

6 The distance of a point from y axis is called x ndashcoordinate or abscissa and

the distance of the point from x ndashaxis is called y ndash coordinate or Ordinate

7 The Origin has zero distance from both x-axis and y-axis so that its

abscissa and ordinate both are zero So the coordinate of the origin is (0

0)

8 A point on the x ndashaxis has zero distance from x-axis so coordinate of any

point on the x-axis will be (x 0)

9 A point on the y ndashaxis has zero distance from y-axis so coordinate of any

point on the y-axis will be (0 y)

10 The axes divide the Cartesian plane in to four parts These Four parts arecalled the quadrants

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10

The coordinates of the points in the four quadrants will have sign according to thebelow table

Quadrant x-coordinate y-coordinate

Ist Quadrant + +

IInd quadrant - +

IIIrd quadrant - -

IVth quadrant + -

Sno Terms Descriptions

1 Distance formula Distance between the points AB is given by

Distance of Point A from Origin

2 Section Formula A point P(xy) which divide the line segment AB in

the ratio m1 and m2 is given by

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11

The midpoint P is given by

3 Area of Triangle Area of triangle ABC of coordinates A(x 1y 1 )

B(x 2 y 2 ) and C(x 3y 3 )

12 For point AB and C to be collinear The value of A

should be zero

LINEAR EQUATIONS IN TWO

VARIABLESAn equation of the form ax + by + c = 0 where a b and c are real numbers such

that a andb are not both zero is called a linear equation in two variablesImportant points to Note

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12

Sno Points

1 A linear equation in two variable has infinite solutions

2 The graph of every linear equation in two variable is a straight line

3 x = 0 is the equation of the y-axis and y = 0 is the equation of the x-axis

4 The graph x=a is a line parallel to y -axis

5 The graph y=b is a line parallel to x -axis

6 An equation of the type y = mx represents a line passing through theorigin

7 Every point on the graph of a linear equation in two variables is a solution

of the linear

equation Moreover every solution of the linear equation is a point on the

graph

Sno Type of equation Mathematical

representation

Solutions

1 Linear equation in one Variable ax+b=0 ane0

a and b are real

number

One solution

2 Linear equation in two Variable ax+by+c=0 ane0 and

bne0

a b and c are real

number

Infinite solution

possible

3 Linear equation in three Variable ax+by+cz+d=0 ane0

bne0 and cne0

Infinite solution

possible

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a b c d are realnumber

Simultaneous pair of Linear equation

A pair of Linear equation in two variables

a1x+b1y+c1=0

a2x +b2y+c2=0

Graphically it is represented by two straight lines on Cartesian plane

Simultaneous pair of

Linear equation

Condition Graphical

representation

Algebraic

interpretation

a1x+b1y+c1=0

a2x +b2y+c2=0

Example

x-4y+14=0

3x+2y-14=0

Intersecting lines The

intersecting point

coordinate is the only

solution

One unique solution

only

a1x+b1y+c1=0

a2x +b2y+c2=0

Coincident lines Theany coordinate on the

line is the solution

Infinite solution

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Example

2x+4y=16

3x+6y=24

a1x+b1y+c1=0

a2x +b2y+c2=0

Example

2x+4y=6

4x+8y=18

Parallel Lines No solution

The graphical solution can be obtained by drawing the lines on the Cartesian plane

Algebraic Solution of system of Linear equation

Sno Type of method Working of method

1 Method of elimination by

substitution

1) Suppose the equation are

a1x+b1y+c1=0

a2x +b2y+c2=0

2) Find the value of variable of either x or y in other

variable term in first equation

3) Substitute the value of that variable in second

equation

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4) Now this is a linear equation in one variable Findthe value of the variable

5) Substitute this value in first equation and get the

second variable

2 Method of elimination by

equating the coefficients

1) Suppose the equation are

a1x+b1y+c1=0

a2x +b2y+c2=0

2) Find the LCM of a1 and a2 Let it k

3) Multiple the first equation by the value ka1

4) Multiple the first equation by the value ka2

4) Subtract the equation obtained This way one

variable will be eliminated and we can solve to get the

value of variable y

5) Substitute this value in first equation and get the

second variable

3 Cross Multiplication method 1) Suppose the equation are

a1x+b1y+c1=0

a2x +b2y+c2=0

2) This can be written as

3) This can be written as

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4) Value of x and y can be find using the

x =gt first and last expression

y=gt second and last expression

Quadratic EquationsSno Terms Descriptions

1 Quadratic Polynomial P(x) = ax +bx+c where ane0

2 Quadratic equation ax +bx+c =0 where ane0

3 Solution or root of the

Quadratic equation

A real number α is called the root or solution of the

quadratic equation if

aα2 +bα+c=0

4 zeroes of the

polynomial p(x)

The root of the quadratic equation are called zeroes

5 Maximum roots of

quadratic equations

We know from chapter two that a polynomial of

degree can have max two zeroes So a quadratic

equation can have maximum two roots

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6 Condition for realroots A quadratic equation has real roots if b

2

- 4ac gt 0

How to Solve Quadratic equation

Sno Method Working

1 factorization This method we factorize the equation by splitting themiddle term b

In ax2+bx+c=0

Example

6x2-x-2=0

1) First we need to multiple the coefficient a and cIn this

case =6X-2=-12

2) Splitting the middle term so that multiplication is 12

and difference is the coefficient b

6x2 +3x-4x-2=0

3x( 2x+1) -2(2x+1)=0

(3x-2) (2x+1)=0

3) Roots of the equation can be find equating the factorsto zero

3x-2=0 =gt x=32

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2x+1=0 =gt x=-12

2 Square method In this method we create square on LHS and RHS and

then find the value

ax2 +bx+c=0

1) x2 +(ba) x+(ca)=0

2) ( x+b2a)2 ndash(b2a)2 +(ca)=0

3) ( x+b2a)2=(b2-4ac)4a2

4) radic

Example

x2 +4x-5=0

1) (x+2)2 -4-5=0

2) (x+2)2=9

3) Roots of the equation can be find using square root on

both the sides

x+2 =-3 =gt x=-5

x+2=3=gt x=1

3 Quadratic

method

For quadratic equation

ax2 +bx+c=0

roots are given by radic radic

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For b2 -4ac gt 0 Quadratic equation has two real roots of

different value

For b2-4ac =0 quadratic equation has one real root

For b2-4ac lt 0 no real roots for quadratic equation

Nature of roots of Quadratic equation

Sno Condition Nature of roots

1 b -4ac gt 0 Two distinct real roots

2 b -4ac =0 One real root

3 b

2

-4ac lt 0 No real roots

Triangles

Sno Terms Descriptions

1 Congruence Two Geometric figure are said to be congruence ifthey are exactly same size and shapeSymbol used is cong Two angles are congruent if they are equalTwo circle are congruent if they have equal radiiTwo squares are congruent if the sides are equal

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20

2 Triangle Congruencebull

Two triangles are congruent if three sidesand three angles of one triangle is

congruent to the corresponding sides and

angles of the other

bull Corresponding sides are equal

AB=DE BC=EF AC=DF

bull Corresponding angles are equalang angang angang ang

bull We write this as

cong

bull The above six equalities are between the

corresponding parts of the two congruent

triangles In short form this is called

CPCT

bull We should keep the letters in correct order

on both sides

3 Inequalities in

Triangles

1) In a triangle angle opposite to longer side is

larger

2) In a triangle side opposite to larger angle is

larger

3) The sum of any two sides of the triangle is

greater than the third side

983105

983107983106

983108

983109 983110

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21

In triangle ABC

AB +BC gt AC

Different Criterion for Congruence of the triangles

N Criterion Description Figures and

expression

1 Side angle

Side (SAS)

congruence

bull Two triangles are congruent if the

two sides and included angles of

one triangle is equal to the two

sides and included angle

bull It is an axiom as it cannot be

proved so it is an accepted truth

bull ASS and SSA type two triangles

may not be congruent alwaysIf following

condition

983105

983110983109

983108

983107983106

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983151983150983148983161

22

AB=DE BC=EFang ang

Then

cong

2 Angle side

angle (ASA)

congruence

bull Two triangles are congruent if the

two angles and included side of

one triangle is equal to the

corresponding angles and side

bull It is a theorem and can be proved

If following

condition

BC=EF

ang angang ang

Then

cong

3 Angle angle

side( AAS)

congruence

bull Two triangles are congruent if the

any two pair of angles and any

side of one triangle is equal to the

corresponding angles and side

bull It is a theorem and can be proved

983110983109

983108

983107983106

983105

983110983109

983108

983107983106

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983151983150983148983161

2983091

If followingcondition

BC=EF

ang angang ang

Then

cong

4 Side-Side-Side

(SSS)congruence

bull Two triangles are congruent if the

three sides of one triangle is equalto the three sides of the another

If following

condition

BC=EFAB=DEDF

=AC

Then

cong

5 Right angle ndash

hypotenuse-

side(RHS)

bull Two right triangles are congruent if

the hypotenuse and a side of the

one triangle are equal to

983110983109

983108

983107983106

983105

983105 983108

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983151983150983148983161

2983092

congruence corresponding hypotenuse and sideof the another

If following

condition

AC=DFBC=EF

Then

cong

Some Important points on Triangles

Terms Description

Orthocenter Point of intersection of the three altitudeof the triangle

Equilateral triangle whose all sides are equal and allangles are equal to 600

Median A line Segment joining the corner of thetriangle to the midpoint of the oppositeside of the triangle

Altitude A line Segment from the corner of thetriangle and perpendicular to theopposite side of the triangle

Isosceles A triangle whose two sides are equalCentroid Point of intersection of the three median

of the triangle is called the centroid of

the triangleIn center All the angle bisector of the trianglepasses through same point

Circumcenter The perpendicular bisector of the sidesof the triangles passes through samepoint

983106 983107 983109 983110

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2983093

Scalene triangle Triangle having no equal angles and noequal sidesRight Triangle Right triangle has one angle equal to 900 Obtuse Triangle One angle is obtuse angle while other

two are acute anglesAcute Triangle All the angles are acute

Similarity of Triangles

Sno Points

1 Two figures having the same shape but not necessarily the same size are called similarfigures

2 All the congruent figures are similar but the converse is not true

3 If a line is drawn parallel to one side of a triangle to intersect the other two sides in

distinct points then the other two sides are divided in the same ratio

4 If a line divides any two sides of a triangle in the same ratio then the line is parallel tothe third side

Different Criterion for Similarity of the triangles

N Criterion Description Expression

1 Angle Angle

angle(AAA)

similarity

bull Two triangles are similar if

corresponding angle are equal

If following

condition

ang ang

ang ang

ang ang

Then

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983151983150983148983161

2983094

Then 983166

2 Angle angle

(AA) similarity

bull Two triangles are similar if the two

corresponding angles are equal as

by angle property third angle will

be also equal

If following

condition

ang ang

ang ang

Then

ang ang

Then

Then

983166

3 Side side

side(SSS)

Similarity

Two triangles are similar if the

sides of one triangle is

proportional to the sides of other

triangle

If following

condition

Then

ang ang

ang ang

ang ang

Then

cong

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983151983150983148983161

2983095

4 Side-Angle-Side (SAS)

similarity

bull

Two triangles are similar if the oneangle of a triangle is equal to one

angle of other triangles and sides

including that angle is proportional

If followingcondition

And ang ang

Then

cong

Area of Similar triangles

If the two triangle ABC and DEF are similar

cong

Then

Pythagoras Theorem

Sno Points

1 If a perpendicular is drawn from the vertex of the right angle of a right triangle to thehypotenuse then the triangles on both sides of the perpendicular are similar to the

whole triangle and also to each other

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983151983150983148983161

2983096

2 In a right triangle the square of the hypotenuse is equal to the sum of the squares of the

other two sides (Pythagoras Theorem)

(hyp)2 =(base)

2 + (perp)

2

3 If in a triangle square of one side is equal to the sum of the squares of the other twosides then the angle opposite the first side is a right angle

Arithmetic Progression

Sno Terms Descriptions

1 ArithmeticProgression

An arithmetic progression is a sequence ofnumbers such that the difference of any two

successive members is a constant

Examples

1) 1591317hellip

2) 12345hellip

2 common difference of

the AP

the difference between any successive members is a

constant and it is called the common difference of AP

1) If a1 a2a3a4a5 are the terms in AP then

D=a2 -a1 =a3 - a2 =a4 ndash a3=a5 ndasha4

2) We can represent the general form of AP inthe form

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2983097

aa+da+2da+3da+4d

Where a is first term and d is the commondifference

3 nth term of Arithmetic Progression

n term = a + (n - 1)d

4 Sum of nth item in ArithmeticProgression

Sn =(n2)[a + (n-1)d]

Or

Sn =(n2)[t1+ tn]

Trigonometry

Sno Terms Descriptions

1 What is

Trigonometry

Trigonometry from Greek trigotildenon triangle and

metron measure) is a branch of mathematics

that studies relationships involving lengths and

angles of triangles The field emerged during the

3rd century BC from applications of geometry to

astronomical studies

Trigonometry is most simply associated with

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9830910

planar right angle triangles (each of which is atwo-dimensional triangle with one angle equal to

90 degrees) The applicability to non-right-angle

triangles exists but since any non-right-angle

triangle (on a flat plane) can be bisected to

create two right-angle triangles most problems

can be reduced to calculations on right-angle

triangles Thus the majority of applications relate

to right-angle triangles

2 Trigonometric

Ratiorsquos

In a right angle triangle ABC where B=90deg

We can define following term for angle A

Base Side adjacent to angle

Perpendicular Side Opposite of angleHypotenuse Side opposite to right angle

We can define the trigonometric ratios for

angle A as

sin A= PerpendicularHypotenuse =BCAC

cosec A= HypotenusePerpendicular

=ACBC

cos A= BaseHypotenuse =ABAC

sec A= HypotenuseBase=ACAB

tan A= PerpendicularBase =BCAB

cot A= BasePerpendicular=ABBC

Notice that each ratio in the right-hand column is

the inverse or the reciprocal of the ratio in the

left-hand column

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9830911

3 Reciprocal offunctions The reciprocal of sin A is cosec A and vice-versa

The reciprocal of cos A is sec A

And the reciprocal of tan A is cot A

These are valid for acute angles

We can define tan A = sin Acos A

And Cot A =cos A Sin A

4 Value of of sin and

cos

Is always less 1

5 Trigonometric ration

from another angle

We can define the trigonometric ratios for angle Cas

sin C= PerpendicularHypotenuse =ABACcosec C= HypotenusePerpendicular =ACABBcos C= BaseHypotenuse =BCACsec C= HypotenuseBase=ACBCtan A= PerpendicularBase =ABBCcot A= BasePerpendicular=BCAB

6 Trigonometric ratios

of complimentary

Sin (90-A) =cos(A)

Cos(90-A) = sin A

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983151983150983148983161

9830912

angles Tan(90-A) =cot A

Sec(90-A)= cosec A

Cosec (90-A) =sec A

Cot(90- A) =tan A7 Trigonometric

identities

Sin2 A + cos2 A =0

1 + tan2 A =sec2 A

1 + cot2 A =cosec2 A

Trigonometric Ratios of Common angles

We can find the values of trigonometric ratiorsquos various angle

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983091983091

Area of Circles

Sno Points

1 A circle is a collection of all the points in a plane which are equidistant

from a fixed point in the plane

2 Equal chords of a circle (or of congruent circles) subtend equal angles at

the center

3 If the angles subtended by two chords of a circle (or of congruent circles)

at the center (corresponding center) are equal the chords are equal

4 The perpendicular from the center of a circle to a chord bisects the chord

5 The line drawn through the center of a circle to bisect a chord is

perpendicular to the chord

6 There is one and only one circle passing through three non-collinear points

7 Equal chords of a circle (or of congruent circles) are equidistant from the

center (or corresponding centers)

8 Chords equidistant from the center (or corresponding centers) of a circle

(or of congruent circles) are equal

9 If two arcs of a circle are congruent then their corresponding chords are

equal and conversely if two chords of a circle are equal then their

corresponding arcs (minor major) are congruent

10 Congruent arcs of a circle subtend equal angles at the center

11 The angle subtended by an arc at the center is double the angle subtended

by it at any point on the remaining part of the circle

12 Angles in the same segment of a circle are equal

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983091983092

13 Angle in a semicircle is a right angle

14 If a line segment joining two points subtends equal angles at two other

points lying on the same side of the line containing the line segment the

four points lie on a circle

15 The sum of either pair of opposite angles of a cyclic quadrilateral is 180deg

16 If the sum of a pair of opposite angles of a quadrilateral is 180deg then the

quadrilateral is cyclic

Sno Terms Descriptions

1 Circumference of a circle 2 π r

2 Area of circle π r 2

3 Length of the arc of

the sector of angle

Length of the arc of the sector of angle θ 2

4 Area of the sector of

angle

Area of the sector of angle θ

5 Area of segment of a

circle

Area of the corresponding sector ndash Area of the

corresponding triangle

Surface Area and Volume

Sno Term Description

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983091983093

1 Mensuration It is branch of mathematics which is concernedabout the measurement of length area and

Volume of plane and Solid figure

2 Perimeter a)The perimeter of plane figure is defined as thelength of the boundaryb)It units is same as that of length ie m cmkm

3 Area a)The area of the plane figure is the surfaceenclosed by its boundaryb) It unit is square of length unit ie m2 km2

4 Volume Volume is the measure of the amount of spaceinside of a solid figure like a cube ball cylinderor pyramid Its units are always cubic that isthe number of little element cubes that fit insidethe figure

Volume Unit conversion

1 cm3 1mL 1000 mm3

1 Litre 1000ml 1000 cm3

1 m3 106 cm3 1000 L

1 dm3 1000 cm3 1 L

Surface Area and Volume of Cube and Cuboid

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Type Measurement

Surface Area of Cuboid of Length L

Breadth B and Height H

2(LB + BH + LH)

Lateral surface area of the cuboids 2( L + B ) H

Diagonal of the cuboids Volume of a cuboids LBH

Length of all 12 edges of the cuboids 4 (L+B+H)

Surface Area of Cube of side L 6L2

Lateral surface area of the cube 4L2

Diagonal of the cube radic3 Volume of a cube L3

Surface Area and Volume of Right circular cylinder

Radius The radius (r) of the circular base is called the radius of the

cylinder

Height The length of the axis of the cylinder is called the height (h) of the

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cylinder

Lateral

Surface

The curved surface joining the two base of a right circular cylinder is

called Lateral Surface

Type Measurement

Curved or lateral Surface Area of

cylinder

2πrh

Total surface area of cylinder 2πr (h+r)

Volume of Cylinder π r2h

Surface Area and Volume of Right circular cone

Radius The radius (r) of the circular base is called the radius of the

cone

Height The length of the line segment joining the vertex to the center of

base is called the height (h) of the cone

Slant

Height

The length of the segment joining the vertex to any point on the

circular edge of the base is called the slant height (L) of the cone

Lateral

surface

Area

The curved surface joining the base and uppermost point of a right

circular cone is called Lateral Surface

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983124983144983145983155 983149983137983156983141983154983145983137983148 983145983155 983139983154983141983137983156983141983140 983138983161 983144983156983156983152983098983152983144983161983155983145983139983155983139983137983156983137983148983161983155983156983139983151983149 983137983150983140 983145983155 983142983151983154 983161983151983157983154 983152983141983154983155983151983150983137983148 983137983150983140 983150983151983150983085983139983151983149983149983141983154983139983145983137983148 983157983155983141

983151983150983148983161

983091983096

Type Measurement

Curved or lateral Surface Area of

cone

πrL

Total surface area of cone πr (L+r)

Volume of Cone

Surface Area and Volume of sphere and hemisphere

Sphere A sphere can also be considered as a solid obtained on

rotating a circle About its diameter

Hemisphere A plane through the centre of the sphere divides the sphere into two

equal parts each of which is called a hemisphere

radius The radius of the circle by which it is formed

SphericalShell

The difference of two solid concentric spheres is called a sphericalshell

Lateral

Surface

Area for

Total surface area of the sphere

983123983152983144983141983154983141

983112983141983149983145983155983152983144983141983154983141

7252019 Maths Formula Class10

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983151983150983148983161

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Sphere

Lateral

Surface

area of

Hemisphere

It is the curved surface area leaving the circular base

Type Measurement

Surface area of Sphere 4πr2

Volume of Sphere 43 Curved Surface area of hemisphere 2πr2

Total Surface area of hemisphere 3πr2

Volume of hemisphere 23 Volume of the spherical shell whose

outer and inner radii and lsquoRrsquo and lsquorrsquorespectively

43

How the Surface area and Volume are determined

Area of Circle The circumference of a circle is 2πr

This is the definition of π (pi) Divide

the circle into many triangular

segments The area of the triangles

is 12 times the sum of their bases

2πr (the circumference) times their

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983151983150983148983161

9830920

height r

12 2 Surface Area of cylinder This can be imagined as unwrapping the

surface into a rectangle

Surface area of cone This can be achieved by divide the

surface of the cone into its triangles or

the surface of the cone into many thin

triangles The area of the triangles is 12

times the sum of their bases p times

their height

12 2

Surface Area and Volume of frustum of cone

h = vertical height of the frustum

l = slant height of the frustum

r 1 and r 2 are radii of the two bases (ends) of the frustum

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983151983150983148983161

9830921

Type Measurement

983126983151983148983157983149983141 983151983142 983137 983142983154983157983155983156983157983149 983151983142 983137 983139983151983150983141 983123983148983137983150983156 983144983141983145983143983144983156 983151983142 983142983154983157983155983156983157983149 983151983142 983137 983139983151983150983141 983107983157983154983158983141983140 983155983157983154983142983137983139983141 983137983154983141983137 983151983142 983137 983142983154983157983155983156983157983149 983151983142 983137 983139983151983150983141 983124983151983156983137983148 983155983157983154983142983137983139983141 983137983154983141983137 983151983142 983142983154983157983155983156983157983149 983151983142 983137 983139983151983150983141

Statistics

Sno Term Description

1 Statistics Statistics is a broad mathematical disciplinewhich studies ways to collect summarize and

draw conclusions from data

2 Data A systematic record of facts or different values of

a quantity is called data

Data is of two types - Primary data and Secondary

data

Primary Data The data collected by a researcher

with a specific purpose in mind is called primarydata

Secondary Data The data gathered from a

source where it already exists is called secondary

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983151983150983148983161

9830922

data

3 Features of data bull Statistics deals with collectionpresentation analysis and interpretation ofnumerical data

bull Arranging data in an order to study theirsalient features is called presentation ofdata

bull Data arranged in ascending or descendingorder is called arrayed data or an array

bull Range of the data is the differencebetween the maximum and the minimum

values of the observationsbull Table that shows the frequency of different

values in the given data is called afrequency distribution table

bull A frequency distribution table that showsthe frequency of each individual value in thegiven data is called an ungrouped frequencydistribution table

bull A table that shows the frequency of groupsof values in the given data is called agrouped frequency distribution table

bull

The groupings used to group the values ingiven data are called classes or class-intervals The number of values that eachclass contains is called the class size orclass width The lower value in a class iscalled the lower class limit The higher valuein a class is called the upper class limit

bull Class mark of a class is the mid value ofthe two limits of that class

bull A frequency distribution in which the upperlimit of one class differs from the lower limitof the succeeding class is called an

Inclusive or discontinuous FrequencyDistribution

bull A frequency distribution in which the upperlimit of one class coincides from the lowerlimit of the succeeding class is called anexclusive or continuous Frequency

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983151983150983148983161

983092983091

Distribution bull

4 Bar graph A bar graph is a pictorial representation of data inwhich rectangular bars of uniform width are drawnwith equal spacing between them on one axisusually the x axis The value of the variable isshown on the other axis that is the y axis

5 Histogram A histogram is a set of adjacent rectangles whoseareas are proportional to the frequencies of agiven continuous frequency distribution

6 Mean The mean value of a variable is defined as thesum of all the values of the variable divided by thenumber of values

4 sum

7 Median The median of a set of data values is the middle

value of the data set when it has been arranged inascending order That is from the smallest valueto the highest value

Median is calculated as

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983151983150983148983161

983092983092

12 1 Where n is the number of values in the data

If the number of values in the data set is eventhen the median is the average of the two middlevalues

8 Mode Mode of a statistical data is the value of thatvariable which has the maximum frequency

Sno Term Description

1983117983141983137983150 983142983151983154

983125983150983143983154983151983157983152

983110983154983141983153983157983141983150983139983161 983156983137983138983148983141

Mean is given by

sum sum

2983117983141983137983150 983142983151983154 983143983154983151983157983152

983110983154983141983153983157983141983150983139983161 983156983137983138983148983141

In these distribution it is assumed that frequency of each classinterval is centered around its mid-point ie class marks

2

983117983141983137983150 983139983137983150 983138983141 983139983137983148983139983157983148983137983156983141983140 983157983155983145983150983143 983156983144983154983141983141 983149983141983156983144983151983140

983137983081 983108983145983154983141983139983156 983149983141983156983144983151983140

sum

sum

983138983081 983105983155983155983157983149983141983140 983149983141983137983150 983149983141983156983144983151983140

sum sum

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983151983150983148983161

983092983093

Where

a=gt Assumed mean

di =gt xi ndasha

983139983081 983123983156983141983152 983140983141983158983145983137983156983145983151983150 983117983141983156983144983151983140

sum sum

Where

a=gt Assumed mean

ui =gt (xi ndasha)h

3983117983151983140983141 983142983151983154

983143983154983151983157983152983141983140

983142983154983141983153983157983141983150983139983161 983156983137983138983148983141

Modal class The class interval having highest frequency is

called the modal class and Mode is obtained using the modal

class

2

983127983144983141983154983141

l = lower limit of the modal classh = size of the class interval (assuming all class sizes tobe equal)f 1 = frequency of the modal classf 0 = frequency of the class preceding the modal classf 2 = frequency of the class succeeding the modal class

4983117983141983140983145983137983150 983151983142 983137

983143983154983151983157983152983141983140 983140983137983156983137

983142983154983141983153983157983141983150983139983161 983156983137983138983148983141

For the given data we need to have class interval frequency

distribution and cumulative frequency distribution

Median is calculated as

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983151983150983148983161

983092983094

2

983127983144983141983154983141

983148 983101 983148983151983159983141983154 983148983145983149983145983156 983151983142 983149983141983140983145983137983150 983139983148983137983155983155

983150 983101 983150983157983149983138983141983154 983151983142 983151983138983155983141983154983158983137983156983145983151983150983155

983139983142 983101 983139983157983149983157983148983137983156983145983158983141 983142983154983141983153983157983141983150983139983161 983151983142 983139983148983137983155983155 983152983154983141983139983141983140983145983150983143 983156983144983141 983149983141983140983145983137983150 983139983148983137983155983155

983142 983101 983142983154983141983153983157983141983150983139983161 983151983142 983149983141983140983145983137983150 983139983148983137983155983155

983144 983101 983139983148983137983155983155 983155983145983162983141 (983137983155983155983157983149983145983150983143 983139983148983137983155983155 983155983145983162983141 983156983151 983138983141 983141983153983157983137983148)

5983109983149983152983145983154983145983139983137983148

983110983151983154983149983157983148983137 983138983141983156983159983141983141983150

983117983151983140983141983084 983117983141983137983150 983137983150983140

983117983141983140983145983137983150

3 Median=Mode +2 Mean

ProbabilitySn

o

Term Description

1 Empirical

probability

It is a probability of event which is calculated based on

experiments

Example

A coin is tossed 1000 times we get 499 times head and

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983151983150983148983161

983092983095

501 times tail

So empirical or experimental probability of getting head is

calculated as

4991000 499

Empirical probability depends on experiment and

different will get different values based on the

experiment

2 Important point

about events

If the event A B C covers the entire possible outcome inthe experiment Then

P (A) +P (B) +P(C)=1

3 impossible

event

The probability of an event (U) which is impossible tooccur is 0 Such an event is called an impossible event

P (U)=0

4 Sure or certain

event

The probability of an event (X) which is sure (or certain)to occur is 1 Such an event is called a sure event or acertain event

P(X) =1

5 Probability of

any event

Probability of any event can be as

0 1

Sn

o

Term Description

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983092983096

1

983124983144983141983151983154983141983156983145983139983137983148

983120983154983151983138983137983138983145983148983145983156983161

The theoretical probability or the classical probability of the event isdefined as

2 Elementary

events

983105983150 983141983158983141983150983156 983144983137983158983145983150983143 983151983150983148983161 983151983150983141 983151983157983156983139983151983149983141 983151983142 983156983144983141 983141983160983152983141983154983145983149983141983150983156 983145983155 983139983137983148983148983141983140 983137983150

983141983148983141983149983141983150983156983137983154983161 983141983158983141983150983156

991260983124983144983141 983155983157983149 983151983142 983156983144983141 983152983154983151983138983137983138983145983148983145983156983145983141983155 983151983142 983137983148983148 983156983144983141 983141983148983141983149983141983150983156983137983154983161 983141983158983141983150983156983155 983151983142983137983150 983141983160983152983141983154983145983149983141983150983156 983145983155 1991261

983113983141 983113983142 983159983141 983156983144983154983141983141 983141983148983141983149983141983150983156983137983154983161 983141983158983141983150983156 983105983106983107 983145983150 983156983144983141 983141983160983152983141983154983145983149983141983150983156 983156983144983141983150

983120(983105)+983120(983106) +983120(983107)9831011

3 Complementar

y events

983124983144983141 983141983158983141983150983156 Ᾱ 983154983141983152983154983141983155983141983150983156983145983150983143 991256983150983151983156 983105991257 983145983155 983139983137983148983148983141983140 983156983144983141 983139983151983149983152983148983141983149983141983150983156 983151983142 983156983144983141 983141983158983141983150983156

983105 983127983141 983137983148983155983151 983155983137983161 983156983144983137983156 Ᾱ 983137983150983140 983105 983137983154983141 983139983151983149983152983148983141983149983141983150983156983137983154983161 983141983158983141983150983156983155 983105983148983155983151

983120(983105) +983120(Ᾱ)9831011

4 Sure or certain

event

The probability of an event (X) which is sure (or certain)

to occur is 1 Such an event is called a sure event or acertain event

P(X) =1

5 Probability of

any event

Probability of any event can be as

0 1

Page 5: Maths Formula Class10

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983151983150983148983161

983092

2 n is called the degree of the polynomial

3 when an an-1 an-2 hellipa1a0 all are zero it is called zero polynomial

4 A constant polynomial is the polynomial with zero degree it is a constant

value polynomial

5 A polynomial of one item is called monomial two items binomial and three

items as trinomial

6 A polynomial of one degree is called linear polynomial two degree as

quadratic polynomial and degree three as cubic polynomial

Important concepts on Polynomial

Concept Description

Zerorsquos or roots

of the

polynomial

It is a solution to the polynomial equation S(x)=0 ie a number

a is said to be a zero of a polynomial if S(a) = 0

If we draw the graph of S(x) =0 the values where the curve

cuts the X-axis are called Zeroes of the polynomial

Remainder

Theoremrsquos

If p(x) is an polynomial of degree greater than or equal to 1

and p(x) is divided by the expression (x-a)then the remainder

will be p(a)

FactorrsquosTheoremrsquos

If x-a is a factor of polynomial p(x) then p(a)=0 or if p(a)=0x-a is the factor the polynomial p(x)

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983093

Geometric Meaning of the Zeroes of the polynomial

Letrsquos us assume

y= p(x) where p(x) is the polynomial of any form

Now we can plot the equation y=p(x) on the Cartesian plane by taking various values of x and y obtained

by putting the values The plot or graph obtained can be of any shapes

The zeroes of the polynomial are the points where the graph meet x axis in the Cartesian plane If the

graph does not meet x axis then the polynomial does not have any zerorsquos

Let us take some useful polynomial and shapes obtained on the Cartesian plane

Sno y=p(x) Graph obtained Name of the

graph

Name of the

equation

1 y=ax+b where a and b can be

any values (ane0)

Example y=2x+3

Straight line

It intersect the x-

axis at ( -ba 0)

Example ( -320)

Linear polynomial

2 y=ax2+bx+c

where

b2-4ac gt 0 and ane0 and agt 0

Example

y=x2-7x+12

Parabola

It intersect the x-

axis at two points

Example

(30) and (40)

Quadratic

polynomial

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3 y=ax2+bx+c

where

b2-4ac gt 0 and ane0 and a lt 0

Example

y=-x2+2x+8

Parabola

It intersect the x-

axis at two points

Example (-20)

and (40)

Quadratic

polynomial

4 y=ax +bx+c

where

b2-4ac = 0 and ane0 a gt 0

Example

y=(x-2)2

Parabola

It intersect the x-

axis at one points

Quadratic

polynomial

5 y=ax2+bx+c

where

b

2

-4ac lt 0 and ane0 a gt 0

Example

y=x2-2x+6

Parabola

It does not

intersect the x-axis

It has no zerorsquos

Quadratic

polynomial

6 y=ax +bx+c

where

b2-4ac lt 0 and ane0 a lt 0

Example

y=-x2-2x-6

Parabola

It does not

intersect the x-axis

It has no zerorsquos

Quadratic

polynomial

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7 y=ax3 +bx

2+cx+d

where ane0

It can be of any

shape

It will cut the x-axis

at the most 3 times

Cubic Polynomial

8 ⋯ 991270 991270 991270

Where an ne0

It can be of any

shape

It will cut the x-axis

at the most n times

Polynomial of n

degree

Relation between coefficient and zeroes of the Polynomial

Sno Type of Polynomial General form Zerorsquos Relationship between Zerorsquos

and coefficients

1 Linear polynomial ax+b ane0 1

2 Quadratic ax +bx+c ane0 2

3 Cubic ax +bx +cx+d ane0 3

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Formation of polynomial when the zeroes are given

Type ofpolynomial

Zerorsquos Polynomial Formed

Linear k=a (x-a)

Quadratic k1=a and

k2=b

(x-a)(x-b)

Or

x2-( a+b)x +ab

Or

x2

-( Sum of the zerorsquos)x +product of the zerorsquos

Cubic k1=a k2=b

and k3=c

(x-a)(x-b)(x-c)

Division algorithm for Polynomial

Letlsquos p(x) and q(x) are any two polynomial with q(x) ne0 then we can find polynomial s(x) and r(x) suchthat

P(x)=s(x) q(x) + r(x)

Where r(x) can be zero or degree of r(x) lt degree of g(x)

983108983145983158983145983140983141983150983140 983101983121983157983151983156983145983141983150983156 983128 983108983145983158983145983155983151983154 + 983122983141983149983137983145983150983140983141983154

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COORDINATE GEOMETRYSno Points

1 We require two perpendicular axes to locate a point in the plane One of

them is horizontal and other is Vertical

2 The plane is called Cartesian plane and axis are called the coordinates axis

3 The horizontal axis is called x-axis and Vertical axis is called Y-axis

4 The point of intersection of axis is called origin

5 The distance of a point from y axis is called x ndashcoordinate or abscissa and

the distance of the point from x ndashaxis is called y ndash coordinate or Ordinate

6 The distance of a point from y axis is called x ndashcoordinate or abscissa and

the distance of the point from x ndashaxis is called y ndash coordinate or Ordinate

7 The Origin has zero distance from both x-axis and y-axis so that its

abscissa and ordinate both are zero So the coordinate of the origin is (0

0)

8 A point on the x ndashaxis has zero distance from x-axis so coordinate of any

point on the x-axis will be (x 0)

9 A point on the y ndashaxis has zero distance from y-axis so coordinate of any

point on the y-axis will be (0 y)

10 The axes divide the Cartesian plane in to four parts These Four parts arecalled the quadrants

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10

The coordinates of the points in the four quadrants will have sign according to thebelow table

Quadrant x-coordinate y-coordinate

Ist Quadrant + +

IInd quadrant - +

IIIrd quadrant - -

IVth quadrant + -

Sno Terms Descriptions

1 Distance formula Distance between the points AB is given by

Distance of Point A from Origin

2 Section Formula A point P(xy) which divide the line segment AB in

the ratio m1 and m2 is given by

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11

The midpoint P is given by

3 Area of Triangle Area of triangle ABC of coordinates A(x 1y 1 )

B(x 2 y 2 ) and C(x 3y 3 )

12 For point AB and C to be collinear The value of A

should be zero

LINEAR EQUATIONS IN TWO

VARIABLESAn equation of the form ax + by + c = 0 where a b and c are real numbers such

that a andb are not both zero is called a linear equation in two variablesImportant points to Note

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12

Sno Points

1 A linear equation in two variable has infinite solutions

2 The graph of every linear equation in two variable is a straight line

3 x = 0 is the equation of the y-axis and y = 0 is the equation of the x-axis

4 The graph x=a is a line parallel to y -axis

5 The graph y=b is a line parallel to x -axis

6 An equation of the type y = mx represents a line passing through theorigin

7 Every point on the graph of a linear equation in two variables is a solution

of the linear

equation Moreover every solution of the linear equation is a point on the

graph

Sno Type of equation Mathematical

representation

Solutions

1 Linear equation in one Variable ax+b=0 ane0

a and b are real

number

One solution

2 Linear equation in two Variable ax+by+c=0 ane0 and

bne0

a b and c are real

number

Infinite solution

possible

3 Linear equation in three Variable ax+by+cz+d=0 ane0

bne0 and cne0

Infinite solution

possible

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a b c d are realnumber

Simultaneous pair of Linear equation

A pair of Linear equation in two variables

a1x+b1y+c1=0

a2x +b2y+c2=0

Graphically it is represented by two straight lines on Cartesian plane

Simultaneous pair of

Linear equation

Condition Graphical

representation

Algebraic

interpretation

a1x+b1y+c1=0

a2x +b2y+c2=0

Example

x-4y+14=0

3x+2y-14=0

Intersecting lines The

intersecting point

coordinate is the only

solution

One unique solution

only

a1x+b1y+c1=0

a2x +b2y+c2=0

Coincident lines Theany coordinate on the

line is the solution

Infinite solution

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Example

2x+4y=16

3x+6y=24

a1x+b1y+c1=0

a2x +b2y+c2=0

Example

2x+4y=6

4x+8y=18

Parallel Lines No solution

The graphical solution can be obtained by drawing the lines on the Cartesian plane

Algebraic Solution of system of Linear equation

Sno Type of method Working of method

1 Method of elimination by

substitution

1) Suppose the equation are

a1x+b1y+c1=0

a2x +b2y+c2=0

2) Find the value of variable of either x or y in other

variable term in first equation

3) Substitute the value of that variable in second

equation

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4) Now this is a linear equation in one variable Findthe value of the variable

5) Substitute this value in first equation and get the

second variable

2 Method of elimination by

equating the coefficients

1) Suppose the equation are

a1x+b1y+c1=0

a2x +b2y+c2=0

2) Find the LCM of a1 and a2 Let it k

3) Multiple the first equation by the value ka1

4) Multiple the first equation by the value ka2

4) Subtract the equation obtained This way one

variable will be eliminated and we can solve to get the

value of variable y

5) Substitute this value in first equation and get the

second variable

3 Cross Multiplication method 1) Suppose the equation are

a1x+b1y+c1=0

a2x +b2y+c2=0

2) This can be written as

3) This can be written as

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4) Value of x and y can be find using the

x =gt first and last expression

y=gt second and last expression

Quadratic EquationsSno Terms Descriptions

1 Quadratic Polynomial P(x) = ax +bx+c where ane0

2 Quadratic equation ax +bx+c =0 where ane0

3 Solution or root of the

Quadratic equation

A real number α is called the root or solution of the

quadratic equation if

aα2 +bα+c=0

4 zeroes of the

polynomial p(x)

The root of the quadratic equation are called zeroes

5 Maximum roots of

quadratic equations

We know from chapter two that a polynomial of

degree can have max two zeroes So a quadratic

equation can have maximum two roots

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6 Condition for realroots A quadratic equation has real roots if b

2

- 4ac gt 0

How to Solve Quadratic equation

Sno Method Working

1 factorization This method we factorize the equation by splitting themiddle term b

In ax2+bx+c=0

Example

6x2-x-2=0

1) First we need to multiple the coefficient a and cIn this

case =6X-2=-12

2) Splitting the middle term so that multiplication is 12

and difference is the coefficient b

6x2 +3x-4x-2=0

3x( 2x+1) -2(2x+1)=0

(3x-2) (2x+1)=0

3) Roots of the equation can be find equating the factorsto zero

3x-2=0 =gt x=32

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2x+1=0 =gt x=-12

2 Square method In this method we create square on LHS and RHS and

then find the value

ax2 +bx+c=0

1) x2 +(ba) x+(ca)=0

2) ( x+b2a)2 ndash(b2a)2 +(ca)=0

3) ( x+b2a)2=(b2-4ac)4a2

4) radic

Example

x2 +4x-5=0

1) (x+2)2 -4-5=0

2) (x+2)2=9

3) Roots of the equation can be find using square root on

both the sides

x+2 =-3 =gt x=-5

x+2=3=gt x=1

3 Quadratic

method

For quadratic equation

ax2 +bx+c=0

roots are given by radic radic

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For b2 -4ac gt 0 Quadratic equation has two real roots of

different value

For b2-4ac =0 quadratic equation has one real root

For b2-4ac lt 0 no real roots for quadratic equation

Nature of roots of Quadratic equation

Sno Condition Nature of roots

1 b -4ac gt 0 Two distinct real roots

2 b -4ac =0 One real root

3 b

2

-4ac lt 0 No real roots

Triangles

Sno Terms Descriptions

1 Congruence Two Geometric figure are said to be congruence ifthey are exactly same size and shapeSymbol used is cong Two angles are congruent if they are equalTwo circle are congruent if they have equal radiiTwo squares are congruent if the sides are equal

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20

2 Triangle Congruencebull

Two triangles are congruent if three sidesand three angles of one triangle is

congruent to the corresponding sides and

angles of the other

bull Corresponding sides are equal

AB=DE BC=EF AC=DF

bull Corresponding angles are equalang angang angang ang

bull We write this as

cong

bull The above six equalities are between the

corresponding parts of the two congruent

triangles In short form this is called

CPCT

bull We should keep the letters in correct order

on both sides

3 Inequalities in

Triangles

1) In a triangle angle opposite to longer side is

larger

2) In a triangle side opposite to larger angle is

larger

3) The sum of any two sides of the triangle is

greater than the third side

983105

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983108

983109 983110

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983151983150983148983161

21

In triangle ABC

AB +BC gt AC

Different Criterion for Congruence of the triangles

N Criterion Description Figures and

expression

1 Side angle

Side (SAS)

congruence

bull Two triangles are congruent if the

two sides and included angles of

one triangle is equal to the two

sides and included angle

bull It is an axiom as it cannot be

proved so it is an accepted truth

bull ASS and SSA type two triangles

may not be congruent alwaysIf following

condition

983105

983110983109

983108

983107983106

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983151983150983148983161

22

AB=DE BC=EFang ang

Then

cong

2 Angle side

angle (ASA)

congruence

bull Two triangles are congruent if the

two angles and included side of

one triangle is equal to the

corresponding angles and side

bull It is a theorem and can be proved

If following

condition

BC=EF

ang angang ang

Then

cong

3 Angle angle

side( AAS)

congruence

bull Two triangles are congruent if the

any two pair of angles and any

side of one triangle is equal to the

corresponding angles and side

bull It is a theorem and can be proved

983110983109

983108

983107983106

983105

983110983109

983108

983107983106

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2983091

If followingcondition

BC=EF

ang angang ang

Then

cong

4 Side-Side-Side

(SSS)congruence

bull Two triangles are congruent if the

three sides of one triangle is equalto the three sides of the another

If following

condition

BC=EFAB=DEDF

=AC

Then

cong

5 Right angle ndash

hypotenuse-

side(RHS)

bull Two right triangles are congruent if

the hypotenuse and a side of the

one triangle are equal to

983110983109

983108

983107983106

983105

983105 983108

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2983092

congruence corresponding hypotenuse and sideof the another

If following

condition

AC=DFBC=EF

Then

cong

Some Important points on Triangles

Terms Description

Orthocenter Point of intersection of the three altitudeof the triangle

Equilateral triangle whose all sides are equal and allangles are equal to 600

Median A line Segment joining the corner of thetriangle to the midpoint of the oppositeside of the triangle

Altitude A line Segment from the corner of thetriangle and perpendicular to theopposite side of the triangle

Isosceles A triangle whose two sides are equalCentroid Point of intersection of the three median

of the triangle is called the centroid of

the triangleIn center All the angle bisector of the trianglepasses through same point

Circumcenter The perpendicular bisector of the sidesof the triangles passes through samepoint

983106 983107 983109 983110

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2983093

Scalene triangle Triangle having no equal angles and noequal sidesRight Triangle Right triangle has one angle equal to 900 Obtuse Triangle One angle is obtuse angle while other

two are acute anglesAcute Triangle All the angles are acute

Similarity of Triangles

Sno Points

1 Two figures having the same shape but not necessarily the same size are called similarfigures

2 All the congruent figures are similar but the converse is not true

3 If a line is drawn parallel to one side of a triangle to intersect the other two sides in

distinct points then the other two sides are divided in the same ratio

4 If a line divides any two sides of a triangle in the same ratio then the line is parallel tothe third side

Different Criterion for Similarity of the triangles

N Criterion Description Expression

1 Angle Angle

angle(AAA)

similarity

bull Two triangles are similar if

corresponding angle are equal

If following

condition

ang ang

ang ang

ang ang

Then

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2983094

Then 983166

2 Angle angle

(AA) similarity

bull Two triangles are similar if the two

corresponding angles are equal as

by angle property third angle will

be also equal

If following

condition

ang ang

ang ang

Then

ang ang

Then

Then

983166

3 Side side

side(SSS)

Similarity

Two triangles are similar if the

sides of one triangle is

proportional to the sides of other

triangle

If following

condition

Then

ang ang

ang ang

ang ang

Then

cong

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2983095

4 Side-Angle-Side (SAS)

similarity

bull

Two triangles are similar if the oneangle of a triangle is equal to one

angle of other triangles and sides

including that angle is proportional

If followingcondition

And ang ang

Then

cong

Area of Similar triangles

If the two triangle ABC and DEF are similar

cong

Then

Pythagoras Theorem

Sno Points

1 If a perpendicular is drawn from the vertex of the right angle of a right triangle to thehypotenuse then the triangles on both sides of the perpendicular are similar to the

whole triangle and also to each other

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2 In a right triangle the square of the hypotenuse is equal to the sum of the squares of the

other two sides (Pythagoras Theorem)

(hyp)2 =(base)

2 + (perp)

2

3 If in a triangle square of one side is equal to the sum of the squares of the other twosides then the angle opposite the first side is a right angle

Arithmetic Progression

Sno Terms Descriptions

1 ArithmeticProgression

An arithmetic progression is a sequence ofnumbers such that the difference of any two

successive members is a constant

Examples

1) 1591317hellip

2) 12345hellip

2 common difference of

the AP

the difference between any successive members is a

constant and it is called the common difference of AP

1) If a1 a2a3a4a5 are the terms in AP then

D=a2 -a1 =a3 - a2 =a4 ndash a3=a5 ndasha4

2) We can represent the general form of AP inthe form

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2983097

aa+da+2da+3da+4d

Where a is first term and d is the commondifference

3 nth term of Arithmetic Progression

n term = a + (n - 1)d

4 Sum of nth item in ArithmeticProgression

Sn =(n2)[a + (n-1)d]

Or

Sn =(n2)[t1+ tn]

Trigonometry

Sno Terms Descriptions

1 What is

Trigonometry

Trigonometry from Greek trigotildenon triangle and

metron measure) is a branch of mathematics

that studies relationships involving lengths and

angles of triangles The field emerged during the

3rd century BC from applications of geometry to

astronomical studies

Trigonometry is most simply associated with

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9830910

planar right angle triangles (each of which is atwo-dimensional triangle with one angle equal to

90 degrees) The applicability to non-right-angle

triangles exists but since any non-right-angle

triangle (on a flat plane) can be bisected to

create two right-angle triangles most problems

can be reduced to calculations on right-angle

triangles Thus the majority of applications relate

to right-angle triangles

2 Trigonometric

Ratiorsquos

In a right angle triangle ABC where B=90deg

We can define following term for angle A

Base Side adjacent to angle

Perpendicular Side Opposite of angleHypotenuse Side opposite to right angle

We can define the trigonometric ratios for

angle A as

sin A= PerpendicularHypotenuse =BCAC

cosec A= HypotenusePerpendicular

=ACBC

cos A= BaseHypotenuse =ABAC

sec A= HypotenuseBase=ACAB

tan A= PerpendicularBase =BCAB

cot A= BasePerpendicular=ABBC

Notice that each ratio in the right-hand column is

the inverse or the reciprocal of the ratio in the

left-hand column

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9830911

3 Reciprocal offunctions The reciprocal of sin A is cosec A and vice-versa

The reciprocal of cos A is sec A

And the reciprocal of tan A is cot A

These are valid for acute angles

We can define tan A = sin Acos A

And Cot A =cos A Sin A

4 Value of of sin and

cos

Is always less 1

5 Trigonometric ration

from another angle

We can define the trigonometric ratios for angle Cas

sin C= PerpendicularHypotenuse =ABACcosec C= HypotenusePerpendicular =ACABBcos C= BaseHypotenuse =BCACsec C= HypotenuseBase=ACBCtan A= PerpendicularBase =ABBCcot A= BasePerpendicular=BCAB

6 Trigonometric ratios

of complimentary

Sin (90-A) =cos(A)

Cos(90-A) = sin A

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9830912

angles Tan(90-A) =cot A

Sec(90-A)= cosec A

Cosec (90-A) =sec A

Cot(90- A) =tan A7 Trigonometric

identities

Sin2 A + cos2 A =0

1 + tan2 A =sec2 A

1 + cot2 A =cosec2 A

Trigonometric Ratios of Common angles

We can find the values of trigonometric ratiorsquos various angle

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983091983091

Area of Circles

Sno Points

1 A circle is a collection of all the points in a plane which are equidistant

from a fixed point in the plane

2 Equal chords of a circle (or of congruent circles) subtend equal angles at

the center

3 If the angles subtended by two chords of a circle (or of congruent circles)

at the center (corresponding center) are equal the chords are equal

4 The perpendicular from the center of a circle to a chord bisects the chord

5 The line drawn through the center of a circle to bisect a chord is

perpendicular to the chord

6 There is one and only one circle passing through three non-collinear points

7 Equal chords of a circle (or of congruent circles) are equidistant from the

center (or corresponding centers)

8 Chords equidistant from the center (or corresponding centers) of a circle

(or of congruent circles) are equal

9 If two arcs of a circle are congruent then their corresponding chords are

equal and conversely if two chords of a circle are equal then their

corresponding arcs (minor major) are congruent

10 Congruent arcs of a circle subtend equal angles at the center

11 The angle subtended by an arc at the center is double the angle subtended

by it at any point on the remaining part of the circle

12 Angles in the same segment of a circle are equal

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983091983092

13 Angle in a semicircle is a right angle

14 If a line segment joining two points subtends equal angles at two other

points lying on the same side of the line containing the line segment the

four points lie on a circle

15 The sum of either pair of opposite angles of a cyclic quadrilateral is 180deg

16 If the sum of a pair of opposite angles of a quadrilateral is 180deg then the

quadrilateral is cyclic

Sno Terms Descriptions

1 Circumference of a circle 2 π r

2 Area of circle π r 2

3 Length of the arc of

the sector of angle

Length of the arc of the sector of angle θ 2

4 Area of the sector of

angle

Area of the sector of angle θ

5 Area of segment of a

circle

Area of the corresponding sector ndash Area of the

corresponding triangle

Surface Area and Volume

Sno Term Description

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983091983093

1 Mensuration It is branch of mathematics which is concernedabout the measurement of length area and

Volume of plane and Solid figure

2 Perimeter a)The perimeter of plane figure is defined as thelength of the boundaryb)It units is same as that of length ie m cmkm

3 Area a)The area of the plane figure is the surfaceenclosed by its boundaryb) It unit is square of length unit ie m2 km2

4 Volume Volume is the measure of the amount of spaceinside of a solid figure like a cube ball cylinderor pyramid Its units are always cubic that isthe number of little element cubes that fit insidethe figure

Volume Unit conversion

1 cm3 1mL 1000 mm3

1 Litre 1000ml 1000 cm3

1 m3 106 cm3 1000 L

1 dm3 1000 cm3 1 L

Surface Area and Volume of Cube and Cuboid

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Type Measurement

Surface Area of Cuboid of Length L

Breadth B and Height H

2(LB + BH + LH)

Lateral surface area of the cuboids 2( L + B ) H

Diagonal of the cuboids Volume of a cuboids LBH

Length of all 12 edges of the cuboids 4 (L+B+H)

Surface Area of Cube of side L 6L2

Lateral surface area of the cube 4L2

Diagonal of the cube radic3 Volume of a cube L3

Surface Area and Volume of Right circular cylinder

Radius The radius (r) of the circular base is called the radius of the

cylinder

Height The length of the axis of the cylinder is called the height (h) of the

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983091983095

cylinder

Lateral

Surface

The curved surface joining the two base of a right circular cylinder is

called Lateral Surface

Type Measurement

Curved or lateral Surface Area of

cylinder

2πrh

Total surface area of cylinder 2πr (h+r)

Volume of Cylinder π r2h

Surface Area and Volume of Right circular cone

Radius The radius (r) of the circular base is called the radius of the

cone

Height The length of the line segment joining the vertex to the center of

base is called the height (h) of the cone

Slant

Height

The length of the segment joining the vertex to any point on the

circular edge of the base is called the slant height (L) of the cone

Lateral

surface

Area

The curved surface joining the base and uppermost point of a right

circular cone is called Lateral Surface

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Type Measurement

Curved or lateral Surface Area of

cone

πrL

Total surface area of cone πr (L+r)

Volume of Cone

Surface Area and Volume of sphere and hemisphere

Sphere A sphere can also be considered as a solid obtained on

rotating a circle About its diameter

Hemisphere A plane through the centre of the sphere divides the sphere into two

equal parts each of which is called a hemisphere

radius The radius of the circle by which it is formed

SphericalShell

The difference of two solid concentric spheres is called a sphericalshell

Lateral

Surface

Area for

Total surface area of the sphere

983123983152983144983141983154983141

983112983141983149983145983155983152983144983141983154983141

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Sphere

Lateral

Surface

area of

Hemisphere

It is the curved surface area leaving the circular base

Type Measurement

Surface area of Sphere 4πr2

Volume of Sphere 43 Curved Surface area of hemisphere 2πr2

Total Surface area of hemisphere 3πr2

Volume of hemisphere 23 Volume of the spherical shell whose

outer and inner radii and lsquoRrsquo and lsquorrsquorespectively

43

How the Surface area and Volume are determined

Area of Circle The circumference of a circle is 2πr

This is the definition of π (pi) Divide

the circle into many triangular

segments The area of the triangles

is 12 times the sum of their bases

2πr (the circumference) times their

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9830920

height r

12 2 Surface Area of cylinder This can be imagined as unwrapping the

surface into a rectangle

Surface area of cone This can be achieved by divide the

surface of the cone into its triangles or

the surface of the cone into many thin

triangles The area of the triangles is 12

times the sum of their bases p times

their height

12 2

Surface Area and Volume of frustum of cone

h = vertical height of the frustum

l = slant height of the frustum

r 1 and r 2 are radii of the two bases (ends) of the frustum

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9830921

Type Measurement

983126983151983148983157983149983141 983151983142 983137 983142983154983157983155983156983157983149 983151983142 983137 983139983151983150983141 983123983148983137983150983156 983144983141983145983143983144983156 983151983142 983142983154983157983155983156983157983149 983151983142 983137 983139983151983150983141 983107983157983154983158983141983140 983155983157983154983142983137983139983141 983137983154983141983137 983151983142 983137 983142983154983157983155983156983157983149 983151983142 983137 983139983151983150983141 983124983151983156983137983148 983155983157983154983142983137983139983141 983137983154983141983137 983151983142 983142983154983157983155983156983157983149 983151983142 983137 983139983151983150983141

Statistics

Sno Term Description

1 Statistics Statistics is a broad mathematical disciplinewhich studies ways to collect summarize and

draw conclusions from data

2 Data A systematic record of facts or different values of

a quantity is called data

Data is of two types - Primary data and Secondary

data

Primary Data The data collected by a researcher

with a specific purpose in mind is called primarydata

Secondary Data The data gathered from a

source where it already exists is called secondary

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9830922

data

3 Features of data bull Statistics deals with collectionpresentation analysis and interpretation ofnumerical data

bull Arranging data in an order to study theirsalient features is called presentation ofdata

bull Data arranged in ascending or descendingorder is called arrayed data or an array

bull Range of the data is the differencebetween the maximum and the minimum

values of the observationsbull Table that shows the frequency of different

values in the given data is called afrequency distribution table

bull A frequency distribution table that showsthe frequency of each individual value in thegiven data is called an ungrouped frequencydistribution table

bull A table that shows the frequency of groupsof values in the given data is called agrouped frequency distribution table

bull

The groupings used to group the values ingiven data are called classes or class-intervals The number of values that eachclass contains is called the class size orclass width The lower value in a class iscalled the lower class limit The higher valuein a class is called the upper class limit

bull Class mark of a class is the mid value ofthe two limits of that class

bull A frequency distribution in which the upperlimit of one class differs from the lower limitof the succeeding class is called an

Inclusive or discontinuous FrequencyDistribution

bull A frequency distribution in which the upperlimit of one class coincides from the lowerlimit of the succeeding class is called anexclusive or continuous Frequency

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983092983091

Distribution bull

4 Bar graph A bar graph is a pictorial representation of data inwhich rectangular bars of uniform width are drawnwith equal spacing between them on one axisusually the x axis The value of the variable isshown on the other axis that is the y axis

5 Histogram A histogram is a set of adjacent rectangles whoseareas are proportional to the frequencies of agiven continuous frequency distribution

6 Mean The mean value of a variable is defined as thesum of all the values of the variable divided by thenumber of values

4 sum

7 Median The median of a set of data values is the middle

value of the data set when it has been arranged inascending order That is from the smallest valueto the highest value

Median is calculated as

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983151983150983148983161

983092983092

12 1 Where n is the number of values in the data

If the number of values in the data set is eventhen the median is the average of the two middlevalues

8 Mode Mode of a statistical data is the value of thatvariable which has the maximum frequency

Sno Term Description

1983117983141983137983150 983142983151983154

983125983150983143983154983151983157983152

983110983154983141983153983157983141983150983139983161 983156983137983138983148983141

Mean is given by

sum sum

2983117983141983137983150 983142983151983154 983143983154983151983157983152

983110983154983141983153983157983141983150983139983161 983156983137983138983148983141

In these distribution it is assumed that frequency of each classinterval is centered around its mid-point ie class marks

2

983117983141983137983150 983139983137983150 983138983141 983139983137983148983139983157983148983137983156983141983140 983157983155983145983150983143 983156983144983154983141983141 983149983141983156983144983151983140

983137983081 983108983145983154983141983139983156 983149983141983156983144983151983140

sum

sum

983138983081 983105983155983155983157983149983141983140 983149983141983137983150 983149983141983156983144983151983140

sum sum

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983151983150983148983161

983092983093

Where

a=gt Assumed mean

di =gt xi ndasha

983139983081 983123983156983141983152 983140983141983158983145983137983156983145983151983150 983117983141983156983144983151983140

sum sum

Where

a=gt Assumed mean

ui =gt (xi ndasha)h

3983117983151983140983141 983142983151983154

983143983154983151983157983152983141983140

983142983154983141983153983157983141983150983139983161 983156983137983138983148983141

Modal class The class interval having highest frequency is

called the modal class and Mode is obtained using the modal

class

2

983127983144983141983154983141

l = lower limit of the modal classh = size of the class interval (assuming all class sizes tobe equal)f 1 = frequency of the modal classf 0 = frequency of the class preceding the modal classf 2 = frequency of the class succeeding the modal class

4983117983141983140983145983137983150 983151983142 983137

983143983154983151983157983152983141983140 983140983137983156983137

983142983154983141983153983157983141983150983139983161 983156983137983138983148983141

For the given data we need to have class interval frequency

distribution and cumulative frequency distribution

Median is calculated as

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983151983150983148983161

983092983094

2

983127983144983141983154983141

983148 983101 983148983151983159983141983154 983148983145983149983145983156 983151983142 983149983141983140983145983137983150 983139983148983137983155983155

983150 983101 983150983157983149983138983141983154 983151983142 983151983138983155983141983154983158983137983156983145983151983150983155

983139983142 983101 983139983157983149983157983148983137983156983145983158983141 983142983154983141983153983157983141983150983139983161 983151983142 983139983148983137983155983155 983152983154983141983139983141983140983145983150983143 983156983144983141 983149983141983140983145983137983150 983139983148983137983155983155

983142 983101 983142983154983141983153983157983141983150983139983161 983151983142 983149983141983140983145983137983150 983139983148983137983155983155

983144 983101 983139983148983137983155983155 983155983145983162983141 (983137983155983155983157983149983145983150983143 983139983148983137983155983155 983155983145983162983141 983156983151 983138983141 983141983153983157983137983148)

5983109983149983152983145983154983145983139983137983148

983110983151983154983149983157983148983137 983138983141983156983159983141983141983150

983117983151983140983141983084 983117983141983137983150 983137983150983140

983117983141983140983145983137983150

3 Median=Mode +2 Mean

ProbabilitySn

o

Term Description

1 Empirical

probability

It is a probability of event which is calculated based on

experiments

Example

A coin is tossed 1000 times we get 499 times head and

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983151983150983148983161

983092983095

501 times tail

So empirical or experimental probability of getting head is

calculated as

4991000 499

Empirical probability depends on experiment and

different will get different values based on the

experiment

2 Important point

about events

If the event A B C covers the entire possible outcome inthe experiment Then

P (A) +P (B) +P(C)=1

3 impossible

event

The probability of an event (U) which is impossible tooccur is 0 Such an event is called an impossible event

P (U)=0

4 Sure or certain

event

The probability of an event (X) which is sure (or certain)to occur is 1 Such an event is called a sure event or acertain event

P(X) =1

5 Probability of

any event

Probability of any event can be as

0 1

Sn

o

Term Description

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983092983096

1

983124983144983141983151983154983141983156983145983139983137983148

983120983154983151983138983137983138983145983148983145983156983161

The theoretical probability or the classical probability of the event isdefined as

2 Elementary

events

983105983150 983141983158983141983150983156 983144983137983158983145983150983143 983151983150983148983161 983151983150983141 983151983157983156983139983151983149983141 983151983142 983156983144983141 983141983160983152983141983154983145983149983141983150983156 983145983155 983139983137983148983148983141983140 983137983150

983141983148983141983149983141983150983156983137983154983161 983141983158983141983150983156

991260983124983144983141 983155983157983149 983151983142 983156983144983141 983152983154983151983138983137983138983145983148983145983156983145983141983155 983151983142 983137983148983148 983156983144983141 983141983148983141983149983141983150983156983137983154983161 983141983158983141983150983156983155 983151983142983137983150 983141983160983152983141983154983145983149983141983150983156 983145983155 1991261

983113983141 983113983142 983159983141 983156983144983154983141983141 983141983148983141983149983141983150983156983137983154983161 983141983158983141983150983156 983105983106983107 983145983150 983156983144983141 983141983160983152983141983154983145983149983141983150983156 983156983144983141983150

983120(983105)+983120(983106) +983120(983107)9831011

3 Complementar

y events

983124983144983141 983141983158983141983150983156 Ᾱ 983154983141983152983154983141983155983141983150983156983145983150983143 991256983150983151983156 983105991257 983145983155 983139983137983148983148983141983140 983156983144983141 983139983151983149983152983148983141983149983141983150983156 983151983142 983156983144983141 983141983158983141983150983156

983105 983127983141 983137983148983155983151 983155983137983161 983156983144983137983156 Ᾱ 983137983150983140 983105 983137983154983141 983139983151983149983152983148983141983149983141983150983156983137983154983161 983141983158983141983150983156983155 983105983148983155983151

983120(983105) +983120(Ᾱ)9831011

4 Sure or certain

event

The probability of an event (X) which is sure (or certain)

to occur is 1 Such an event is called a sure event or acertain event

P(X) =1

5 Probability of

any event

Probability of any event can be as

0 1

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983151983150983148983161

983093

Geometric Meaning of the Zeroes of the polynomial

Letrsquos us assume

y= p(x) where p(x) is the polynomial of any form

Now we can plot the equation y=p(x) on the Cartesian plane by taking various values of x and y obtained

by putting the values The plot or graph obtained can be of any shapes

The zeroes of the polynomial are the points where the graph meet x axis in the Cartesian plane If the

graph does not meet x axis then the polynomial does not have any zerorsquos

Let us take some useful polynomial and shapes obtained on the Cartesian plane

Sno y=p(x) Graph obtained Name of the

graph

Name of the

equation

1 y=ax+b where a and b can be

any values (ane0)

Example y=2x+3

Straight line

It intersect the x-

axis at ( -ba 0)

Example ( -320)

Linear polynomial

2 y=ax2+bx+c

where

b2-4ac gt 0 and ane0 and agt 0

Example

y=x2-7x+12

Parabola

It intersect the x-

axis at two points

Example

(30) and (40)

Quadratic

polynomial

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3 y=ax2+bx+c

where

b2-4ac gt 0 and ane0 and a lt 0

Example

y=-x2+2x+8

Parabola

It intersect the x-

axis at two points

Example (-20)

and (40)

Quadratic

polynomial

4 y=ax +bx+c

where

b2-4ac = 0 and ane0 a gt 0

Example

y=(x-2)2

Parabola

It intersect the x-

axis at one points

Quadratic

polynomial

5 y=ax2+bx+c

where

b

2

-4ac lt 0 and ane0 a gt 0

Example

y=x2-2x+6

Parabola

It does not

intersect the x-axis

It has no zerorsquos

Quadratic

polynomial

6 y=ax +bx+c

where

b2-4ac lt 0 and ane0 a lt 0

Example

y=-x2-2x-6

Parabola

It does not

intersect the x-axis

It has no zerorsquos

Quadratic

polynomial

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7 y=ax3 +bx

2+cx+d

where ane0

It can be of any

shape

It will cut the x-axis

at the most 3 times

Cubic Polynomial

8 ⋯ 991270 991270 991270

Where an ne0

It can be of any

shape

It will cut the x-axis

at the most n times

Polynomial of n

degree

Relation between coefficient and zeroes of the Polynomial

Sno Type of Polynomial General form Zerorsquos Relationship between Zerorsquos

and coefficients

1 Linear polynomial ax+b ane0 1

2 Quadratic ax +bx+c ane0 2

3 Cubic ax +bx +cx+d ane0 3

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Formation of polynomial when the zeroes are given

Type ofpolynomial

Zerorsquos Polynomial Formed

Linear k=a (x-a)

Quadratic k1=a and

k2=b

(x-a)(x-b)

Or

x2-( a+b)x +ab

Or

x2

-( Sum of the zerorsquos)x +product of the zerorsquos

Cubic k1=a k2=b

and k3=c

(x-a)(x-b)(x-c)

Division algorithm for Polynomial

Letlsquos p(x) and q(x) are any two polynomial with q(x) ne0 then we can find polynomial s(x) and r(x) suchthat

P(x)=s(x) q(x) + r(x)

Where r(x) can be zero or degree of r(x) lt degree of g(x)

983108983145983158983145983140983141983150983140 983101983121983157983151983156983145983141983150983156 983128 983108983145983158983145983155983151983154 + 983122983141983149983137983145983150983140983141983154

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COORDINATE GEOMETRYSno Points

1 We require two perpendicular axes to locate a point in the plane One of

them is horizontal and other is Vertical

2 The plane is called Cartesian plane and axis are called the coordinates axis

3 The horizontal axis is called x-axis and Vertical axis is called Y-axis

4 The point of intersection of axis is called origin

5 The distance of a point from y axis is called x ndashcoordinate or abscissa and

the distance of the point from x ndashaxis is called y ndash coordinate or Ordinate

6 The distance of a point from y axis is called x ndashcoordinate or abscissa and

the distance of the point from x ndashaxis is called y ndash coordinate or Ordinate

7 The Origin has zero distance from both x-axis and y-axis so that its

abscissa and ordinate both are zero So the coordinate of the origin is (0

0)

8 A point on the x ndashaxis has zero distance from x-axis so coordinate of any

point on the x-axis will be (x 0)

9 A point on the y ndashaxis has zero distance from y-axis so coordinate of any

point on the y-axis will be (0 y)

10 The axes divide the Cartesian plane in to four parts These Four parts arecalled the quadrants

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10

The coordinates of the points in the four quadrants will have sign according to thebelow table

Quadrant x-coordinate y-coordinate

Ist Quadrant + +

IInd quadrant - +

IIIrd quadrant - -

IVth quadrant + -

Sno Terms Descriptions

1 Distance formula Distance between the points AB is given by

Distance of Point A from Origin

2 Section Formula A point P(xy) which divide the line segment AB in

the ratio m1 and m2 is given by

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11

The midpoint P is given by

3 Area of Triangle Area of triangle ABC of coordinates A(x 1y 1 )

B(x 2 y 2 ) and C(x 3y 3 )

12 For point AB and C to be collinear The value of A

should be zero

LINEAR EQUATIONS IN TWO

VARIABLESAn equation of the form ax + by + c = 0 where a b and c are real numbers such

that a andb are not both zero is called a linear equation in two variablesImportant points to Note

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12

Sno Points

1 A linear equation in two variable has infinite solutions

2 The graph of every linear equation in two variable is a straight line

3 x = 0 is the equation of the y-axis and y = 0 is the equation of the x-axis

4 The graph x=a is a line parallel to y -axis

5 The graph y=b is a line parallel to x -axis

6 An equation of the type y = mx represents a line passing through theorigin

7 Every point on the graph of a linear equation in two variables is a solution

of the linear

equation Moreover every solution of the linear equation is a point on the

graph

Sno Type of equation Mathematical

representation

Solutions

1 Linear equation in one Variable ax+b=0 ane0

a and b are real

number

One solution

2 Linear equation in two Variable ax+by+c=0 ane0 and

bne0

a b and c are real

number

Infinite solution

possible

3 Linear equation in three Variable ax+by+cz+d=0 ane0

bne0 and cne0

Infinite solution

possible

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a b c d are realnumber

Simultaneous pair of Linear equation

A pair of Linear equation in two variables

a1x+b1y+c1=0

a2x +b2y+c2=0

Graphically it is represented by two straight lines on Cartesian plane

Simultaneous pair of

Linear equation

Condition Graphical

representation

Algebraic

interpretation

a1x+b1y+c1=0

a2x +b2y+c2=0

Example

x-4y+14=0

3x+2y-14=0

Intersecting lines The

intersecting point

coordinate is the only

solution

One unique solution

only

a1x+b1y+c1=0

a2x +b2y+c2=0

Coincident lines Theany coordinate on the

line is the solution

Infinite solution

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Example

2x+4y=16

3x+6y=24

a1x+b1y+c1=0

a2x +b2y+c2=0

Example

2x+4y=6

4x+8y=18

Parallel Lines No solution

The graphical solution can be obtained by drawing the lines on the Cartesian plane

Algebraic Solution of system of Linear equation

Sno Type of method Working of method

1 Method of elimination by

substitution

1) Suppose the equation are

a1x+b1y+c1=0

a2x +b2y+c2=0

2) Find the value of variable of either x or y in other

variable term in first equation

3) Substitute the value of that variable in second

equation

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4) Now this is a linear equation in one variable Findthe value of the variable

5) Substitute this value in first equation and get the

second variable

2 Method of elimination by

equating the coefficients

1) Suppose the equation are

a1x+b1y+c1=0

a2x +b2y+c2=0

2) Find the LCM of a1 and a2 Let it k

3) Multiple the first equation by the value ka1

4) Multiple the first equation by the value ka2

4) Subtract the equation obtained This way one

variable will be eliminated and we can solve to get the

value of variable y

5) Substitute this value in first equation and get the

second variable

3 Cross Multiplication method 1) Suppose the equation are

a1x+b1y+c1=0

a2x +b2y+c2=0

2) This can be written as

3) This can be written as

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4) Value of x and y can be find using the

x =gt first and last expression

y=gt second and last expression

Quadratic EquationsSno Terms Descriptions

1 Quadratic Polynomial P(x) = ax +bx+c where ane0

2 Quadratic equation ax +bx+c =0 where ane0

3 Solution or root of the

Quadratic equation

A real number α is called the root or solution of the

quadratic equation if

aα2 +bα+c=0

4 zeroes of the

polynomial p(x)

The root of the quadratic equation are called zeroes

5 Maximum roots of

quadratic equations

We know from chapter two that a polynomial of

degree can have max two zeroes So a quadratic

equation can have maximum two roots

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6 Condition for realroots A quadratic equation has real roots if b

2

- 4ac gt 0

How to Solve Quadratic equation

Sno Method Working

1 factorization This method we factorize the equation by splitting themiddle term b

In ax2+bx+c=0

Example

6x2-x-2=0

1) First we need to multiple the coefficient a and cIn this

case =6X-2=-12

2) Splitting the middle term so that multiplication is 12

and difference is the coefficient b

6x2 +3x-4x-2=0

3x( 2x+1) -2(2x+1)=0

(3x-2) (2x+1)=0

3) Roots of the equation can be find equating the factorsto zero

3x-2=0 =gt x=32

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2x+1=0 =gt x=-12

2 Square method In this method we create square on LHS and RHS and

then find the value

ax2 +bx+c=0

1) x2 +(ba) x+(ca)=0

2) ( x+b2a)2 ndash(b2a)2 +(ca)=0

3) ( x+b2a)2=(b2-4ac)4a2

4) radic

Example

x2 +4x-5=0

1) (x+2)2 -4-5=0

2) (x+2)2=9

3) Roots of the equation can be find using square root on

both the sides

x+2 =-3 =gt x=-5

x+2=3=gt x=1

3 Quadratic

method

For quadratic equation

ax2 +bx+c=0

roots are given by radic radic

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For b2 -4ac gt 0 Quadratic equation has two real roots of

different value

For b2-4ac =0 quadratic equation has one real root

For b2-4ac lt 0 no real roots for quadratic equation

Nature of roots of Quadratic equation

Sno Condition Nature of roots

1 b -4ac gt 0 Two distinct real roots

2 b -4ac =0 One real root

3 b

2

-4ac lt 0 No real roots

Triangles

Sno Terms Descriptions

1 Congruence Two Geometric figure are said to be congruence ifthey are exactly same size and shapeSymbol used is cong Two angles are congruent if they are equalTwo circle are congruent if they have equal radiiTwo squares are congruent if the sides are equal

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20

2 Triangle Congruencebull

Two triangles are congruent if three sidesand three angles of one triangle is

congruent to the corresponding sides and

angles of the other

bull Corresponding sides are equal

AB=DE BC=EF AC=DF

bull Corresponding angles are equalang angang angang ang

bull We write this as

cong

bull The above six equalities are between the

corresponding parts of the two congruent

triangles In short form this is called

CPCT

bull We should keep the letters in correct order

on both sides

3 Inequalities in

Triangles

1) In a triangle angle opposite to longer side is

larger

2) In a triangle side opposite to larger angle is

larger

3) The sum of any two sides of the triangle is

greater than the third side

983105

983107983106

983108

983109 983110

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21

In triangle ABC

AB +BC gt AC

Different Criterion for Congruence of the triangles

N Criterion Description Figures and

expression

1 Side angle

Side (SAS)

congruence

bull Two triangles are congruent if the

two sides and included angles of

one triangle is equal to the two

sides and included angle

bull It is an axiom as it cannot be

proved so it is an accepted truth

bull ASS and SSA type two triangles

may not be congruent alwaysIf following

condition

983105

983110983109

983108

983107983106

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983151983150983148983161

22

AB=DE BC=EFang ang

Then

cong

2 Angle side

angle (ASA)

congruence

bull Two triangles are congruent if the

two angles and included side of

one triangle is equal to the

corresponding angles and side

bull It is a theorem and can be proved

If following

condition

BC=EF

ang angang ang

Then

cong

3 Angle angle

side( AAS)

congruence

bull Two triangles are congruent if the

any two pair of angles and any

side of one triangle is equal to the

corresponding angles and side

bull It is a theorem and can be proved

983110983109

983108

983107983106

983105

983110983109

983108

983107983106

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983151983150983148983161

2983091

If followingcondition

BC=EF

ang angang ang

Then

cong

4 Side-Side-Side

(SSS)congruence

bull Two triangles are congruent if the

three sides of one triangle is equalto the three sides of the another

If following

condition

BC=EFAB=DEDF

=AC

Then

cong

5 Right angle ndash

hypotenuse-

side(RHS)

bull Two right triangles are congruent if

the hypotenuse and a side of the

one triangle are equal to

983110983109

983108

983107983106

983105

983105 983108

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983151983150983148983161

2983092

congruence corresponding hypotenuse and sideof the another

If following

condition

AC=DFBC=EF

Then

cong

Some Important points on Triangles

Terms Description

Orthocenter Point of intersection of the three altitudeof the triangle

Equilateral triangle whose all sides are equal and allangles are equal to 600

Median A line Segment joining the corner of thetriangle to the midpoint of the oppositeside of the triangle

Altitude A line Segment from the corner of thetriangle and perpendicular to theopposite side of the triangle

Isosceles A triangle whose two sides are equalCentroid Point of intersection of the three median

of the triangle is called the centroid of

the triangleIn center All the angle bisector of the trianglepasses through same point

Circumcenter The perpendicular bisector of the sidesof the triangles passes through samepoint

983106 983107 983109 983110

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2983093

Scalene triangle Triangle having no equal angles and noequal sidesRight Triangle Right triangle has one angle equal to 900 Obtuse Triangle One angle is obtuse angle while other

two are acute anglesAcute Triangle All the angles are acute

Similarity of Triangles

Sno Points

1 Two figures having the same shape but not necessarily the same size are called similarfigures

2 All the congruent figures are similar but the converse is not true

3 If a line is drawn parallel to one side of a triangle to intersect the other two sides in

distinct points then the other two sides are divided in the same ratio

4 If a line divides any two sides of a triangle in the same ratio then the line is parallel tothe third side

Different Criterion for Similarity of the triangles

N Criterion Description Expression

1 Angle Angle

angle(AAA)

similarity

bull Two triangles are similar if

corresponding angle are equal

If following

condition

ang ang

ang ang

ang ang

Then

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2983094

Then 983166

2 Angle angle

(AA) similarity

bull Two triangles are similar if the two

corresponding angles are equal as

by angle property third angle will

be also equal

If following

condition

ang ang

ang ang

Then

ang ang

Then

Then

983166

3 Side side

side(SSS)

Similarity

Two triangles are similar if the

sides of one triangle is

proportional to the sides of other

triangle

If following

condition

Then

ang ang

ang ang

ang ang

Then

cong

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2983095

4 Side-Angle-Side (SAS)

similarity

bull

Two triangles are similar if the oneangle of a triangle is equal to one

angle of other triangles and sides

including that angle is proportional

If followingcondition

And ang ang

Then

cong

Area of Similar triangles

If the two triangle ABC and DEF are similar

cong

Then

Pythagoras Theorem

Sno Points

1 If a perpendicular is drawn from the vertex of the right angle of a right triangle to thehypotenuse then the triangles on both sides of the perpendicular are similar to the

whole triangle and also to each other

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2983096

2 In a right triangle the square of the hypotenuse is equal to the sum of the squares of the

other two sides (Pythagoras Theorem)

(hyp)2 =(base)

2 + (perp)

2

3 If in a triangle square of one side is equal to the sum of the squares of the other twosides then the angle opposite the first side is a right angle

Arithmetic Progression

Sno Terms Descriptions

1 ArithmeticProgression

An arithmetic progression is a sequence ofnumbers such that the difference of any two

successive members is a constant

Examples

1) 1591317hellip

2) 12345hellip

2 common difference of

the AP

the difference between any successive members is a

constant and it is called the common difference of AP

1) If a1 a2a3a4a5 are the terms in AP then

D=a2 -a1 =a3 - a2 =a4 ndash a3=a5 ndasha4

2) We can represent the general form of AP inthe form

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2983097

aa+da+2da+3da+4d

Where a is first term and d is the commondifference

3 nth term of Arithmetic Progression

n term = a + (n - 1)d

4 Sum of nth item in ArithmeticProgression

Sn =(n2)[a + (n-1)d]

Or

Sn =(n2)[t1+ tn]

Trigonometry

Sno Terms Descriptions

1 What is

Trigonometry

Trigonometry from Greek trigotildenon triangle and

metron measure) is a branch of mathematics

that studies relationships involving lengths and

angles of triangles The field emerged during the

3rd century BC from applications of geometry to

astronomical studies

Trigonometry is most simply associated with

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9830910

planar right angle triangles (each of which is atwo-dimensional triangle with one angle equal to

90 degrees) The applicability to non-right-angle

triangles exists but since any non-right-angle

triangle (on a flat plane) can be bisected to

create two right-angle triangles most problems

can be reduced to calculations on right-angle

triangles Thus the majority of applications relate

to right-angle triangles

2 Trigonometric

Ratiorsquos

In a right angle triangle ABC where B=90deg

We can define following term for angle A

Base Side adjacent to angle

Perpendicular Side Opposite of angleHypotenuse Side opposite to right angle

We can define the trigonometric ratios for

angle A as

sin A= PerpendicularHypotenuse =BCAC

cosec A= HypotenusePerpendicular

=ACBC

cos A= BaseHypotenuse =ABAC

sec A= HypotenuseBase=ACAB

tan A= PerpendicularBase =BCAB

cot A= BasePerpendicular=ABBC

Notice that each ratio in the right-hand column is

the inverse or the reciprocal of the ratio in the

left-hand column

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9830911

3 Reciprocal offunctions The reciprocal of sin A is cosec A and vice-versa

The reciprocal of cos A is sec A

And the reciprocal of tan A is cot A

These are valid for acute angles

We can define tan A = sin Acos A

And Cot A =cos A Sin A

4 Value of of sin and

cos

Is always less 1

5 Trigonometric ration

from another angle

We can define the trigonometric ratios for angle Cas

sin C= PerpendicularHypotenuse =ABACcosec C= HypotenusePerpendicular =ACABBcos C= BaseHypotenuse =BCACsec C= HypotenuseBase=ACBCtan A= PerpendicularBase =ABBCcot A= BasePerpendicular=BCAB

6 Trigonometric ratios

of complimentary

Sin (90-A) =cos(A)

Cos(90-A) = sin A

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9830912

angles Tan(90-A) =cot A

Sec(90-A)= cosec A

Cosec (90-A) =sec A

Cot(90- A) =tan A7 Trigonometric

identities

Sin2 A + cos2 A =0

1 + tan2 A =sec2 A

1 + cot2 A =cosec2 A

Trigonometric Ratios of Common angles

We can find the values of trigonometric ratiorsquos various angle

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983091983091

Area of Circles

Sno Points

1 A circle is a collection of all the points in a plane which are equidistant

from a fixed point in the plane

2 Equal chords of a circle (or of congruent circles) subtend equal angles at

the center

3 If the angles subtended by two chords of a circle (or of congruent circles)

at the center (corresponding center) are equal the chords are equal

4 The perpendicular from the center of a circle to a chord bisects the chord

5 The line drawn through the center of a circle to bisect a chord is

perpendicular to the chord

6 There is one and only one circle passing through three non-collinear points

7 Equal chords of a circle (or of congruent circles) are equidistant from the

center (or corresponding centers)

8 Chords equidistant from the center (or corresponding centers) of a circle

(or of congruent circles) are equal

9 If two arcs of a circle are congruent then their corresponding chords are

equal and conversely if two chords of a circle are equal then their

corresponding arcs (minor major) are congruent

10 Congruent arcs of a circle subtend equal angles at the center

11 The angle subtended by an arc at the center is double the angle subtended

by it at any point on the remaining part of the circle

12 Angles in the same segment of a circle are equal

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983091983092

13 Angle in a semicircle is a right angle

14 If a line segment joining two points subtends equal angles at two other

points lying on the same side of the line containing the line segment the

four points lie on a circle

15 The sum of either pair of opposite angles of a cyclic quadrilateral is 180deg

16 If the sum of a pair of opposite angles of a quadrilateral is 180deg then the

quadrilateral is cyclic

Sno Terms Descriptions

1 Circumference of a circle 2 π r

2 Area of circle π r 2

3 Length of the arc of

the sector of angle

Length of the arc of the sector of angle θ 2

4 Area of the sector of

angle

Area of the sector of angle θ

5 Area of segment of a

circle

Area of the corresponding sector ndash Area of the

corresponding triangle

Surface Area and Volume

Sno Term Description

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983091983093

1 Mensuration It is branch of mathematics which is concernedabout the measurement of length area and

Volume of plane and Solid figure

2 Perimeter a)The perimeter of plane figure is defined as thelength of the boundaryb)It units is same as that of length ie m cmkm

3 Area a)The area of the plane figure is the surfaceenclosed by its boundaryb) It unit is square of length unit ie m2 km2

4 Volume Volume is the measure of the amount of spaceinside of a solid figure like a cube ball cylinderor pyramid Its units are always cubic that isthe number of little element cubes that fit insidethe figure

Volume Unit conversion

1 cm3 1mL 1000 mm3

1 Litre 1000ml 1000 cm3

1 m3 106 cm3 1000 L

1 dm3 1000 cm3 1 L

Surface Area and Volume of Cube and Cuboid

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Type Measurement

Surface Area of Cuboid of Length L

Breadth B and Height H

2(LB + BH + LH)

Lateral surface area of the cuboids 2( L + B ) H

Diagonal of the cuboids Volume of a cuboids LBH

Length of all 12 edges of the cuboids 4 (L+B+H)

Surface Area of Cube of side L 6L2

Lateral surface area of the cube 4L2

Diagonal of the cube radic3 Volume of a cube L3

Surface Area and Volume of Right circular cylinder

Radius The radius (r) of the circular base is called the radius of the

cylinder

Height The length of the axis of the cylinder is called the height (h) of the

983107983157983138983141 983107983157983138983151983145983140

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cylinder

Lateral

Surface

The curved surface joining the two base of a right circular cylinder is

called Lateral Surface

Type Measurement

Curved or lateral Surface Area of

cylinder

2πrh

Total surface area of cylinder 2πr (h+r)

Volume of Cylinder π r2h

Surface Area and Volume of Right circular cone

Radius The radius (r) of the circular base is called the radius of the

cone

Height The length of the line segment joining the vertex to the center of

base is called the height (h) of the cone

Slant

Height

The length of the segment joining the vertex to any point on the

circular edge of the base is called the slant height (L) of the cone

Lateral

surface

Area

The curved surface joining the base and uppermost point of a right

circular cone is called Lateral Surface

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Type Measurement

Curved or lateral Surface Area of

cone

πrL

Total surface area of cone πr (L+r)

Volume of Cone

Surface Area and Volume of sphere and hemisphere

Sphere A sphere can also be considered as a solid obtained on

rotating a circle About its diameter

Hemisphere A plane through the centre of the sphere divides the sphere into two

equal parts each of which is called a hemisphere

radius The radius of the circle by which it is formed

SphericalShell

The difference of two solid concentric spheres is called a sphericalshell

Lateral

Surface

Area for

Total surface area of the sphere

983123983152983144983141983154983141

983112983141983149983145983155983152983144983141983154983141

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Sphere

Lateral

Surface

area of

Hemisphere

It is the curved surface area leaving the circular base

Type Measurement

Surface area of Sphere 4πr2

Volume of Sphere 43 Curved Surface area of hemisphere 2πr2

Total Surface area of hemisphere 3πr2

Volume of hemisphere 23 Volume of the spherical shell whose

outer and inner radii and lsquoRrsquo and lsquorrsquorespectively

43

How the Surface area and Volume are determined

Area of Circle The circumference of a circle is 2πr

This is the definition of π (pi) Divide

the circle into many triangular

segments The area of the triangles

is 12 times the sum of their bases

2πr (the circumference) times their

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height r

12 2 Surface Area of cylinder This can be imagined as unwrapping the

surface into a rectangle

Surface area of cone This can be achieved by divide the

surface of the cone into its triangles or

the surface of the cone into many thin

triangles The area of the triangles is 12

times the sum of their bases p times

their height

12 2

Surface Area and Volume of frustum of cone

h = vertical height of the frustum

l = slant height of the frustum

r 1 and r 2 are radii of the two bases (ends) of the frustum

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Type Measurement

983126983151983148983157983149983141 983151983142 983137 983142983154983157983155983156983157983149 983151983142 983137 983139983151983150983141 983123983148983137983150983156 983144983141983145983143983144983156 983151983142 983142983154983157983155983156983157983149 983151983142 983137 983139983151983150983141 983107983157983154983158983141983140 983155983157983154983142983137983139983141 983137983154983141983137 983151983142 983137 983142983154983157983155983156983157983149 983151983142 983137 983139983151983150983141 983124983151983156983137983148 983155983157983154983142983137983139983141 983137983154983141983137 983151983142 983142983154983157983155983156983157983149 983151983142 983137 983139983151983150983141

Statistics

Sno Term Description

1 Statistics Statistics is a broad mathematical disciplinewhich studies ways to collect summarize and

draw conclusions from data

2 Data A systematic record of facts or different values of

a quantity is called data

Data is of two types - Primary data and Secondary

data

Primary Data The data collected by a researcher

with a specific purpose in mind is called primarydata

Secondary Data The data gathered from a

source where it already exists is called secondary

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data

3 Features of data bull Statistics deals with collectionpresentation analysis and interpretation ofnumerical data

bull Arranging data in an order to study theirsalient features is called presentation ofdata

bull Data arranged in ascending or descendingorder is called arrayed data or an array

bull Range of the data is the differencebetween the maximum and the minimum

values of the observationsbull Table that shows the frequency of different

values in the given data is called afrequency distribution table

bull A frequency distribution table that showsthe frequency of each individual value in thegiven data is called an ungrouped frequencydistribution table

bull A table that shows the frequency of groupsof values in the given data is called agrouped frequency distribution table

bull

The groupings used to group the values ingiven data are called classes or class-intervals The number of values that eachclass contains is called the class size orclass width The lower value in a class iscalled the lower class limit The higher valuein a class is called the upper class limit

bull Class mark of a class is the mid value ofthe two limits of that class

bull A frequency distribution in which the upperlimit of one class differs from the lower limitof the succeeding class is called an

Inclusive or discontinuous FrequencyDistribution

bull A frequency distribution in which the upperlimit of one class coincides from the lowerlimit of the succeeding class is called anexclusive or continuous Frequency

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983092983091

Distribution bull

4 Bar graph A bar graph is a pictorial representation of data inwhich rectangular bars of uniform width are drawnwith equal spacing between them on one axisusually the x axis The value of the variable isshown on the other axis that is the y axis

5 Histogram A histogram is a set of adjacent rectangles whoseareas are proportional to the frequencies of agiven continuous frequency distribution

6 Mean The mean value of a variable is defined as thesum of all the values of the variable divided by thenumber of values

4 sum

7 Median The median of a set of data values is the middle

value of the data set when it has been arranged inascending order That is from the smallest valueto the highest value

Median is calculated as

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983092983092

12 1 Where n is the number of values in the data

If the number of values in the data set is eventhen the median is the average of the two middlevalues

8 Mode Mode of a statistical data is the value of thatvariable which has the maximum frequency

Sno Term Description

1983117983141983137983150 983142983151983154

983125983150983143983154983151983157983152

983110983154983141983153983157983141983150983139983161 983156983137983138983148983141

Mean is given by

sum sum

2983117983141983137983150 983142983151983154 983143983154983151983157983152

983110983154983141983153983157983141983150983139983161 983156983137983138983148983141

In these distribution it is assumed that frequency of each classinterval is centered around its mid-point ie class marks

2

983117983141983137983150 983139983137983150 983138983141 983139983137983148983139983157983148983137983156983141983140 983157983155983145983150983143 983156983144983154983141983141 983149983141983156983144983151983140

983137983081 983108983145983154983141983139983156 983149983141983156983144983151983140

sum

sum

983138983081 983105983155983155983157983149983141983140 983149983141983137983150 983149983141983156983144983151983140

sum sum

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983151983150983148983161

983092983093

Where

a=gt Assumed mean

di =gt xi ndasha

983139983081 983123983156983141983152 983140983141983158983145983137983156983145983151983150 983117983141983156983144983151983140

sum sum

Where

a=gt Assumed mean

ui =gt (xi ndasha)h

3983117983151983140983141 983142983151983154

983143983154983151983157983152983141983140

983142983154983141983153983157983141983150983139983161 983156983137983138983148983141

Modal class The class interval having highest frequency is

called the modal class and Mode is obtained using the modal

class

2

983127983144983141983154983141

l = lower limit of the modal classh = size of the class interval (assuming all class sizes tobe equal)f 1 = frequency of the modal classf 0 = frequency of the class preceding the modal classf 2 = frequency of the class succeeding the modal class

4983117983141983140983145983137983150 983151983142 983137

983143983154983151983157983152983141983140 983140983137983156983137

983142983154983141983153983157983141983150983139983161 983156983137983138983148983141

For the given data we need to have class interval frequency

distribution and cumulative frequency distribution

Median is calculated as

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983151983150983148983161

983092983094

2

983127983144983141983154983141

983148 983101 983148983151983159983141983154 983148983145983149983145983156 983151983142 983149983141983140983145983137983150 983139983148983137983155983155

983150 983101 983150983157983149983138983141983154 983151983142 983151983138983155983141983154983158983137983156983145983151983150983155

983139983142 983101 983139983157983149983157983148983137983156983145983158983141 983142983154983141983153983157983141983150983139983161 983151983142 983139983148983137983155983155 983152983154983141983139983141983140983145983150983143 983156983144983141 983149983141983140983145983137983150 983139983148983137983155983155

983142 983101 983142983154983141983153983157983141983150983139983161 983151983142 983149983141983140983145983137983150 983139983148983137983155983155

983144 983101 983139983148983137983155983155 983155983145983162983141 (983137983155983155983157983149983145983150983143 983139983148983137983155983155 983155983145983162983141 983156983151 983138983141 983141983153983157983137983148)

5983109983149983152983145983154983145983139983137983148

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983117983141983140983145983137983150

3 Median=Mode +2 Mean

ProbabilitySn

o

Term Description

1 Empirical

probability

It is a probability of event which is calculated based on

experiments

Example

A coin is tossed 1000 times we get 499 times head and

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983092983095

501 times tail

So empirical or experimental probability of getting head is

calculated as

4991000 499

Empirical probability depends on experiment and

different will get different values based on the

experiment

2 Important point

about events

If the event A B C covers the entire possible outcome inthe experiment Then

P (A) +P (B) +P(C)=1

3 impossible

event

The probability of an event (U) which is impossible tooccur is 0 Such an event is called an impossible event

P (U)=0

4 Sure or certain

event

The probability of an event (X) which is sure (or certain)to occur is 1 Such an event is called a sure event or acertain event

P(X) =1

5 Probability of

any event

Probability of any event can be as

0 1

Sn

o

Term Description

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983092983096

1

983124983144983141983151983154983141983156983145983139983137983148

983120983154983151983138983137983138983145983148983145983156983161

The theoretical probability or the classical probability of the event isdefined as

2 Elementary

events

983105983150 983141983158983141983150983156 983144983137983158983145983150983143 983151983150983148983161 983151983150983141 983151983157983156983139983151983149983141 983151983142 983156983144983141 983141983160983152983141983154983145983149983141983150983156 983145983155 983139983137983148983148983141983140 983137983150

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983120(983105)+983120(983106) +983120(983107)9831011

3 Complementar

y events

983124983144983141 983141983158983141983150983156 Ᾱ 983154983141983152983154983141983155983141983150983156983145983150983143 991256983150983151983156 983105991257 983145983155 983139983137983148983148983141983140 983156983144983141 983139983151983149983152983148983141983149983141983150983156 983151983142 983156983144983141 983141983158983141983150983156

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983120(983105) +983120(Ᾱ)9831011

4 Sure or certain

event

The probability of an event (X) which is sure (or certain)

to occur is 1 Such an event is called a sure event or acertain event

P(X) =1

5 Probability of

any event

Probability of any event can be as

0 1

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983151983150983148983161

983094

3 y=ax2+bx+c

where

b2-4ac gt 0 and ane0 and a lt 0

Example

y=-x2+2x+8

Parabola

It intersect the x-

axis at two points

Example (-20)

and (40)

Quadratic

polynomial

4 y=ax +bx+c

where

b2-4ac = 0 and ane0 a gt 0

Example

y=(x-2)2

Parabola

It intersect the x-

axis at one points

Quadratic

polynomial

5 y=ax2+bx+c

where

b

2

-4ac lt 0 and ane0 a gt 0

Example

y=x2-2x+6

Parabola

It does not

intersect the x-axis

It has no zerorsquos

Quadratic

polynomial

6 y=ax +bx+c

where

b2-4ac lt 0 and ane0 a lt 0

Example

y=-x2-2x-6

Parabola

It does not

intersect the x-axis

It has no zerorsquos

Quadratic

polynomial

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7 y=ax3 +bx

2+cx+d

where ane0

It can be of any

shape

It will cut the x-axis

at the most 3 times

Cubic Polynomial

8 ⋯ 991270 991270 991270

Where an ne0

It can be of any

shape

It will cut the x-axis

at the most n times

Polynomial of n

degree

Relation between coefficient and zeroes of the Polynomial

Sno Type of Polynomial General form Zerorsquos Relationship between Zerorsquos

and coefficients

1 Linear polynomial ax+b ane0 1

2 Quadratic ax +bx+c ane0 2

3 Cubic ax +bx +cx+d ane0 3

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Formation of polynomial when the zeroes are given

Type ofpolynomial

Zerorsquos Polynomial Formed

Linear k=a (x-a)

Quadratic k1=a and

k2=b

(x-a)(x-b)

Or

x2-( a+b)x +ab

Or

x2

-( Sum of the zerorsquos)x +product of the zerorsquos

Cubic k1=a k2=b

and k3=c

(x-a)(x-b)(x-c)

Division algorithm for Polynomial

Letlsquos p(x) and q(x) are any two polynomial with q(x) ne0 then we can find polynomial s(x) and r(x) suchthat

P(x)=s(x) q(x) + r(x)

Where r(x) can be zero or degree of r(x) lt degree of g(x)

983108983145983158983145983140983141983150983140 983101983121983157983151983156983145983141983150983156 983128 983108983145983158983145983155983151983154 + 983122983141983149983137983145983150983140983141983154

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COORDINATE GEOMETRYSno Points

1 We require two perpendicular axes to locate a point in the plane One of

them is horizontal and other is Vertical

2 The plane is called Cartesian plane and axis are called the coordinates axis

3 The horizontal axis is called x-axis and Vertical axis is called Y-axis

4 The point of intersection of axis is called origin

5 The distance of a point from y axis is called x ndashcoordinate or abscissa and

the distance of the point from x ndashaxis is called y ndash coordinate or Ordinate

6 The distance of a point from y axis is called x ndashcoordinate or abscissa and

the distance of the point from x ndashaxis is called y ndash coordinate or Ordinate

7 The Origin has zero distance from both x-axis and y-axis so that its

abscissa and ordinate both are zero So the coordinate of the origin is (0

0)

8 A point on the x ndashaxis has zero distance from x-axis so coordinate of any

point on the x-axis will be (x 0)

9 A point on the y ndashaxis has zero distance from y-axis so coordinate of any

point on the y-axis will be (0 y)

10 The axes divide the Cartesian plane in to four parts These Four parts arecalled the quadrants

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10

The coordinates of the points in the four quadrants will have sign according to thebelow table

Quadrant x-coordinate y-coordinate

Ist Quadrant + +

IInd quadrant - +

IIIrd quadrant - -

IVth quadrant + -

Sno Terms Descriptions

1 Distance formula Distance between the points AB is given by

Distance of Point A from Origin

2 Section Formula A point P(xy) which divide the line segment AB in

the ratio m1 and m2 is given by

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11

The midpoint P is given by

3 Area of Triangle Area of triangle ABC of coordinates A(x 1y 1 )

B(x 2 y 2 ) and C(x 3y 3 )

12 For point AB and C to be collinear The value of A

should be zero

LINEAR EQUATIONS IN TWO

VARIABLESAn equation of the form ax + by + c = 0 where a b and c are real numbers such

that a andb are not both zero is called a linear equation in two variablesImportant points to Note

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12

Sno Points

1 A linear equation in two variable has infinite solutions

2 The graph of every linear equation in two variable is a straight line

3 x = 0 is the equation of the y-axis and y = 0 is the equation of the x-axis

4 The graph x=a is a line parallel to y -axis

5 The graph y=b is a line parallel to x -axis

6 An equation of the type y = mx represents a line passing through theorigin

7 Every point on the graph of a linear equation in two variables is a solution

of the linear

equation Moreover every solution of the linear equation is a point on the

graph

Sno Type of equation Mathematical

representation

Solutions

1 Linear equation in one Variable ax+b=0 ane0

a and b are real

number

One solution

2 Linear equation in two Variable ax+by+c=0 ane0 and

bne0

a b and c are real

number

Infinite solution

possible

3 Linear equation in three Variable ax+by+cz+d=0 ane0

bne0 and cne0

Infinite solution

possible

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a b c d are realnumber

Simultaneous pair of Linear equation

A pair of Linear equation in two variables

a1x+b1y+c1=0

a2x +b2y+c2=0

Graphically it is represented by two straight lines on Cartesian plane

Simultaneous pair of

Linear equation

Condition Graphical

representation

Algebraic

interpretation

a1x+b1y+c1=0

a2x +b2y+c2=0

Example

x-4y+14=0

3x+2y-14=0

Intersecting lines The

intersecting point

coordinate is the only

solution

One unique solution

only

a1x+b1y+c1=0

a2x +b2y+c2=0

Coincident lines Theany coordinate on the

line is the solution

Infinite solution

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Example

2x+4y=16

3x+6y=24

a1x+b1y+c1=0

a2x +b2y+c2=0

Example

2x+4y=6

4x+8y=18

Parallel Lines No solution

The graphical solution can be obtained by drawing the lines on the Cartesian plane

Algebraic Solution of system of Linear equation

Sno Type of method Working of method

1 Method of elimination by

substitution

1) Suppose the equation are

a1x+b1y+c1=0

a2x +b2y+c2=0

2) Find the value of variable of either x or y in other

variable term in first equation

3) Substitute the value of that variable in second

equation

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4) Now this is a linear equation in one variable Findthe value of the variable

5) Substitute this value in first equation and get the

second variable

2 Method of elimination by

equating the coefficients

1) Suppose the equation are

a1x+b1y+c1=0

a2x +b2y+c2=0

2) Find the LCM of a1 and a2 Let it k

3) Multiple the first equation by the value ka1

4) Multiple the first equation by the value ka2

4) Subtract the equation obtained This way one

variable will be eliminated and we can solve to get the

value of variable y

5) Substitute this value in first equation and get the

second variable

3 Cross Multiplication method 1) Suppose the equation are

a1x+b1y+c1=0

a2x +b2y+c2=0

2) This can be written as

3) This can be written as

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1983094

4) Value of x and y can be find using the

x =gt first and last expression

y=gt second and last expression

Quadratic EquationsSno Terms Descriptions

1 Quadratic Polynomial P(x) = ax +bx+c where ane0

2 Quadratic equation ax +bx+c =0 where ane0

3 Solution or root of the

Quadratic equation

A real number α is called the root or solution of the

quadratic equation if

aα2 +bα+c=0

4 zeroes of the

polynomial p(x)

The root of the quadratic equation are called zeroes

5 Maximum roots of

quadratic equations

We know from chapter two that a polynomial of

degree can have max two zeroes So a quadratic

equation can have maximum two roots

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6 Condition for realroots A quadratic equation has real roots if b

2

- 4ac gt 0

How to Solve Quadratic equation

Sno Method Working

1 factorization This method we factorize the equation by splitting themiddle term b

In ax2+bx+c=0

Example

6x2-x-2=0

1) First we need to multiple the coefficient a and cIn this

case =6X-2=-12

2) Splitting the middle term so that multiplication is 12

and difference is the coefficient b

6x2 +3x-4x-2=0

3x( 2x+1) -2(2x+1)=0

(3x-2) (2x+1)=0

3) Roots of the equation can be find equating the factorsto zero

3x-2=0 =gt x=32

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2x+1=0 =gt x=-12

2 Square method In this method we create square on LHS and RHS and

then find the value

ax2 +bx+c=0

1) x2 +(ba) x+(ca)=0

2) ( x+b2a)2 ndash(b2a)2 +(ca)=0

3) ( x+b2a)2=(b2-4ac)4a2

4) radic

Example

x2 +4x-5=0

1) (x+2)2 -4-5=0

2) (x+2)2=9

3) Roots of the equation can be find using square root on

both the sides

x+2 =-3 =gt x=-5

x+2=3=gt x=1

3 Quadratic

method

For quadratic equation

ax2 +bx+c=0

roots are given by radic radic

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For b2 -4ac gt 0 Quadratic equation has two real roots of

different value

For b2-4ac =0 quadratic equation has one real root

For b2-4ac lt 0 no real roots for quadratic equation

Nature of roots of Quadratic equation

Sno Condition Nature of roots

1 b -4ac gt 0 Two distinct real roots

2 b -4ac =0 One real root

3 b

2

-4ac lt 0 No real roots

Triangles

Sno Terms Descriptions

1 Congruence Two Geometric figure are said to be congruence ifthey are exactly same size and shapeSymbol used is cong Two angles are congruent if they are equalTwo circle are congruent if they have equal radiiTwo squares are congruent if the sides are equal

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20

2 Triangle Congruencebull

Two triangles are congruent if three sidesand three angles of one triangle is

congruent to the corresponding sides and

angles of the other

bull Corresponding sides are equal

AB=DE BC=EF AC=DF

bull Corresponding angles are equalang angang angang ang

bull We write this as

cong

bull The above six equalities are between the

corresponding parts of the two congruent

triangles In short form this is called

CPCT

bull We should keep the letters in correct order

on both sides

3 Inequalities in

Triangles

1) In a triangle angle opposite to longer side is

larger

2) In a triangle side opposite to larger angle is

larger

3) The sum of any two sides of the triangle is

greater than the third side

983105

983107983106

983108

983109 983110

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21

In triangle ABC

AB +BC gt AC

Different Criterion for Congruence of the triangles

N Criterion Description Figures and

expression

1 Side angle

Side (SAS)

congruence

bull Two triangles are congruent if the

two sides and included angles of

one triangle is equal to the two

sides and included angle

bull It is an axiom as it cannot be

proved so it is an accepted truth

bull ASS and SSA type two triangles

may not be congruent alwaysIf following

condition

983105

983110983109

983108

983107983106

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22

AB=DE BC=EFang ang

Then

cong

2 Angle side

angle (ASA)

congruence

bull Two triangles are congruent if the

two angles and included side of

one triangle is equal to the

corresponding angles and side

bull It is a theorem and can be proved

If following

condition

BC=EF

ang angang ang

Then

cong

3 Angle angle

side( AAS)

congruence

bull Two triangles are congruent if the

any two pair of angles and any

side of one triangle is equal to the

corresponding angles and side

bull It is a theorem and can be proved

983110983109

983108

983107983106

983105

983110983109

983108

983107983106

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983151983150983148983161

2983091

If followingcondition

BC=EF

ang angang ang

Then

cong

4 Side-Side-Side

(SSS)congruence

bull Two triangles are congruent if the

three sides of one triangle is equalto the three sides of the another

If following

condition

BC=EFAB=DEDF

=AC

Then

cong

5 Right angle ndash

hypotenuse-

side(RHS)

bull Two right triangles are congruent if

the hypotenuse and a side of the

one triangle are equal to

983110983109

983108

983107983106

983105

983105 983108

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983151983150983148983161

2983092

congruence corresponding hypotenuse and sideof the another

If following

condition

AC=DFBC=EF

Then

cong

Some Important points on Triangles

Terms Description

Orthocenter Point of intersection of the three altitudeof the triangle

Equilateral triangle whose all sides are equal and allangles are equal to 600

Median A line Segment joining the corner of thetriangle to the midpoint of the oppositeside of the triangle

Altitude A line Segment from the corner of thetriangle and perpendicular to theopposite side of the triangle

Isosceles A triangle whose two sides are equalCentroid Point of intersection of the three median

of the triangle is called the centroid of

the triangleIn center All the angle bisector of the trianglepasses through same point

Circumcenter The perpendicular bisector of the sidesof the triangles passes through samepoint

983106 983107 983109 983110

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2983093

Scalene triangle Triangle having no equal angles and noequal sidesRight Triangle Right triangle has one angle equal to 900 Obtuse Triangle One angle is obtuse angle while other

two are acute anglesAcute Triangle All the angles are acute

Similarity of Triangles

Sno Points

1 Two figures having the same shape but not necessarily the same size are called similarfigures

2 All the congruent figures are similar but the converse is not true

3 If a line is drawn parallel to one side of a triangle to intersect the other two sides in

distinct points then the other two sides are divided in the same ratio

4 If a line divides any two sides of a triangle in the same ratio then the line is parallel tothe third side

Different Criterion for Similarity of the triangles

N Criterion Description Expression

1 Angle Angle

angle(AAA)

similarity

bull Two triangles are similar if

corresponding angle are equal

If following

condition

ang ang

ang ang

ang ang

Then

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983151983150983148983161

2983094

Then 983166

2 Angle angle

(AA) similarity

bull Two triangles are similar if the two

corresponding angles are equal as

by angle property third angle will

be also equal

If following

condition

ang ang

ang ang

Then

ang ang

Then

Then

983166

3 Side side

side(SSS)

Similarity

Two triangles are similar if the

sides of one triangle is

proportional to the sides of other

triangle

If following

condition

Then

ang ang

ang ang

ang ang

Then

cong

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983151983150983148983161

2983095

4 Side-Angle-Side (SAS)

similarity

bull

Two triangles are similar if the oneangle of a triangle is equal to one

angle of other triangles and sides

including that angle is proportional

If followingcondition

And ang ang

Then

cong

Area of Similar triangles

If the two triangle ABC and DEF are similar

cong

Then

Pythagoras Theorem

Sno Points

1 If a perpendicular is drawn from the vertex of the right angle of a right triangle to thehypotenuse then the triangles on both sides of the perpendicular are similar to the

whole triangle and also to each other

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2983096

2 In a right triangle the square of the hypotenuse is equal to the sum of the squares of the

other two sides (Pythagoras Theorem)

(hyp)2 =(base)

2 + (perp)

2

3 If in a triangle square of one side is equal to the sum of the squares of the other twosides then the angle opposite the first side is a right angle

Arithmetic Progression

Sno Terms Descriptions

1 ArithmeticProgression

An arithmetic progression is a sequence ofnumbers such that the difference of any two

successive members is a constant

Examples

1) 1591317hellip

2) 12345hellip

2 common difference of

the AP

the difference between any successive members is a

constant and it is called the common difference of AP

1) If a1 a2a3a4a5 are the terms in AP then

D=a2 -a1 =a3 - a2 =a4 ndash a3=a5 ndasha4

2) We can represent the general form of AP inthe form

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2983097

aa+da+2da+3da+4d

Where a is first term and d is the commondifference

3 nth term of Arithmetic Progression

n term = a + (n - 1)d

4 Sum of nth item in ArithmeticProgression

Sn =(n2)[a + (n-1)d]

Or

Sn =(n2)[t1+ tn]

Trigonometry

Sno Terms Descriptions

1 What is

Trigonometry

Trigonometry from Greek trigotildenon triangle and

metron measure) is a branch of mathematics

that studies relationships involving lengths and

angles of triangles The field emerged during the

3rd century BC from applications of geometry to

astronomical studies

Trigonometry is most simply associated with

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9830910

planar right angle triangles (each of which is atwo-dimensional triangle with one angle equal to

90 degrees) The applicability to non-right-angle

triangles exists but since any non-right-angle

triangle (on a flat plane) can be bisected to

create two right-angle triangles most problems

can be reduced to calculations on right-angle

triangles Thus the majority of applications relate

to right-angle triangles

2 Trigonometric

Ratiorsquos

In a right angle triangle ABC where B=90deg

We can define following term for angle A

Base Side adjacent to angle

Perpendicular Side Opposite of angleHypotenuse Side opposite to right angle

We can define the trigonometric ratios for

angle A as

sin A= PerpendicularHypotenuse =BCAC

cosec A= HypotenusePerpendicular

=ACBC

cos A= BaseHypotenuse =ABAC

sec A= HypotenuseBase=ACAB

tan A= PerpendicularBase =BCAB

cot A= BasePerpendicular=ABBC

Notice that each ratio in the right-hand column is

the inverse or the reciprocal of the ratio in the

left-hand column

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9830911

3 Reciprocal offunctions The reciprocal of sin A is cosec A and vice-versa

The reciprocal of cos A is sec A

And the reciprocal of tan A is cot A

These are valid for acute angles

We can define tan A = sin Acos A

And Cot A =cos A Sin A

4 Value of of sin and

cos

Is always less 1

5 Trigonometric ration

from another angle

We can define the trigonometric ratios for angle Cas

sin C= PerpendicularHypotenuse =ABACcosec C= HypotenusePerpendicular =ACABBcos C= BaseHypotenuse =BCACsec C= HypotenuseBase=ACBCtan A= PerpendicularBase =ABBCcot A= BasePerpendicular=BCAB

6 Trigonometric ratios

of complimentary

Sin (90-A) =cos(A)

Cos(90-A) = sin A

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983151983150983148983161

9830912

angles Tan(90-A) =cot A

Sec(90-A)= cosec A

Cosec (90-A) =sec A

Cot(90- A) =tan A7 Trigonometric

identities

Sin2 A + cos2 A =0

1 + tan2 A =sec2 A

1 + cot2 A =cosec2 A

Trigonometric Ratios of Common angles

We can find the values of trigonometric ratiorsquos various angle

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983091983091

Area of Circles

Sno Points

1 A circle is a collection of all the points in a plane which are equidistant

from a fixed point in the plane

2 Equal chords of a circle (or of congruent circles) subtend equal angles at

the center

3 If the angles subtended by two chords of a circle (or of congruent circles)

at the center (corresponding center) are equal the chords are equal

4 The perpendicular from the center of a circle to a chord bisects the chord

5 The line drawn through the center of a circle to bisect a chord is

perpendicular to the chord

6 There is one and only one circle passing through three non-collinear points

7 Equal chords of a circle (or of congruent circles) are equidistant from the

center (or corresponding centers)

8 Chords equidistant from the center (or corresponding centers) of a circle

(or of congruent circles) are equal

9 If two arcs of a circle are congruent then their corresponding chords are

equal and conversely if two chords of a circle are equal then their

corresponding arcs (minor major) are congruent

10 Congruent arcs of a circle subtend equal angles at the center

11 The angle subtended by an arc at the center is double the angle subtended

by it at any point on the remaining part of the circle

12 Angles in the same segment of a circle are equal

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983091983092

13 Angle in a semicircle is a right angle

14 If a line segment joining two points subtends equal angles at two other

points lying on the same side of the line containing the line segment the

four points lie on a circle

15 The sum of either pair of opposite angles of a cyclic quadrilateral is 180deg

16 If the sum of a pair of opposite angles of a quadrilateral is 180deg then the

quadrilateral is cyclic

Sno Terms Descriptions

1 Circumference of a circle 2 π r

2 Area of circle π r 2

3 Length of the arc of

the sector of angle

Length of the arc of the sector of angle θ 2

4 Area of the sector of

angle

Area of the sector of angle θ

5 Area of segment of a

circle

Area of the corresponding sector ndash Area of the

corresponding triangle

Surface Area and Volume

Sno Term Description

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983091983093

1 Mensuration It is branch of mathematics which is concernedabout the measurement of length area and

Volume of plane and Solid figure

2 Perimeter a)The perimeter of plane figure is defined as thelength of the boundaryb)It units is same as that of length ie m cmkm

3 Area a)The area of the plane figure is the surfaceenclosed by its boundaryb) It unit is square of length unit ie m2 km2

4 Volume Volume is the measure of the amount of spaceinside of a solid figure like a cube ball cylinderor pyramid Its units are always cubic that isthe number of little element cubes that fit insidethe figure

Volume Unit conversion

1 cm3 1mL 1000 mm3

1 Litre 1000ml 1000 cm3

1 m3 106 cm3 1000 L

1 dm3 1000 cm3 1 L

Surface Area and Volume of Cube and Cuboid

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Type Measurement

Surface Area of Cuboid of Length L

Breadth B and Height H

2(LB + BH + LH)

Lateral surface area of the cuboids 2( L + B ) H

Diagonal of the cuboids Volume of a cuboids LBH

Length of all 12 edges of the cuboids 4 (L+B+H)

Surface Area of Cube of side L 6L2

Lateral surface area of the cube 4L2

Diagonal of the cube radic3 Volume of a cube L3

Surface Area and Volume of Right circular cylinder

Radius The radius (r) of the circular base is called the radius of the

cylinder

Height The length of the axis of the cylinder is called the height (h) of the

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cylinder

Lateral

Surface

The curved surface joining the two base of a right circular cylinder is

called Lateral Surface

Type Measurement

Curved or lateral Surface Area of

cylinder

2πrh

Total surface area of cylinder 2πr (h+r)

Volume of Cylinder π r2h

Surface Area and Volume of Right circular cone

Radius The radius (r) of the circular base is called the radius of the

cone

Height The length of the line segment joining the vertex to the center of

base is called the height (h) of the cone

Slant

Height

The length of the segment joining the vertex to any point on the

circular edge of the base is called the slant height (L) of the cone

Lateral

surface

Area

The curved surface joining the base and uppermost point of a right

circular cone is called Lateral Surface

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983091983096

Type Measurement

Curved or lateral Surface Area of

cone

πrL

Total surface area of cone πr (L+r)

Volume of Cone

Surface Area and Volume of sphere and hemisphere

Sphere A sphere can also be considered as a solid obtained on

rotating a circle About its diameter

Hemisphere A plane through the centre of the sphere divides the sphere into two

equal parts each of which is called a hemisphere

radius The radius of the circle by which it is formed

SphericalShell

The difference of two solid concentric spheres is called a sphericalshell

Lateral

Surface

Area for

Total surface area of the sphere

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Sphere

Lateral

Surface

area of

Hemisphere

It is the curved surface area leaving the circular base

Type Measurement

Surface area of Sphere 4πr2

Volume of Sphere 43 Curved Surface area of hemisphere 2πr2

Total Surface area of hemisphere 3πr2

Volume of hemisphere 23 Volume of the spherical shell whose

outer and inner radii and lsquoRrsquo and lsquorrsquorespectively

43

How the Surface area and Volume are determined

Area of Circle The circumference of a circle is 2πr

This is the definition of π (pi) Divide

the circle into many triangular

segments The area of the triangles

is 12 times the sum of their bases

2πr (the circumference) times their

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9830920

height r

12 2 Surface Area of cylinder This can be imagined as unwrapping the

surface into a rectangle

Surface area of cone This can be achieved by divide the

surface of the cone into its triangles or

the surface of the cone into many thin

triangles The area of the triangles is 12

times the sum of their bases p times

their height

12 2

Surface Area and Volume of frustum of cone

h = vertical height of the frustum

l = slant height of the frustum

r 1 and r 2 are radii of the two bases (ends) of the frustum

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Type Measurement

983126983151983148983157983149983141 983151983142 983137 983142983154983157983155983156983157983149 983151983142 983137 983139983151983150983141 983123983148983137983150983156 983144983141983145983143983144983156 983151983142 983142983154983157983155983156983157983149 983151983142 983137 983139983151983150983141 983107983157983154983158983141983140 983155983157983154983142983137983139983141 983137983154983141983137 983151983142 983137 983142983154983157983155983156983157983149 983151983142 983137 983139983151983150983141 983124983151983156983137983148 983155983157983154983142983137983139983141 983137983154983141983137 983151983142 983142983154983157983155983156983157983149 983151983142 983137 983139983151983150983141

Statistics

Sno Term Description

1 Statistics Statistics is a broad mathematical disciplinewhich studies ways to collect summarize and

draw conclusions from data

2 Data A systematic record of facts or different values of

a quantity is called data

Data is of two types - Primary data and Secondary

data

Primary Data The data collected by a researcher

with a specific purpose in mind is called primarydata

Secondary Data The data gathered from a

source where it already exists is called secondary

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data

3 Features of data bull Statistics deals with collectionpresentation analysis and interpretation ofnumerical data

bull Arranging data in an order to study theirsalient features is called presentation ofdata

bull Data arranged in ascending or descendingorder is called arrayed data or an array

bull Range of the data is the differencebetween the maximum and the minimum

values of the observationsbull Table that shows the frequency of different

values in the given data is called afrequency distribution table

bull A frequency distribution table that showsthe frequency of each individual value in thegiven data is called an ungrouped frequencydistribution table

bull A table that shows the frequency of groupsof values in the given data is called agrouped frequency distribution table

bull

The groupings used to group the values ingiven data are called classes or class-intervals The number of values that eachclass contains is called the class size orclass width The lower value in a class iscalled the lower class limit The higher valuein a class is called the upper class limit

bull Class mark of a class is the mid value ofthe two limits of that class

bull A frequency distribution in which the upperlimit of one class differs from the lower limitof the succeeding class is called an

Inclusive or discontinuous FrequencyDistribution

bull A frequency distribution in which the upperlimit of one class coincides from the lowerlimit of the succeeding class is called anexclusive or continuous Frequency

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Distribution bull

4 Bar graph A bar graph is a pictorial representation of data inwhich rectangular bars of uniform width are drawnwith equal spacing between them on one axisusually the x axis The value of the variable isshown on the other axis that is the y axis

5 Histogram A histogram is a set of adjacent rectangles whoseareas are proportional to the frequencies of agiven continuous frequency distribution

6 Mean The mean value of a variable is defined as thesum of all the values of the variable divided by thenumber of values

4 sum

7 Median The median of a set of data values is the middle

value of the data set when it has been arranged inascending order That is from the smallest valueto the highest value

Median is calculated as

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983092983092

12 1 Where n is the number of values in the data

If the number of values in the data set is eventhen the median is the average of the two middlevalues

8 Mode Mode of a statistical data is the value of thatvariable which has the maximum frequency

Sno Term Description

1983117983141983137983150 983142983151983154

983125983150983143983154983151983157983152

983110983154983141983153983157983141983150983139983161 983156983137983138983148983141

Mean is given by

sum sum

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983110983154983141983153983157983141983150983139983161 983156983137983138983148983141

In these distribution it is assumed that frequency of each classinterval is centered around its mid-point ie class marks

2

983117983141983137983150 983139983137983150 983138983141 983139983137983148983139983157983148983137983156983141983140 983157983155983145983150983143 983156983144983154983141983141 983149983141983156983144983151983140

983137983081 983108983145983154983141983139983156 983149983141983156983144983151983140

sum

sum

983138983081 983105983155983155983157983149983141983140 983149983141983137983150 983149983141983156983144983151983140

sum sum

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983151983150983148983161

983092983093

Where

a=gt Assumed mean

di =gt xi ndasha

983139983081 983123983156983141983152 983140983141983158983145983137983156983145983151983150 983117983141983156983144983151983140

sum sum

Where

a=gt Assumed mean

ui =gt (xi ndasha)h

3983117983151983140983141 983142983151983154

983143983154983151983157983152983141983140

983142983154983141983153983157983141983150983139983161 983156983137983138983148983141

Modal class The class interval having highest frequency is

called the modal class and Mode is obtained using the modal

class

2

983127983144983141983154983141

l = lower limit of the modal classh = size of the class interval (assuming all class sizes tobe equal)f 1 = frequency of the modal classf 0 = frequency of the class preceding the modal classf 2 = frequency of the class succeeding the modal class

4983117983141983140983145983137983150 983151983142 983137

983143983154983151983157983152983141983140 983140983137983156983137

983142983154983141983153983157983141983150983139983161 983156983137983138983148983141

For the given data we need to have class interval frequency

distribution and cumulative frequency distribution

Median is calculated as

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983151983150983148983161

983092983094

2

983127983144983141983154983141

983148 983101 983148983151983159983141983154 983148983145983149983145983156 983151983142 983149983141983140983145983137983150 983139983148983137983155983155

983150 983101 983150983157983149983138983141983154 983151983142 983151983138983155983141983154983158983137983156983145983151983150983155

983139983142 983101 983139983157983149983157983148983137983156983145983158983141 983142983154983141983153983157983141983150983139983161 983151983142 983139983148983137983155983155 983152983154983141983139983141983140983145983150983143 983156983144983141 983149983141983140983145983137983150 983139983148983137983155983155

983142 983101 983142983154983141983153983157983141983150983139983161 983151983142 983149983141983140983145983137983150 983139983148983137983155983155

983144 983101 983139983148983137983155983155 983155983145983162983141 (983137983155983155983157983149983145983150983143 983139983148983137983155983155 983155983145983162983141 983156983151 983138983141 983141983153983157983137983148)

5983109983149983152983145983154983145983139983137983148

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983117983141983140983145983137983150

3 Median=Mode +2 Mean

ProbabilitySn

o

Term Description

1 Empirical

probability

It is a probability of event which is calculated based on

experiments

Example

A coin is tossed 1000 times we get 499 times head and

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501 times tail

So empirical or experimental probability of getting head is

calculated as

4991000 499

Empirical probability depends on experiment and

different will get different values based on the

experiment

2 Important point

about events

If the event A B C covers the entire possible outcome inthe experiment Then

P (A) +P (B) +P(C)=1

3 impossible

event

The probability of an event (U) which is impossible tooccur is 0 Such an event is called an impossible event

P (U)=0

4 Sure or certain

event

The probability of an event (X) which is sure (or certain)to occur is 1 Such an event is called a sure event or acertain event

P(X) =1

5 Probability of

any event

Probability of any event can be as

0 1

Sn

o

Term Description

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983092983096

1

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The theoretical probability or the classical probability of the event isdefined as

2 Elementary

events

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983120(983105)+983120(983106) +983120(983107)9831011

3 Complementar

y events

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983120(983105) +983120(Ᾱ)9831011

4 Sure or certain

event

The probability of an event (X) which is sure (or certain)

to occur is 1 Such an event is called a sure event or acertain event

P(X) =1

5 Probability of

any event

Probability of any event can be as

0 1

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983095

7 y=ax3 +bx

2+cx+d

where ane0

It can be of any

shape

It will cut the x-axis

at the most 3 times

Cubic Polynomial

8 ⋯ 991270 991270 991270

Where an ne0

It can be of any

shape

It will cut the x-axis

at the most n times

Polynomial of n

degree

Relation between coefficient and zeroes of the Polynomial

Sno Type of Polynomial General form Zerorsquos Relationship between Zerorsquos

and coefficients

1 Linear polynomial ax+b ane0 1

2 Quadratic ax +bx+c ane0 2

3 Cubic ax +bx +cx+d ane0 3

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983096

Formation of polynomial when the zeroes are given

Type ofpolynomial

Zerorsquos Polynomial Formed

Linear k=a (x-a)

Quadratic k1=a and

k2=b

(x-a)(x-b)

Or

x2-( a+b)x +ab

Or

x2

-( Sum of the zerorsquos)x +product of the zerorsquos

Cubic k1=a k2=b

and k3=c

(x-a)(x-b)(x-c)

Division algorithm for Polynomial

Letlsquos p(x) and q(x) are any two polynomial with q(x) ne0 then we can find polynomial s(x) and r(x) suchthat

P(x)=s(x) q(x) + r(x)

Where r(x) can be zero or degree of r(x) lt degree of g(x)

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983097

COORDINATE GEOMETRYSno Points

1 We require two perpendicular axes to locate a point in the plane One of

them is horizontal and other is Vertical

2 The plane is called Cartesian plane and axis are called the coordinates axis

3 The horizontal axis is called x-axis and Vertical axis is called Y-axis

4 The point of intersection of axis is called origin

5 The distance of a point from y axis is called x ndashcoordinate or abscissa and

the distance of the point from x ndashaxis is called y ndash coordinate or Ordinate

6 The distance of a point from y axis is called x ndashcoordinate or abscissa and

the distance of the point from x ndashaxis is called y ndash coordinate or Ordinate

7 The Origin has zero distance from both x-axis and y-axis so that its

abscissa and ordinate both are zero So the coordinate of the origin is (0

0)

8 A point on the x ndashaxis has zero distance from x-axis so coordinate of any

point on the x-axis will be (x 0)

9 A point on the y ndashaxis has zero distance from y-axis so coordinate of any

point on the y-axis will be (0 y)

10 The axes divide the Cartesian plane in to four parts These Four parts arecalled the quadrants

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The coordinates of the points in the four quadrants will have sign according to thebelow table

Quadrant x-coordinate y-coordinate

Ist Quadrant + +

IInd quadrant - +

IIIrd quadrant - -

IVth quadrant + -

Sno Terms Descriptions

1 Distance formula Distance between the points AB is given by

Distance of Point A from Origin

2 Section Formula A point P(xy) which divide the line segment AB in

the ratio m1 and m2 is given by

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The midpoint P is given by

3 Area of Triangle Area of triangle ABC of coordinates A(x 1y 1 )

B(x 2 y 2 ) and C(x 3y 3 )

12 For point AB and C to be collinear The value of A

should be zero

LINEAR EQUATIONS IN TWO

VARIABLESAn equation of the form ax + by + c = 0 where a b and c are real numbers such

that a andb are not both zero is called a linear equation in two variablesImportant points to Note

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Sno Points

1 A linear equation in two variable has infinite solutions

2 The graph of every linear equation in two variable is a straight line

3 x = 0 is the equation of the y-axis and y = 0 is the equation of the x-axis

4 The graph x=a is a line parallel to y -axis

5 The graph y=b is a line parallel to x -axis

6 An equation of the type y = mx represents a line passing through theorigin

7 Every point on the graph of a linear equation in two variables is a solution

of the linear

equation Moreover every solution of the linear equation is a point on the

graph

Sno Type of equation Mathematical

representation

Solutions

1 Linear equation in one Variable ax+b=0 ane0

a and b are real

number

One solution

2 Linear equation in two Variable ax+by+c=0 ane0 and

bne0

a b and c are real

number

Infinite solution

possible

3 Linear equation in three Variable ax+by+cz+d=0 ane0

bne0 and cne0

Infinite solution

possible

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a b c d are realnumber

Simultaneous pair of Linear equation

A pair of Linear equation in two variables

a1x+b1y+c1=0

a2x +b2y+c2=0

Graphically it is represented by two straight lines on Cartesian plane

Simultaneous pair of

Linear equation

Condition Graphical

representation

Algebraic

interpretation

a1x+b1y+c1=0

a2x +b2y+c2=0

Example

x-4y+14=0

3x+2y-14=0

Intersecting lines The

intersecting point

coordinate is the only

solution

One unique solution

only

a1x+b1y+c1=0

a2x +b2y+c2=0

Coincident lines Theany coordinate on the

line is the solution

Infinite solution

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Example

2x+4y=16

3x+6y=24

a1x+b1y+c1=0

a2x +b2y+c2=0

Example

2x+4y=6

4x+8y=18

Parallel Lines No solution

The graphical solution can be obtained by drawing the lines on the Cartesian plane

Algebraic Solution of system of Linear equation

Sno Type of method Working of method

1 Method of elimination by

substitution

1) Suppose the equation are

a1x+b1y+c1=0

a2x +b2y+c2=0

2) Find the value of variable of either x or y in other

variable term in first equation

3) Substitute the value of that variable in second

equation

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4) Now this is a linear equation in one variable Findthe value of the variable

5) Substitute this value in first equation and get the

second variable

2 Method of elimination by

equating the coefficients

1) Suppose the equation are

a1x+b1y+c1=0

a2x +b2y+c2=0

2) Find the LCM of a1 and a2 Let it k

3) Multiple the first equation by the value ka1

4) Multiple the first equation by the value ka2

4) Subtract the equation obtained This way one

variable will be eliminated and we can solve to get the

value of variable y

5) Substitute this value in first equation and get the

second variable

3 Cross Multiplication method 1) Suppose the equation are

a1x+b1y+c1=0

a2x +b2y+c2=0

2) This can be written as

3) This can be written as

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4) Value of x and y can be find using the

x =gt first and last expression

y=gt second and last expression

Quadratic EquationsSno Terms Descriptions

1 Quadratic Polynomial P(x) = ax +bx+c where ane0

2 Quadratic equation ax +bx+c =0 where ane0

3 Solution or root of the

Quadratic equation

A real number α is called the root or solution of the

quadratic equation if

aα2 +bα+c=0

4 zeroes of the

polynomial p(x)

The root of the quadratic equation are called zeroes

5 Maximum roots of

quadratic equations

We know from chapter two that a polynomial of

degree can have max two zeroes So a quadratic

equation can have maximum two roots

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6 Condition for realroots A quadratic equation has real roots if b

2

- 4ac gt 0

How to Solve Quadratic equation

Sno Method Working

1 factorization This method we factorize the equation by splitting themiddle term b

In ax2+bx+c=0

Example

6x2-x-2=0

1) First we need to multiple the coefficient a and cIn this

case =6X-2=-12

2) Splitting the middle term so that multiplication is 12

and difference is the coefficient b

6x2 +3x-4x-2=0

3x( 2x+1) -2(2x+1)=0

(3x-2) (2x+1)=0

3) Roots of the equation can be find equating the factorsto zero

3x-2=0 =gt x=32

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2x+1=0 =gt x=-12

2 Square method In this method we create square on LHS and RHS and

then find the value

ax2 +bx+c=0

1) x2 +(ba) x+(ca)=0

2) ( x+b2a)2 ndash(b2a)2 +(ca)=0

3) ( x+b2a)2=(b2-4ac)4a2

4) radic

Example

x2 +4x-5=0

1) (x+2)2 -4-5=0

2) (x+2)2=9

3) Roots of the equation can be find using square root on

both the sides

x+2 =-3 =gt x=-5

x+2=3=gt x=1

3 Quadratic

method

For quadratic equation

ax2 +bx+c=0

roots are given by radic radic

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For b2 -4ac gt 0 Quadratic equation has two real roots of

different value

For b2-4ac =0 quadratic equation has one real root

For b2-4ac lt 0 no real roots for quadratic equation

Nature of roots of Quadratic equation

Sno Condition Nature of roots

1 b -4ac gt 0 Two distinct real roots

2 b -4ac =0 One real root

3 b

2

-4ac lt 0 No real roots

Triangles

Sno Terms Descriptions

1 Congruence Two Geometric figure are said to be congruence ifthey are exactly same size and shapeSymbol used is cong Two angles are congruent if they are equalTwo circle are congruent if they have equal radiiTwo squares are congruent if the sides are equal

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2 Triangle Congruencebull

Two triangles are congruent if three sidesand three angles of one triangle is

congruent to the corresponding sides and

angles of the other

bull Corresponding sides are equal

AB=DE BC=EF AC=DF

bull Corresponding angles are equalang angang angang ang

bull We write this as

cong

bull The above six equalities are between the

corresponding parts of the two congruent

triangles In short form this is called

CPCT

bull We should keep the letters in correct order

on both sides

3 Inequalities in

Triangles

1) In a triangle angle opposite to longer side is

larger

2) In a triangle side opposite to larger angle is

larger

3) The sum of any two sides of the triangle is

greater than the third side

983105

983107983106

983108

983109 983110

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In triangle ABC

AB +BC gt AC

Different Criterion for Congruence of the triangles

N Criterion Description Figures and

expression

1 Side angle

Side (SAS)

congruence

bull Two triangles are congruent if the

two sides and included angles of

one triangle is equal to the two

sides and included angle

bull It is an axiom as it cannot be

proved so it is an accepted truth

bull ASS and SSA type two triangles

may not be congruent alwaysIf following

condition

983105

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983108

983107983106

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AB=DE BC=EFang ang

Then

cong

2 Angle side

angle (ASA)

congruence

bull Two triangles are congruent if the

two angles and included side of

one triangle is equal to the

corresponding angles and side

bull It is a theorem and can be proved

If following

condition

BC=EF

ang angang ang

Then

cong

3 Angle angle

side( AAS)

congruence

bull Two triangles are congruent if the

any two pair of angles and any

side of one triangle is equal to the

corresponding angles and side

bull It is a theorem and can be proved

983110983109

983108

983107983106

983105

983110983109

983108

983107983106

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If followingcondition

BC=EF

ang angang ang

Then

cong

4 Side-Side-Side

(SSS)congruence

bull Two triangles are congruent if the

three sides of one triangle is equalto the three sides of the another

If following

condition

BC=EFAB=DEDF

=AC

Then

cong

5 Right angle ndash

hypotenuse-

side(RHS)

bull Two right triangles are congruent if

the hypotenuse and a side of the

one triangle are equal to

983110983109

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983107983106

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983105 983108

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congruence corresponding hypotenuse and sideof the another

If following

condition

AC=DFBC=EF

Then

cong

Some Important points on Triangles

Terms Description

Orthocenter Point of intersection of the three altitudeof the triangle

Equilateral triangle whose all sides are equal and allangles are equal to 600

Median A line Segment joining the corner of thetriangle to the midpoint of the oppositeside of the triangle

Altitude A line Segment from the corner of thetriangle and perpendicular to theopposite side of the triangle

Isosceles A triangle whose two sides are equalCentroid Point of intersection of the three median

of the triangle is called the centroid of

the triangleIn center All the angle bisector of the trianglepasses through same point

Circumcenter The perpendicular bisector of the sidesof the triangles passes through samepoint

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Scalene triangle Triangle having no equal angles and noequal sidesRight Triangle Right triangle has one angle equal to 900 Obtuse Triangle One angle is obtuse angle while other

two are acute anglesAcute Triangle All the angles are acute

Similarity of Triangles

Sno Points

1 Two figures having the same shape but not necessarily the same size are called similarfigures

2 All the congruent figures are similar but the converse is not true

3 If a line is drawn parallel to one side of a triangle to intersect the other two sides in

distinct points then the other two sides are divided in the same ratio

4 If a line divides any two sides of a triangle in the same ratio then the line is parallel tothe third side

Different Criterion for Similarity of the triangles

N Criterion Description Expression

1 Angle Angle

angle(AAA)

similarity

bull Two triangles are similar if

corresponding angle are equal

If following

condition

ang ang

ang ang

ang ang

Then

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2983094

Then 983166

2 Angle angle

(AA) similarity

bull Two triangles are similar if the two

corresponding angles are equal as

by angle property third angle will

be also equal

If following

condition

ang ang

ang ang

Then

ang ang

Then

Then

983166

3 Side side

side(SSS)

Similarity

Two triangles are similar if the

sides of one triangle is

proportional to the sides of other

triangle

If following

condition

Then

ang ang

ang ang

ang ang

Then

cong

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983151983150983148983161

2983095

4 Side-Angle-Side (SAS)

similarity

bull

Two triangles are similar if the oneangle of a triangle is equal to one

angle of other triangles and sides

including that angle is proportional

If followingcondition

And ang ang

Then

cong

Area of Similar triangles

If the two triangle ABC and DEF are similar

cong

Then

Pythagoras Theorem

Sno Points

1 If a perpendicular is drawn from the vertex of the right angle of a right triangle to thehypotenuse then the triangles on both sides of the perpendicular are similar to the

whole triangle and also to each other

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983151983150983148983161

2983096

2 In a right triangle the square of the hypotenuse is equal to the sum of the squares of the

other two sides (Pythagoras Theorem)

(hyp)2 =(base)

2 + (perp)

2

3 If in a triangle square of one side is equal to the sum of the squares of the other twosides then the angle opposite the first side is a right angle

Arithmetic Progression

Sno Terms Descriptions

1 ArithmeticProgression

An arithmetic progression is a sequence ofnumbers such that the difference of any two

successive members is a constant

Examples

1) 1591317hellip

2) 12345hellip

2 common difference of

the AP

the difference between any successive members is a

constant and it is called the common difference of AP

1) If a1 a2a3a4a5 are the terms in AP then

D=a2 -a1 =a3 - a2 =a4 ndash a3=a5 ndasha4

2) We can represent the general form of AP inthe form

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983151983150983148983161

2983097

aa+da+2da+3da+4d

Where a is first term and d is the commondifference

3 nth term of Arithmetic Progression

n term = a + (n - 1)d

4 Sum of nth item in ArithmeticProgression

Sn =(n2)[a + (n-1)d]

Or

Sn =(n2)[t1+ tn]

Trigonometry

Sno Terms Descriptions

1 What is

Trigonometry

Trigonometry from Greek trigotildenon triangle and

metron measure) is a branch of mathematics

that studies relationships involving lengths and

angles of triangles The field emerged during the

3rd century BC from applications of geometry to

astronomical studies

Trigonometry is most simply associated with

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9830910

planar right angle triangles (each of which is atwo-dimensional triangle with one angle equal to

90 degrees) The applicability to non-right-angle

triangles exists but since any non-right-angle

triangle (on a flat plane) can be bisected to

create two right-angle triangles most problems

can be reduced to calculations on right-angle

triangles Thus the majority of applications relate

to right-angle triangles

2 Trigonometric

Ratiorsquos

In a right angle triangle ABC where B=90deg

We can define following term for angle A

Base Side adjacent to angle

Perpendicular Side Opposite of angleHypotenuse Side opposite to right angle

We can define the trigonometric ratios for

angle A as

sin A= PerpendicularHypotenuse =BCAC

cosec A= HypotenusePerpendicular

=ACBC

cos A= BaseHypotenuse =ABAC

sec A= HypotenuseBase=ACAB

tan A= PerpendicularBase =BCAB

cot A= BasePerpendicular=ABBC

Notice that each ratio in the right-hand column is

the inverse or the reciprocal of the ratio in the

left-hand column

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9830911

3 Reciprocal offunctions The reciprocal of sin A is cosec A and vice-versa

The reciprocal of cos A is sec A

And the reciprocal of tan A is cot A

These are valid for acute angles

We can define tan A = sin Acos A

And Cot A =cos A Sin A

4 Value of of sin and

cos

Is always less 1

5 Trigonometric ration

from another angle

We can define the trigonometric ratios for angle Cas

sin C= PerpendicularHypotenuse =ABACcosec C= HypotenusePerpendicular =ACABBcos C= BaseHypotenuse =BCACsec C= HypotenuseBase=ACBCtan A= PerpendicularBase =ABBCcot A= BasePerpendicular=BCAB

6 Trigonometric ratios

of complimentary

Sin (90-A) =cos(A)

Cos(90-A) = sin A

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9830912

angles Tan(90-A) =cot A

Sec(90-A)= cosec A

Cosec (90-A) =sec A

Cot(90- A) =tan A7 Trigonometric

identities

Sin2 A + cos2 A =0

1 + tan2 A =sec2 A

1 + cot2 A =cosec2 A

Trigonometric Ratios of Common angles

We can find the values of trigonometric ratiorsquos various angle

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Area of Circles

Sno Points

1 A circle is a collection of all the points in a plane which are equidistant

from a fixed point in the plane

2 Equal chords of a circle (or of congruent circles) subtend equal angles at

the center

3 If the angles subtended by two chords of a circle (or of congruent circles)

at the center (corresponding center) are equal the chords are equal

4 The perpendicular from the center of a circle to a chord bisects the chord

5 The line drawn through the center of a circle to bisect a chord is

perpendicular to the chord

6 There is one and only one circle passing through three non-collinear points

7 Equal chords of a circle (or of congruent circles) are equidistant from the

center (or corresponding centers)

8 Chords equidistant from the center (or corresponding centers) of a circle

(or of congruent circles) are equal

9 If two arcs of a circle are congruent then their corresponding chords are

equal and conversely if two chords of a circle are equal then their

corresponding arcs (minor major) are congruent

10 Congruent arcs of a circle subtend equal angles at the center

11 The angle subtended by an arc at the center is double the angle subtended

by it at any point on the remaining part of the circle

12 Angles in the same segment of a circle are equal

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983091983092

13 Angle in a semicircle is a right angle

14 If a line segment joining two points subtends equal angles at two other

points lying on the same side of the line containing the line segment the

four points lie on a circle

15 The sum of either pair of opposite angles of a cyclic quadrilateral is 180deg

16 If the sum of a pair of opposite angles of a quadrilateral is 180deg then the

quadrilateral is cyclic

Sno Terms Descriptions

1 Circumference of a circle 2 π r

2 Area of circle π r 2

3 Length of the arc of

the sector of angle

Length of the arc of the sector of angle θ 2

4 Area of the sector of

angle

Area of the sector of angle θ

5 Area of segment of a

circle

Area of the corresponding sector ndash Area of the

corresponding triangle

Surface Area and Volume

Sno Term Description

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983091983093

1 Mensuration It is branch of mathematics which is concernedabout the measurement of length area and

Volume of plane and Solid figure

2 Perimeter a)The perimeter of plane figure is defined as thelength of the boundaryb)It units is same as that of length ie m cmkm

3 Area a)The area of the plane figure is the surfaceenclosed by its boundaryb) It unit is square of length unit ie m2 km2

4 Volume Volume is the measure of the amount of spaceinside of a solid figure like a cube ball cylinderor pyramid Its units are always cubic that isthe number of little element cubes that fit insidethe figure

Volume Unit conversion

1 cm3 1mL 1000 mm3

1 Litre 1000ml 1000 cm3

1 m3 106 cm3 1000 L

1 dm3 1000 cm3 1 L

Surface Area and Volume of Cube and Cuboid

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Type Measurement

Surface Area of Cuboid of Length L

Breadth B and Height H

2(LB + BH + LH)

Lateral surface area of the cuboids 2( L + B ) H

Diagonal of the cuboids Volume of a cuboids LBH

Length of all 12 edges of the cuboids 4 (L+B+H)

Surface Area of Cube of side L 6L2

Lateral surface area of the cube 4L2

Diagonal of the cube radic3 Volume of a cube L3

Surface Area and Volume of Right circular cylinder

Radius The radius (r) of the circular base is called the radius of the

cylinder

Height The length of the axis of the cylinder is called the height (h) of the

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983091983095

cylinder

Lateral

Surface

The curved surface joining the two base of a right circular cylinder is

called Lateral Surface

Type Measurement

Curved or lateral Surface Area of

cylinder

2πrh

Total surface area of cylinder 2πr (h+r)

Volume of Cylinder π r2h

Surface Area and Volume of Right circular cone

Radius The radius (r) of the circular base is called the radius of the

cone

Height The length of the line segment joining the vertex to the center of

base is called the height (h) of the cone

Slant

Height

The length of the segment joining the vertex to any point on the

circular edge of the base is called the slant height (L) of the cone

Lateral

surface

Area

The curved surface joining the base and uppermost point of a right

circular cone is called Lateral Surface

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983091983096

Type Measurement

Curved or lateral Surface Area of

cone

πrL

Total surface area of cone πr (L+r)

Volume of Cone

Surface Area and Volume of sphere and hemisphere

Sphere A sphere can also be considered as a solid obtained on

rotating a circle About its diameter

Hemisphere A plane through the centre of the sphere divides the sphere into two

equal parts each of which is called a hemisphere

radius The radius of the circle by which it is formed

SphericalShell

The difference of two solid concentric spheres is called a sphericalshell

Lateral

Surface

Area for

Total surface area of the sphere

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983091983097

Sphere

Lateral

Surface

area of

Hemisphere

It is the curved surface area leaving the circular base

Type Measurement

Surface area of Sphere 4πr2

Volume of Sphere 43 Curved Surface area of hemisphere 2πr2

Total Surface area of hemisphere 3πr2

Volume of hemisphere 23 Volume of the spherical shell whose

outer and inner radii and lsquoRrsquo and lsquorrsquorespectively

43

How the Surface area and Volume are determined

Area of Circle The circumference of a circle is 2πr

This is the definition of π (pi) Divide

the circle into many triangular

segments The area of the triangles

is 12 times the sum of their bases

2πr (the circumference) times their

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9830920

height r

12 2 Surface Area of cylinder This can be imagined as unwrapping the

surface into a rectangle

Surface area of cone This can be achieved by divide the

surface of the cone into its triangles or

the surface of the cone into many thin

triangles The area of the triangles is 12

times the sum of their bases p times

their height

12 2

Surface Area and Volume of frustum of cone

h = vertical height of the frustum

l = slant height of the frustum

r 1 and r 2 are radii of the two bases (ends) of the frustum

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9830921

Type Measurement

983126983151983148983157983149983141 983151983142 983137 983142983154983157983155983156983157983149 983151983142 983137 983139983151983150983141 983123983148983137983150983156 983144983141983145983143983144983156 983151983142 983142983154983157983155983156983157983149 983151983142 983137 983139983151983150983141 983107983157983154983158983141983140 983155983157983154983142983137983139983141 983137983154983141983137 983151983142 983137 983142983154983157983155983156983157983149 983151983142 983137 983139983151983150983141 983124983151983156983137983148 983155983157983154983142983137983139983141 983137983154983141983137 983151983142 983142983154983157983155983156983157983149 983151983142 983137 983139983151983150983141

Statistics

Sno Term Description

1 Statistics Statistics is a broad mathematical disciplinewhich studies ways to collect summarize and

draw conclusions from data

2 Data A systematic record of facts or different values of

a quantity is called data

Data is of two types - Primary data and Secondary

data

Primary Data The data collected by a researcher

with a specific purpose in mind is called primarydata

Secondary Data The data gathered from a

source where it already exists is called secondary

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983151983150983148983161

9830922

data

3 Features of data bull Statistics deals with collectionpresentation analysis and interpretation ofnumerical data

bull Arranging data in an order to study theirsalient features is called presentation ofdata

bull Data arranged in ascending or descendingorder is called arrayed data or an array

bull Range of the data is the differencebetween the maximum and the minimum

values of the observationsbull Table that shows the frequency of different

values in the given data is called afrequency distribution table

bull A frequency distribution table that showsthe frequency of each individual value in thegiven data is called an ungrouped frequencydistribution table

bull A table that shows the frequency of groupsof values in the given data is called agrouped frequency distribution table

bull

The groupings used to group the values ingiven data are called classes or class-intervals The number of values that eachclass contains is called the class size orclass width The lower value in a class iscalled the lower class limit The higher valuein a class is called the upper class limit

bull Class mark of a class is the mid value ofthe two limits of that class

bull A frequency distribution in which the upperlimit of one class differs from the lower limitof the succeeding class is called an

Inclusive or discontinuous FrequencyDistribution

bull A frequency distribution in which the upperlimit of one class coincides from the lowerlimit of the succeeding class is called anexclusive or continuous Frequency

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983151983150983148983161

983092983091

Distribution bull

4 Bar graph A bar graph is a pictorial representation of data inwhich rectangular bars of uniform width are drawnwith equal spacing between them on one axisusually the x axis The value of the variable isshown on the other axis that is the y axis

5 Histogram A histogram is a set of adjacent rectangles whoseareas are proportional to the frequencies of agiven continuous frequency distribution

6 Mean The mean value of a variable is defined as thesum of all the values of the variable divided by thenumber of values

4 sum

7 Median The median of a set of data values is the middle

value of the data set when it has been arranged inascending order That is from the smallest valueto the highest value

Median is calculated as

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983151983150983148983161

983092983092

12 1 Where n is the number of values in the data

If the number of values in the data set is eventhen the median is the average of the two middlevalues

8 Mode Mode of a statistical data is the value of thatvariable which has the maximum frequency

Sno Term Description

1983117983141983137983150 983142983151983154

983125983150983143983154983151983157983152

983110983154983141983153983157983141983150983139983161 983156983137983138983148983141

Mean is given by

sum sum

2983117983141983137983150 983142983151983154 983143983154983151983157983152

983110983154983141983153983157983141983150983139983161 983156983137983138983148983141

In these distribution it is assumed that frequency of each classinterval is centered around its mid-point ie class marks

2

983117983141983137983150 983139983137983150 983138983141 983139983137983148983139983157983148983137983156983141983140 983157983155983145983150983143 983156983144983154983141983141 983149983141983156983144983151983140

983137983081 983108983145983154983141983139983156 983149983141983156983144983151983140

sum

sum

983138983081 983105983155983155983157983149983141983140 983149983141983137983150 983149983141983156983144983151983140

sum sum

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983151983150983148983161

983092983093

Where

a=gt Assumed mean

di =gt xi ndasha

983139983081 983123983156983141983152 983140983141983158983145983137983156983145983151983150 983117983141983156983144983151983140

sum sum

Where

a=gt Assumed mean

ui =gt (xi ndasha)h

3983117983151983140983141 983142983151983154

983143983154983151983157983152983141983140

983142983154983141983153983157983141983150983139983161 983156983137983138983148983141

Modal class The class interval having highest frequency is

called the modal class and Mode is obtained using the modal

class

2

983127983144983141983154983141

l = lower limit of the modal classh = size of the class interval (assuming all class sizes tobe equal)f 1 = frequency of the modal classf 0 = frequency of the class preceding the modal classf 2 = frequency of the class succeeding the modal class

4983117983141983140983145983137983150 983151983142 983137

983143983154983151983157983152983141983140 983140983137983156983137

983142983154983141983153983157983141983150983139983161 983156983137983138983148983141

For the given data we need to have class interval frequency

distribution and cumulative frequency distribution

Median is calculated as

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983151983150983148983161

983092983094

2

983127983144983141983154983141

983148 983101 983148983151983159983141983154 983148983145983149983145983156 983151983142 983149983141983140983145983137983150 983139983148983137983155983155

983150 983101 983150983157983149983138983141983154 983151983142 983151983138983155983141983154983158983137983156983145983151983150983155

983139983142 983101 983139983157983149983157983148983137983156983145983158983141 983142983154983141983153983157983141983150983139983161 983151983142 983139983148983137983155983155 983152983154983141983139983141983140983145983150983143 983156983144983141 983149983141983140983145983137983150 983139983148983137983155983155

983142 983101 983142983154983141983153983157983141983150983139983161 983151983142 983149983141983140983145983137983150 983139983148983137983155983155

983144 983101 983139983148983137983155983155 983155983145983162983141 (983137983155983155983157983149983145983150983143 983139983148983137983155983155 983155983145983162983141 983156983151 983138983141 983141983153983157983137983148)

5983109983149983152983145983154983145983139983137983148

983110983151983154983149983157983148983137 983138983141983156983159983141983141983150

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983117983141983140983145983137983150

3 Median=Mode +2 Mean

ProbabilitySn

o

Term Description

1 Empirical

probability

It is a probability of event which is calculated based on

experiments

Example

A coin is tossed 1000 times we get 499 times head and

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983092983095

501 times tail

So empirical or experimental probability of getting head is

calculated as

4991000 499

Empirical probability depends on experiment and

different will get different values based on the

experiment

2 Important point

about events

If the event A B C covers the entire possible outcome inthe experiment Then

P (A) +P (B) +P(C)=1

3 impossible

event

The probability of an event (U) which is impossible tooccur is 0 Such an event is called an impossible event

P (U)=0

4 Sure or certain

event

The probability of an event (X) which is sure (or certain)to occur is 1 Such an event is called a sure event or acertain event

P(X) =1

5 Probability of

any event

Probability of any event can be as

0 1

Sn

o

Term Description

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983092983096

1

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The theoretical probability or the classical probability of the event isdefined as

2 Elementary

events

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983120(983105)+983120(983106) +983120(983107)9831011

3 Complementar

y events

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983120(983105) +983120(Ᾱ)9831011

4 Sure or certain

event

The probability of an event (X) which is sure (or certain)

to occur is 1 Such an event is called a sure event or acertain event

P(X) =1

5 Probability of

any event

Probability of any event can be as

0 1

Page 9: Maths Formula Class10

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983096

Formation of polynomial when the zeroes are given

Type ofpolynomial

Zerorsquos Polynomial Formed

Linear k=a (x-a)

Quadratic k1=a and

k2=b

(x-a)(x-b)

Or

x2-( a+b)x +ab

Or

x2

-( Sum of the zerorsquos)x +product of the zerorsquos

Cubic k1=a k2=b

and k3=c

(x-a)(x-b)(x-c)

Division algorithm for Polynomial

Letlsquos p(x) and q(x) are any two polynomial with q(x) ne0 then we can find polynomial s(x) and r(x) suchthat

P(x)=s(x) q(x) + r(x)

Where r(x) can be zero or degree of r(x) lt degree of g(x)

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983097

COORDINATE GEOMETRYSno Points

1 We require two perpendicular axes to locate a point in the plane One of

them is horizontal and other is Vertical

2 The plane is called Cartesian plane and axis are called the coordinates axis

3 The horizontal axis is called x-axis and Vertical axis is called Y-axis

4 The point of intersection of axis is called origin

5 The distance of a point from y axis is called x ndashcoordinate or abscissa and

the distance of the point from x ndashaxis is called y ndash coordinate or Ordinate

6 The distance of a point from y axis is called x ndashcoordinate or abscissa and

the distance of the point from x ndashaxis is called y ndash coordinate or Ordinate

7 The Origin has zero distance from both x-axis and y-axis so that its

abscissa and ordinate both are zero So the coordinate of the origin is (0

0)

8 A point on the x ndashaxis has zero distance from x-axis so coordinate of any

point on the x-axis will be (x 0)

9 A point on the y ndashaxis has zero distance from y-axis so coordinate of any

point on the y-axis will be (0 y)

10 The axes divide the Cartesian plane in to four parts These Four parts arecalled the quadrants

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10

The coordinates of the points in the four quadrants will have sign according to thebelow table

Quadrant x-coordinate y-coordinate

Ist Quadrant + +

IInd quadrant - +

IIIrd quadrant - -

IVth quadrant + -

Sno Terms Descriptions

1 Distance formula Distance between the points AB is given by

Distance of Point A from Origin

2 Section Formula A point P(xy) which divide the line segment AB in

the ratio m1 and m2 is given by

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11

The midpoint P is given by

3 Area of Triangle Area of triangle ABC of coordinates A(x 1y 1 )

B(x 2 y 2 ) and C(x 3y 3 )

12 For point AB and C to be collinear The value of A

should be zero

LINEAR EQUATIONS IN TWO

VARIABLESAn equation of the form ax + by + c = 0 where a b and c are real numbers such

that a andb are not both zero is called a linear equation in two variablesImportant points to Note

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983151983150983148983161

12

Sno Points

1 A linear equation in two variable has infinite solutions

2 The graph of every linear equation in two variable is a straight line

3 x = 0 is the equation of the y-axis and y = 0 is the equation of the x-axis

4 The graph x=a is a line parallel to y -axis

5 The graph y=b is a line parallel to x -axis

6 An equation of the type y = mx represents a line passing through theorigin

7 Every point on the graph of a linear equation in two variables is a solution

of the linear

equation Moreover every solution of the linear equation is a point on the

graph

Sno Type of equation Mathematical

representation

Solutions

1 Linear equation in one Variable ax+b=0 ane0

a and b are real

number

One solution

2 Linear equation in two Variable ax+by+c=0 ane0 and

bne0

a b and c are real

number

Infinite solution

possible

3 Linear equation in three Variable ax+by+cz+d=0 ane0

bne0 and cne0

Infinite solution

possible

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1983091

a b c d are realnumber

Simultaneous pair of Linear equation

A pair of Linear equation in two variables

a1x+b1y+c1=0

a2x +b2y+c2=0

Graphically it is represented by two straight lines on Cartesian plane

Simultaneous pair of

Linear equation

Condition Graphical

representation

Algebraic

interpretation

a1x+b1y+c1=0

a2x +b2y+c2=0

Example

x-4y+14=0

3x+2y-14=0

Intersecting lines The

intersecting point

coordinate is the only

solution

One unique solution

only

a1x+b1y+c1=0

a2x +b2y+c2=0

Coincident lines Theany coordinate on the

line is the solution

Infinite solution

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Example

2x+4y=16

3x+6y=24

a1x+b1y+c1=0

a2x +b2y+c2=0

Example

2x+4y=6

4x+8y=18

Parallel Lines No solution

The graphical solution can be obtained by drawing the lines on the Cartesian plane

Algebraic Solution of system of Linear equation

Sno Type of method Working of method

1 Method of elimination by

substitution

1) Suppose the equation are

a1x+b1y+c1=0

a2x +b2y+c2=0

2) Find the value of variable of either x or y in other

variable term in first equation

3) Substitute the value of that variable in second

equation

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4) Now this is a linear equation in one variable Findthe value of the variable

5) Substitute this value in first equation and get the

second variable

2 Method of elimination by

equating the coefficients

1) Suppose the equation are

a1x+b1y+c1=0

a2x +b2y+c2=0

2) Find the LCM of a1 and a2 Let it k

3) Multiple the first equation by the value ka1

4) Multiple the first equation by the value ka2

4) Subtract the equation obtained This way one

variable will be eliminated and we can solve to get the

value of variable y

5) Substitute this value in first equation and get the

second variable

3 Cross Multiplication method 1) Suppose the equation are

a1x+b1y+c1=0

a2x +b2y+c2=0

2) This can be written as

3) This can be written as

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4) Value of x and y can be find using the

x =gt first and last expression

y=gt second and last expression

Quadratic EquationsSno Terms Descriptions

1 Quadratic Polynomial P(x) = ax +bx+c where ane0

2 Quadratic equation ax +bx+c =0 where ane0

3 Solution or root of the

Quadratic equation

A real number α is called the root or solution of the

quadratic equation if

aα2 +bα+c=0

4 zeroes of the

polynomial p(x)

The root of the quadratic equation are called zeroes

5 Maximum roots of

quadratic equations

We know from chapter two that a polynomial of

degree can have max two zeroes So a quadratic

equation can have maximum two roots

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6 Condition for realroots A quadratic equation has real roots if b

2

- 4ac gt 0

How to Solve Quadratic equation

Sno Method Working

1 factorization This method we factorize the equation by splitting themiddle term b

In ax2+bx+c=0

Example

6x2-x-2=0

1) First we need to multiple the coefficient a and cIn this

case =6X-2=-12

2) Splitting the middle term so that multiplication is 12

and difference is the coefficient b

6x2 +3x-4x-2=0

3x( 2x+1) -2(2x+1)=0

(3x-2) (2x+1)=0

3) Roots of the equation can be find equating the factorsto zero

3x-2=0 =gt x=32

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2x+1=0 =gt x=-12

2 Square method In this method we create square on LHS and RHS and

then find the value

ax2 +bx+c=0

1) x2 +(ba) x+(ca)=0

2) ( x+b2a)2 ndash(b2a)2 +(ca)=0

3) ( x+b2a)2=(b2-4ac)4a2

4) radic

Example

x2 +4x-5=0

1) (x+2)2 -4-5=0

2) (x+2)2=9

3) Roots of the equation can be find using square root on

both the sides

x+2 =-3 =gt x=-5

x+2=3=gt x=1

3 Quadratic

method

For quadratic equation

ax2 +bx+c=0

roots are given by radic radic

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For b2 -4ac gt 0 Quadratic equation has two real roots of

different value

For b2-4ac =0 quadratic equation has one real root

For b2-4ac lt 0 no real roots for quadratic equation

Nature of roots of Quadratic equation

Sno Condition Nature of roots

1 b -4ac gt 0 Two distinct real roots

2 b -4ac =0 One real root

3 b

2

-4ac lt 0 No real roots

Triangles

Sno Terms Descriptions

1 Congruence Two Geometric figure are said to be congruence ifthey are exactly same size and shapeSymbol used is cong Two angles are congruent if they are equalTwo circle are congruent if they have equal radiiTwo squares are congruent if the sides are equal

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20

2 Triangle Congruencebull

Two triangles are congruent if three sidesand three angles of one triangle is

congruent to the corresponding sides and

angles of the other

bull Corresponding sides are equal

AB=DE BC=EF AC=DF

bull Corresponding angles are equalang angang angang ang

bull We write this as

cong

bull The above six equalities are between the

corresponding parts of the two congruent

triangles In short form this is called

CPCT

bull We should keep the letters in correct order

on both sides

3 Inequalities in

Triangles

1) In a triangle angle opposite to longer side is

larger

2) In a triangle side opposite to larger angle is

larger

3) The sum of any two sides of the triangle is

greater than the third side

983105

983107983106

983108

983109 983110

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21

In triangle ABC

AB +BC gt AC

Different Criterion for Congruence of the triangles

N Criterion Description Figures and

expression

1 Side angle

Side (SAS)

congruence

bull Two triangles are congruent if the

two sides and included angles of

one triangle is equal to the two

sides and included angle

bull It is an axiom as it cannot be

proved so it is an accepted truth

bull ASS and SSA type two triangles

may not be congruent alwaysIf following

condition

983105

983110983109

983108

983107983106

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22

AB=DE BC=EFang ang

Then

cong

2 Angle side

angle (ASA)

congruence

bull Two triangles are congruent if the

two angles and included side of

one triangle is equal to the

corresponding angles and side

bull It is a theorem and can be proved

If following

condition

BC=EF

ang angang ang

Then

cong

3 Angle angle

side( AAS)

congruence

bull Two triangles are congruent if the

any two pair of angles and any

side of one triangle is equal to the

corresponding angles and side

bull It is a theorem and can be proved

983110983109

983108

983107983106

983105

983110983109

983108

983107983106

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If followingcondition

BC=EF

ang angang ang

Then

cong

4 Side-Side-Side

(SSS)congruence

bull Two triangles are congruent if the

three sides of one triangle is equalto the three sides of the another

If following

condition

BC=EFAB=DEDF

=AC

Then

cong

5 Right angle ndash

hypotenuse-

side(RHS)

bull Two right triangles are congruent if

the hypotenuse and a side of the

one triangle are equal to

983110983109

983108

983107983106

983105

983105 983108

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congruence corresponding hypotenuse and sideof the another

If following

condition

AC=DFBC=EF

Then

cong

Some Important points on Triangles

Terms Description

Orthocenter Point of intersection of the three altitudeof the triangle

Equilateral triangle whose all sides are equal and allangles are equal to 600

Median A line Segment joining the corner of thetriangle to the midpoint of the oppositeside of the triangle

Altitude A line Segment from the corner of thetriangle and perpendicular to theopposite side of the triangle

Isosceles A triangle whose two sides are equalCentroid Point of intersection of the three median

of the triangle is called the centroid of

the triangleIn center All the angle bisector of the trianglepasses through same point

Circumcenter The perpendicular bisector of the sidesof the triangles passes through samepoint

983106 983107 983109 983110

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Scalene triangle Triangle having no equal angles and noequal sidesRight Triangle Right triangle has one angle equal to 900 Obtuse Triangle One angle is obtuse angle while other

two are acute anglesAcute Triangle All the angles are acute

Similarity of Triangles

Sno Points

1 Two figures having the same shape but not necessarily the same size are called similarfigures

2 All the congruent figures are similar but the converse is not true

3 If a line is drawn parallel to one side of a triangle to intersect the other two sides in

distinct points then the other two sides are divided in the same ratio

4 If a line divides any two sides of a triangle in the same ratio then the line is parallel tothe third side

Different Criterion for Similarity of the triangles

N Criterion Description Expression

1 Angle Angle

angle(AAA)

similarity

bull Two triangles are similar if

corresponding angle are equal

If following

condition

ang ang

ang ang

ang ang

Then

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Then 983166

2 Angle angle

(AA) similarity

bull Two triangles are similar if the two

corresponding angles are equal as

by angle property third angle will

be also equal

If following

condition

ang ang

ang ang

Then

ang ang

Then

Then

983166

3 Side side

side(SSS)

Similarity

Two triangles are similar if the

sides of one triangle is

proportional to the sides of other

triangle

If following

condition

Then

ang ang

ang ang

ang ang

Then

cong

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4 Side-Angle-Side (SAS)

similarity

bull

Two triangles are similar if the oneangle of a triangle is equal to one

angle of other triangles and sides

including that angle is proportional

If followingcondition

And ang ang

Then

cong

Area of Similar triangles

If the two triangle ABC and DEF are similar

cong

Then

Pythagoras Theorem

Sno Points

1 If a perpendicular is drawn from the vertex of the right angle of a right triangle to thehypotenuse then the triangles on both sides of the perpendicular are similar to the

whole triangle and also to each other

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2 In a right triangle the square of the hypotenuse is equal to the sum of the squares of the

other two sides (Pythagoras Theorem)

(hyp)2 =(base)

2 + (perp)

2

3 If in a triangle square of one side is equal to the sum of the squares of the other twosides then the angle opposite the first side is a right angle

Arithmetic Progression

Sno Terms Descriptions

1 ArithmeticProgression

An arithmetic progression is a sequence ofnumbers such that the difference of any two

successive members is a constant

Examples

1) 1591317hellip

2) 12345hellip

2 common difference of

the AP

the difference between any successive members is a

constant and it is called the common difference of AP

1) If a1 a2a3a4a5 are the terms in AP then

D=a2 -a1 =a3 - a2 =a4 ndash a3=a5 ndasha4

2) We can represent the general form of AP inthe form

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aa+da+2da+3da+4d

Where a is first term and d is the commondifference

3 nth term of Arithmetic Progression

n term = a + (n - 1)d

4 Sum of nth item in ArithmeticProgression

Sn =(n2)[a + (n-1)d]

Or

Sn =(n2)[t1+ tn]

Trigonometry

Sno Terms Descriptions

1 What is

Trigonometry

Trigonometry from Greek trigotildenon triangle and

metron measure) is a branch of mathematics

that studies relationships involving lengths and

angles of triangles The field emerged during the

3rd century BC from applications of geometry to

astronomical studies

Trigonometry is most simply associated with

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planar right angle triangles (each of which is atwo-dimensional triangle with one angle equal to

90 degrees) The applicability to non-right-angle

triangles exists but since any non-right-angle

triangle (on a flat plane) can be bisected to

create two right-angle triangles most problems

can be reduced to calculations on right-angle

triangles Thus the majority of applications relate

to right-angle triangles

2 Trigonometric

Ratiorsquos

In a right angle triangle ABC where B=90deg

We can define following term for angle A

Base Side adjacent to angle

Perpendicular Side Opposite of angleHypotenuse Side opposite to right angle

We can define the trigonometric ratios for

angle A as

sin A= PerpendicularHypotenuse =BCAC

cosec A= HypotenusePerpendicular

=ACBC

cos A= BaseHypotenuse =ABAC

sec A= HypotenuseBase=ACAB

tan A= PerpendicularBase =BCAB

cot A= BasePerpendicular=ABBC

Notice that each ratio in the right-hand column is

the inverse or the reciprocal of the ratio in the

left-hand column

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3 Reciprocal offunctions The reciprocal of sin A is cosec A and vice-versa

The reciprocal of cos A is sec A

And the reciprocal of tan A is cot A

These are valid for acute angles

We can define tan A = sin Acos A

And Cot A =cos A Sin A

4 Value of of sin and

cos

Is always less 1

5 Trigonometric ration

from another angle

We can define the trigonometric ratios for angle Cas

sin C= PerpendicularHypotenuse =ABACcosec C= HypotenusePerpendicular =ACABBcos C= BaseHypotenuse =BCACsec C= HypotenuseBase=ACBCtan A= PerpendicularBase =ABBCcot A= BasePerpendicular=BCAB

6 Trigonometric ratios

of complimentary

Sin (90-A) =cos(A)

Cos(90-A) = sin A

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angles Tan(90-A) =cot A

Sec(90-A)= cosec A

Cosec (90-A) =sec A

Cot(90- A) =tan A7 Trigonometric

identities

Sin2 A + cos2 A =0

1 + tan2 A =sec2 A

1 + cot2 A =cosec2 A

Trigonometric Ratios of Common angles

We can find the values of trigonometric ratiorsquos various angle

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Area of Circles

Sno Points

1 A circle is a collection of all the points in a plane which are equidistant

from a fixed point in the plane

2 Equal chords of a circle (or of congruent circles) subtend equal angles at

the center

3 If the angles subtended by two chords of a circle (or of congruent circles)

at the center (corresponding center) are equal the chords are equal

4 The perpendicular from the center of a circle to a chord bisects the chord

5 The line drawn through the center of a circle to bisect a chord is

perpendicular to the chord

6 There is one and only one circle passing through three non-collinear points

7 Equal chords of a circle (or of congruent circles) are equidistant from the

center (or corresponding centers)

8 Chords equidistant from the center (or corresponding centers) of a circle

(or of congruent circles) are equal

9 If two arcs of a circle are congruent then their corresponding chords are

equal and conversely if two chords of a circle are equal then their

corresponding arcs (minor major) are congruent

10 Congruent arcs of a circle subtend equal angles at the center

11 The angle subtended by an arc at the center is double the angle subtended

by it at any point on the remaining part of the circle

12 Angles in the same segment of a circle are equal

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13 Angle in a semicircle is a right angle

14 If a line segment joining two points subtends equal angles at two other

points lying on the same side of the line containing the line segment the

four points lie on a circle

15 The sum of either pair of opposite angles of a cyclic quadrilateral is 180deg

16 If the sum of a pair of opposite angles of a quadrilateral is 180deg then the

quadrilateral is cyclic

Sno Terms Descriptions

1 Circumference of a circle 2 π r

2 Area of circle π r 2

3 Length of the arc of

the sector of angle

Length of the arc of the sector of angle θ 2

4 Area of the sector of

angle

Area of the sector of angle θ

5 Area of segment of a

circle

Area of the corresponding sector ndash Area of the

corresponding triangle

Surface Area and Volume

Sno Term Description

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1 Mensuration It is branch of mathematics which is concernedabout the measurement of length area and

Volume of plane and Solid figure

2 Perimeter a)The perimeter of plane figure is defined as thelength of the boundaryb)It units is same as that of length ie m cmkm

3 Area a)The area of the plane figure is the surfaceenclosed by its boundaryb) It unit is square of length unit ie m2 km2

4 Volume Volume is the measure of the amount of spaceinside of a solid figure like a cube ball cylinderor pyramid Its units are always cubic that isthe number of little element cubes that fit insidethe figure

Volume Unit conversion

1 cm3 1mL 1000 mm3

1 Litre 1000ml 1000 cm3

1 m3 106 cm3 1000 L

1 dm3 1000 cm3 1 L

Surface Area and Volume of Cube and Cuboid

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Type Measurement

Surface Area of Cuboid of Length L

Breadth B and Height H

2(LB + BH + LH)

Lateral surface area of the cuboids 2( L + B ) H

Diagonal of the cuboids Volume of a cuboids LBH

Length of all 12 edges of the cuboids 4 (L+B+H)

Surface Area of Cube of side L 6L2

Lateral surface area of the cube 4L2

Diagonal of the cube radic3 Volume of a cube L3

Surface Area and Volume of Right circular cylinder

Radius The radius (r) of the circular base is called the radius of the

cylinder

Height The length of the axis of the cylinder is called the height (h) of the

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cylinder

Lateral

Surface

The curved surface joining the two base of a right circular cylinder is

called Lateral Surface

Type Measurement

Curved or lateral Surface Area of

cylinder

2πrh

Total surface area of cylinder 2πr (h+r)

Volume of Cylinder π r2h

Surface Area and Volume of Right circular cone

Radius The radius (r) of the circular base is called the radius of the

cone

Height The length of the line segment joining the vertex to the center of

base is called the height (h) of the cone

Slant

Height

The length of the segment joining the vertex to any point on the

circular edge of the base is called the slant height (L) of the cone

Lateral

surface

Area

The curved surface joining the base and uppermost point of a right

circular cone is called Lateral Surface

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Type Measurement

Curved or lateral Surface Area of

cone

πrL

Total surface area of cone πr (L+r)

Volume of Cone

Surface Area and Volume of sphere and hemisphere

Sphere A sphere can also be considered as a solid obtained on

rotating a circle About its diameter

Hemisphere A plane through the centre of the sphere divides the sphere into two

equal parts each of which is called a hemisphere

radius The radius of the circle by which it is formed

SphericalShell

The difference of two solid concentric spheres is called a sphericalshell

Lateral

Surface

Area for

Total surface area of the sphere

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Sphere

Lateral

Surface

area of

Hemisphere

It is the curved surface area leaving the circular base

Type Measurement

Surface area of Sphere 4πr2

Volume of Sphere 43 Curved Surface area of hemisphere 2πr2

Total Surface area of hemisphere 3πr2

Volume of hemisphere 23 Volume of the spherical shell whose

outer and inner radii and lsquoRrsquo and lsquorrsquorespectively

43

How the Surface area and Volume are determined

Area of Circle The circumference of a circle is 2πr

This is the definition of π (pi) Divide

the circle into many triangular

segments The area of the triangles

is 12 times the sum of their bases

2πr (the circumference) times their

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height r

12 2 Surface Area of cylinder This can be imagined as unwrapping the

surface into a rectangle

Surface area of cone This can be achieved by divide the

surface of the cone into its triangles or

the surface of the cone into many thin

triangles The area of the triangles is 12

times the sum of their bases p times

their height

12 2

Surface Area and Volume of frustum of cone

h = vertical height of the frustum

l = slant height of the frustum

r 1 and r 2 are radii of the two bases (ends) of the frustum

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Type Measurement

983126983151983148983157983149983141 983151983142 983137 983142983154983157983155983156983157983149 983151983142 983137 983139983151983150983141 983123983148983137983150983156 983144983141983145983143983144983156 983151983142 983142983154983157983155983156983157983149 983151983142 983137 983139983151983150983141 983107983157983154983158983141983140 983155983157983154983142983137983139983141 983137983154983141983137 983151983142 983137 983142983154983157983155983156983157983149 983151983142 983137 983139983151983150983141 983124983151983156983137983148 983155983157983154983142983137983139983141 983137983154983141983137 983151983142 983142983154983157983155983156983157983149 983151983142 983137 983139983151983150983141

Statistics

Sno Term Description

1 Statistics Statistics is a broad mathematical disciplinewhich studies ways to collect summarize and

draw conclusions from data

2 Data A systematic record of facts or different values of

a quantity is called data

Data is of two types - Primary data and Secondary

data

Primary Data The data collected by a researcher

with a specific purpose in mind is called primarydata

Secondary Data The data gathered from a

source where it already exists is called secondary

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data

3 Features of data bull Statistics deals with collectionpresentation analysis and interpretation ofnumerical data

bull Arranging data in an order to study theirsalient features is called presentation ofdata

bull Data arranged in ascending or descendingorder is called arrayed data or an array

bull Range of the data is the differencebetween the maximum and the minimum

values of the observationsbull Table that shows the frequency of different

values in the given data is called afrequency distribution table

bull A frequency distribution table that showsthe frequency of each individual value in thegiven data is called an ungrouped frequencydistribution table

bull A table that shows the frequency of groupsof values in the given data is called agrouped frequency distribution table

bull

The groupings used to group the values ingiven data are called classes or class-intervals The number of values that eachclass contains is called the class size orclass width The lower value in a class iscalled the lower class limit The higher valuein a class is called the upper class limit

bull Class mark of a class is the mid value ofthe two limits of that class

bull A frequency distribution in which the upperlimit of one class differs from the lower limitof the succeeding class is called an

Inclusive or discontinuous FrequencyDistribution

bull A frequency distribution in which the upperlimit of one class coincides from the lowerlimit of the succeeding class is called anexclusive or continuous Frequency

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Distribution bull

4 Bar graph A bar graph is a pictorial representation of data inwhich rectangular bars of uniform width are drawnwith equal spacing between them on one axisusually the x axis The value of the variable isshown on the other axis that is the y axis

5 Histogram A histogram is a set of adjacent rectangles whoseareas are proportional to the frequencies of agiven continuous frequency distribution

6 Mean The mean value of a variable is defined as thesum of all the values of the variable divided by thenumber of values

4 sum

7 Median The median of a set of data values is the middle

value of the data set when it has been arranged inascending order That is from the smallest valueto the highest value

Median is calculated as

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12 1 Where n is the number of values in the data

If the number of values in the data set is eventhen the median is the average of the two middlevalues

8 Mode Mode of a statistical data is the value of thatvariable which has the maximum frequency

Sno Term Description

1983117983141983137983150 983142983151983154

983125983150983143983154983151983157983152

983110983154983141983153983157983141983150983139983161 983156983137983138983148983141

Mean is given by

sum sum

2983117983141983137983150 983142983151983154 983143983154983151983157983152

983110983154983141983153983157983141983150983139983161 983156983137983138983148983141

In these distribution it is assumed that frequency of each classinterval is centered around its mid-point ie class marks

2

983117983141983137983150 983139983137983150 983138983141 983139983137983148983139983157983148983137983156983141983140 983157983155983145983150983143 983156983144983154983141983141 983149983141983156983144983151983140

983137983081 983108983145983154983141983139983156 983149983141983156983144983151983140

sum

sum

983138983081 983105983155983155983157983149983141983140 983149983141983137983150 983149983141983156983144983151983140

sum sum

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Where

a=gt Assumed mean

di =gt xi ndasha

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sum sum

Where

a=gt Assumed mean

ui =gt (xi ndasha)h

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Modal class The class interval having highest frequency is

called the modal class and Mode is obtained using the modal

class

2

983127983144983141983154983141

l = lower limit of the modal classh = size of the class interval (assuming all class sizes tobe equal)f 1 = frequency of the modal classf 0 = frequency of the class preceding the modal classf 2 = frequency of the class succeeding the modal class

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For the given data we need to have class interval frequency

distribution and cumulative frequency distribution

Median is calculated as

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2

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983144 983101 983139983148983137983155983155 983155983145983162983141 (983137983155983155983157983149983145983150983143 983139983148983137983155983155 983155983145983162983141 983156983151 983138983141 983141983153983157983137983148)

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3 Median=Mode +2 Mean

ProbabilitySn

o

Term Description

1 Empirical

probability

It is a probability of event which is calculated based on

experiments

Example

A coin is tossed 1000 times we get 499 times head and

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501 times tail

So empirical or experimental probability of getting head is

calculated as

4991000 499

Empirical probability depends on experiment and

different will get different values based on the

experiment

2 Important point

about events

If the event A B C covers the entire possible outcome inthe experiment Then

P (A) +P (B) +P(C)=1

3 impossible

event

The probability of an event (U) which is impossible tooccur is 0 Such an event is called an impossible event

P (U)=0

4 Sure or certain

event

The probability of an event (X) which is sure (or certain)to occur is 1 Such an event is called a sure event or acertain event

P(X) =1

5 Probability of

any event

Probability of any event can be as

0 1

Sn

o

Term Description

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983092983096

1

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The theoretical probability or the classical probability of the event isdefined as

2 Elementary

events

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983120(983105)+983120(983106) +983120(983107)9831011

3 Complementar

y events

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983120(983105) +983120(Ᾱ)9831011

4 Sure or certain

event

The probability of an event (X) which is sure (or certain)

to occur is 1 Such an event is called a sure event or acertain event

P(X) =1

5 Probability of

any event

Probability of any event can be as

0 1

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COORDINATE GEOMETRYSno Points

1 We require two perpendicular axes to locate a point in the plane One of

them is horizontal and other is Vertical

2 The plane is called Cartesian plane and axis are called the coordinates axis

3 The horizontal axis is called x-axis and Vertical axis is called Y-axis

4 The point of intersection of axis is called origin

5 The distance of a point from y axis is called x ndashcoordinate or abscissa and

the distance of the point from x ndashaxis is called y ndash coordinate or Ordinate

6 The distance of a point from y axis is called x ndashcoordinate or abscissa and

the distance of the point from x ndashaxis is called y ndash coordinate or Ordinate

7 The Origin has zero distance from both x-axis and y-axis so that its

abscissa and ordinate both are zero So the coordinate of the origin is (0

0)

8 A point on the x ndashaxis has zero distance from x-axis so coordinate of any

point on the x-axis will be (x 0)

9 A point on the y ndashaxis has zero distance from y-axis so coordinate of any

point on the y-axis will be (0 y)

10 The axes divide the Cartesian plane in to four parts These Four parts arecalled the quadrants

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10

The coordinates of the points in the four quadrants will have sign according to thebelow table

Quadrant x-coordinate y-coordinate

Ist Quadrant + +

IInd quadrant - +

IIIrd quadrant - -

IVth quadrant + -

Sno Terms Descriptions

1 Distance formula Distance between the points AB is given by

Distance of Point A from Origin

2 Section Formula A point P(xy) which divide the line segment AB in

the ratio m1 and m2 is given by

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11

The midpoint P is given by

3 Area of Triangle Area of triangle ABC of coordinates A(x 1y 1 )

B(x 2 y 2 ) and C(x 3y 3 )

12 For point AB and C to be collinear The value of A

should be zero

LINEAR EQUATIONS IN TWO

VARIABLESAn equation of the form ax + by + c = 0 where a b and c are real numbers such

that a andb are not both zero is called a linear equation in two variablesImportant points to Note

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12

Sno Points

1 A linear equation in two variable has infinite solutions

2 The graph of every linear equation in two variable is a straight line

3 x = 0 is the equation of the y-axis and y = 0 is the equation of the x-axis

4 The graph x=a is a line parallel to y -axis

5 The graph y=b is a line parallel to x -axis

6 An equation of the type y = mx represents a line passing through theorigin

7 Every point on the graph of a linear equation in two variables is a solution

of the linear

equation Moreover every solution of the linear equation is a point on the

graph

Sno Type of equation Mathematical

representation

Solutions

1 Linear equation in one Variable ax+b=0 ane0

a and b are real

number

One solution

2 Linear equation in two Variable ax+by+c=0 ane0 and

bne0

a b and c are real

number

Infinite solution

possible

3 Linear equation in three Variable ax+by+cz+d=0 ane0

bne0 and cne0

Infinite solution

possible

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a b c d are realnumber

Simultaneous pair of Linear equation

A pair of Linear equation in two variables

a1x+b1y+c1=0

a2x +b2y+c2=0

Graphically it is represented by two straight lines on Cartesian plane

Simultaneous pair of

Linear equation

Condition Graphical

representation

Algebraic

interpretation

a1x+b1y+c1=0

a2x +b2y+c2=0

Example

x-4y+14=0

3x+2y-14=0

Intersecting lines The

intersecting point

coordinate is the only

solution

One unique solution

only

a1x+b1y+c1=0

a2x +b2y+c2=0

Coincident lines Theany coordinate on the

line is the solution

Infinite solution

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Example

2x+4y=16

3x+6y=24

a1x+b1y+c1=0

a2x +b2y+c2=0

Example

2x+4y=6

4x+8y=18

Parallel Lines No solution

The graphical solution can be obtained by drawing the lines on the Cartesian plane

Algebraic Solution of system of Linear equation

Sno Type of method Working of method

1 Method of elimination by

substitution

1) Suppose the equation are

a1x+b1y+c1=0

a2x +b2y+c2=0

2) Find the value of variable of either x or y in other

variable term in first equation

3) Substitute the value of that variable in second

equation

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1983093

4) Now this is a linear equation in one variable Findthe value of the variable

5) Substitute this value in first equation and get the

second variable

2 Method of elimination by

equating the coefficients

1) Suppose the equation are

a1x+b1y+c1=0

a2x +b2y+c2=0

2) Find the LCM of a1 and a2 Let it k

3) Multiple the first equation by the value ka1

4) Multiple the first equation by the value ka2

4) Subtract the equation obtained This way one

variable will be eliminated and we can solve to get the

value of variable y

5) Substitute this value in first equation and get the

second variable

3 Cross Multiplication method 1) Suppose the equation are

a1x+b1y+c1=0

a2x +b2y+c2=0

2) This can be written as

3) This can be written as

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4) Value of x and y can be find using the

x =gt first and last expression

y=gt second and last expression

Quadratic EquationsSno Terms Descriptions

1 Quadratic Polynomial P(x) = ax +bx+c where ane0

2 Quadratic equation ax +bx+c =0 where ane0

3 Solution or root of the

Quadratic equation

A real number α is called the root or solution of the

quadratic equation if

aα2 +bα+c=0

4 zeroes of the

polynomial p(x)

The root of the quadratic equation are called zeroes

5 Maximum roots of

quadratic equations

We know from chapter two that a polynomial of

degree can have max two zeroes So a quadratic

equation can have maximum two roots

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6 Condition for realroots A quadratic equation has real roots if b

2

- 4ac gt 0

How to Solve Quadratic equation

Sno Method Working

1 factorization This method we factorize the equation by splitting themiddle term b

In ax2+bx+c=0

Example

6x2-x-2=0

1) First we need to multiple the coefficient a and cIn this

case =6X-2=-12

2) Splitting the middle term so that multiplication is 12

and difference is the coefficient b

6x2 +3x-4x-2=0

3x( 2x+1) -2(2x+1)=0

(3x-2) (2x+1)=0

3) Roots of the equation can be find equating the factorsto zero

3x-2=0 =gt x=32

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2x+1=0 =gt x=-12

2 Square method In this method we create square on LHS and RHS and

then find the value

ax2 +bx+c=0

1) x2 +(ba) x+(ca)=0

2) ( x+b2a)2 ndash(b2a)2 +(ca)=0

3) ( x+b2a)2=(b2-4ac)4a2

4) radic

Example

x2 +4x-5=0

1) (x+2)2 -4-5=0

2) (x+2)2=9

3) Roots of the equation can be find using square root on

both the sides

x+2 =-3 =gt x=-5

x+2=3=gt x=1

3 Quadratic

method

For quadratic equation

ax2 +bx+c=0

roots are given by radic radic

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1983097

For b2 -4ac gt 0 Quadratic equation has two real roots of

different value

For b2-4ac =0 quadratic equation has one real root

For b2-4ac lt 0 no real roots for quadratic equation

Nature of roots of Quadratic equation

Sno Condition Nature of roots

1 b -4ac gt 0 Two distinct real roots

2 b -4ac =0 One real root

3 b

2

-4ac lt 0 No real roots

Triangles

Sno Terms Descriptions

1 Congruence Two Geometric figure are said to be congruence ifthey are exactly same size and shapeSymbol used is cong Two angles are congruent if they are equalTwo circle are congruent if they have equal radiiTwo squares are congruent if the sides are equal

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983151983150983148983161

20

2 Triangle Congruencebull

Two triangles are congruent if three sidesand three angles of one triangle is

congruent to the corresponding sides and

angles of the other

bull Corresponding sides are equal

AB=DE BC=EF AC=DF

bull Corresponding angles are equalang angang angang ang

bull We write this as

cong

bull The above six equalities are between the

corresponding parts of the two congruent

triangles In short form this is called

CPCT

bull We should keep the letters in correct order

on both sides

3 Inequalities in

Triangles

1) In a triangle angle opposite to longer side is

larger

2) In a triangle side opposite to larger angle is

larger

3) The sum of any two sides of the triangle is

greater than the third side

983105

983107983106

983108

983109 983110

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983151983150983148983161

21

In triangle ABC

AB +BC gt AC

Different Criterion for Congruence of the triangles

N Criterion Description Figures and

expression

1 Side angle

Side (SAS)

congruence

bull Two triangles are congruent if the

two sides and included angles of

one triangle is equal to the two

sides and included angle

bull It is an axiom as it cannot be

proved so it is an accepted truth

bull ASS and SSA type two triangles

may not be congruent alwaysIf following

condition

983105

983110983109

983108

983107983106

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983151983150983148983161

22

AB=DE BC=EFang ang

Then

cong

2 Angle side

angle (ASA)

congruence

bull Two triangles are congruent if the

two angles and included side of

one triangle is equal to the

corresponding angles and side

bull It is a theorem and can be proved

If following

condition

BC=EF

ang angang ang

Then

cong

3 Angle angle

side( AAS)

congruence

bull Two triangles are congruent if the

any two pair of angles and any

side of one triangle is equal to the

corresponding angles and side

bull It is a theorem and can be proved

983110983109

983108

983107983106

983105

983110983109

983108

983107983106

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983151983150983148983161

2983091

If followingcondition

BC=EF

ang angang ang

Then

cong

4 Side-Side-Side

(SSS)congruence

bull Two triangles are congruent if the

three sides of one triangle is equalto the three sides of the another

If following

condition

BC=EFAB=DEDF

=AC

Then

cong

5 Right angle ndash

hypotenuse-

side(RHS)

bull Two right triangles are congruent if

the hypotenuse and a side of the

one triangle are equal to

983110983109

983108

983107983106

983105

983105 983108

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2983092

congruence corresponding hypotenuse and sideof the another

If following

condition

AC=DFBC=EF

Then

cong

Some Important points on Triangles

Terms Description

Orthocenter Point of intersection of the three altitudeof the triangle

Equilateral triangle whose all sides are equal and allangles are equal to 600

Median A line Segment joining the corner of thetriangle to the midpoint of the oppositeside of the triangle

Altitude A line Segment from the corner of thetriangle and perpendicular to theopposite side of the triangle

Isosceles A triangle whose two sides are equalCentroid Point of intersection of the three median

of the triangle is called the centroid of

the triangleIn center All the angle bisector of the trianglepasses through same point

Circumcenter The perpendicular bisector of the sidesof the triangles passes through samepoint

983106 983107 983109 983110

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2983093

Scalene triangle Triangle having no equal angles and noequal sidesRight Triangle Right triangle has one angle equal to 900 Obtuse Triangle One angle is obtuse angle while other

two are acute anglesAcute Triangle All the angles are acute

Similarity of Triangles

Sno Points

1 Two figures having the same shape but not necessarily the same size are called similarfigures

2 All the congruent figures are similar but the converse is not true

3 If a line is drawn parallel to one side of a triangle to intersect the other two sides in

distinct points then the other two sides are divided in the same ratio

4 If a line divides any two sides of a triangle in the same ratio then the line is parallel tothe third side

Different Criterion for Similarity of the triangles

N Criterion Description Expression

1 Angle Angle

angle(AAA)

similarity

bull Two triangles are similar if

corresponding angle are equal

If following

condition

ang ang

ang ang

ang ang

Then

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2983094

Then 983166

2 Angle angle

(AA) similarity

bull Two triangles are similar if the two

corresponding angles are equal as

by angle property third angle will

be also equal

If following

condition

ang ang

ang ang

Then

ang ang

Then

Then

983166

3 Side side

side(SSS)

Similarity

Two triangles are similar if the

sides of one triangle is

proportional to the sides of other

triangle

If following

condition

Then

ang ang

ang ang

ang ang

Then

cong

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2983095

4 Side-Angle-Side (SAS)

similarity

bull

Two triangles are similar if the oneangle of a triangle is equal to one

angle of other triangles and sides

including that angle is proportional

If followingcondition

And ang ang

Then

cong

Area of Similar triangles

If the two triangle ABC and DEF are similar

cong

Then

Pythagoras Theorem

Sno Points

1 If a perpendicular is drawn from the vertex of the right angle of a right triangle to thehypotenuse then the triangles on both sides of the perpendicular are similar to the

whole triangle and also to each other

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2983096

2 In a right triangle the square of the hypotenuse is equal to the sum of the squares of the

other two sides (Pythagoras Theorem)

(hyp)2 =(base)

2 + (perp)

2

3 If in a triangle square of one side is equal to the sum of the squares of the other twosides then the angle opposite the first side is a right angle

Arithmetic Progression

Sno Terms Descriptions

1 ArithmeticProgression

An arithmetic progression is a sequence ofnumbers such that the difference of any two

successive members is a constant

Examples

1) 1591317hellip

2) 12345hellip

2 common difference of

the AP

the difference between any successive members is a

constant and it is called the common difference of AP

1) If a1 a2a3a4a5 are the terms in AP then

D=a2 -a1 =a3 - a2 =a4 ndash a3=a5 ndasha4

2) We can represent the general form of AP inthe form

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2983097

aa+da+2da+3da+4d

Where a is first term and d is the commondifference

3 nth term of Arithmetic Progression

n term = a + (n - 1)d

4 Sum of nth item in ArithmeticProgression

Sn =(n2)[a + (n-1)d]

Or

Sn =(n2)[t1+ tn]

Trigonometry

Sno Terms Descriptions

1 What is

Trigonometry

Trigonometry from Greek trigotildenon triangle and

metron measure) is a branch of mathematics

that studies relationships involving lengths and

angles of triangles The field emerged during the

3rd century BC from applications of geometry to

astronomical studies

Trigonometry is most simply associated with

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planar right angle triangles (each of which is atwo-dimensional triangle with one angle equal to

90 degrees) The applicability to non-right-angle

triangles exists but since any non-right-angle

triangle (on a flat plane) can be bisected to

create two right-angle triangles most problems

can be reduced to calculations on right-angle

triangles Thus the majority of applications relate

to right-angle triangles

2 Trigonometric

Ratiorsquos

In a right angle triangle ABC where B=90deg

We can define following term for angle A

Base Side adjacent to angle

Perpendicular Side Opposite of angleHypotenuse Side opposite to right angle

We can define the trigonometric ratios for

angle A as

sin A= PerpendicularHypotenuse =BCAC

cosec A= HypotenusePerpendicular

=ACBC

cos A= BaseHypotenuse =ABAC

sec A= HypotenuseBase=ACAB

tan A= PerpendicularBase =BCAB

cot A= BasePerpendicular=ABBC

Notice that each ratio in the right-hand column is

the inverse or the reciprocal of the ratio in the

left-hand column

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3 Reciprocal offunctions The reciprocal of sin A is cosec A and vice-versa

The reciprocal of cos A is sec A

And the reciprocal of tan A is cot A

These are valid for acute angles

We can define tan A = sin Acos A

And Cot A =cos A Sin A

4 Value of of sin and

cos

Is always less 1

5 Trigonometric ration

from another angle

We can define the trigonometric ratios for angle Cas

sin C= PerpendicularHypotenuse =ABACcosec C= HypotenusePerpendicular =ACABBcos C= BaseHypotenuse =BCACsec C= HypotenuseBase=ACBCtan A= PerpendicularBase =ABBCcot A= BasePerpendicular=BCAB

6 Trigonometric ratios

of complimentary

Sin (90-A) =cos(A)

Cos(90-A) = sin A

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9830912

angles Tan(90-A) =cot A

Sec(90-A)= cosec A

Cosec (90-A) =sec A

Cot(90- A) =tan A7 Trigonometric

identities

Sin2 A + cos2 A =0

1 + tan2 A =sec2 A

1 + cot2 A =cosec2 A

Trigonometric Ratios of Common angles

We can find the values of trigonometric ratiorsquos various angle

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983091983091

Area of Circles

Sno Points

1 A circle is a collection of all the points in a plane which are equidistant

from a fixed point in the plane

2 Equal chords of a circle (or of congruent circles) subtend equal angles at

the center

3 If the angles subtended by two chords of a circle (or of congruent circles)

at the center (corresponding center) are equal the chords are equal

4 The perpendicular from the center of a circle to a chord bisects the chord

5 The line drawn through the center of a circle to bisect a chord is

perpendicular to the chord

6 There is one and only one circle passing through three non-collinear points

7 Equal chords of a circle (or of congruent circles) are equidistant from the

center (or corresponding centers)

8 Chords equidistant from the center (or corresponding centers) of a circle

(or of congruent circles) are equal

9 If two arcs of a circle are congruent then their corresponding chords are

equal and conversely if two chords of a circle are equal then their

corresponding arcs (minor major) are congruent

10 Congruent arcs of a circle subtend equal angles at the center

11 The angle subtended by an arc at the center is double the angle subtended

by it at any point on the remaining part of the circle

12 Angles in the same segment of a circle are equal

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983091983092

13 Angle in a semicircle is a right angle

14 If a line segment joining two points subtends equal angles at two other

points lying on the same side of the line containing the line segment the

four points lie on a circle

15 The sum of either pair of opposite angles of a cyclic quadrilateral is 180deg

16 If the sum of a pair of opposite angles of a quadrilateral is 180deg then the

quadrilateral is cyclic

Sno Terms Descriptions

1 Circumference of a circle 2 π r

2 Area of circle π r 2

3 Length of the arc of

the sector of angle

Length of the arc of the sector of angle θ 2

4 Area of the sector of

angle

Area of the sector of angle θ

5 Area of segment of a

circle

Area of the corresponding sector ndash Area of the

corresponding triangle

Surface Area and Volume

Sno Term Description

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1 Mensuration It is branch of mathematics which is concernedabout the measurement of length area and

Volume of plane and Solid figure

2 Perimeter a)The perimeter of plane figure is defined as thelength of the boundaryb)It units is same as that of length ie m cmkm

3 Area a)The area of the plane figure is the surfaceenclosed by its boundaryb) It unit is square of length unit ie m2 km2

4 Volume Volume is the measure of the amount of spaceinside of a solid figure like a cube ball cylinderor pyramid Its units are always cubic that isthe number of little element cubes that fit insidethe figure

Volume Unit conversion

1 cm3 1mL 1000 mm3

1 Litre 1000ml 1000 cm3

1 m3 106 cm3 1000 L

1 dm3 1000 cm3 1 L

Surface Area and Volume of Cube and Cuboid

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Type Measurement

Surface Area of Cuboid of Length L

Breadth B and Height H

2(LB + BH + LH)

Lateral surface area of the cuboids 2( L + B ) H

Diagonal of the cuboids Volume of a cuboids LBH

Length of all 12 edges of the cuboids 4 (L+B+H)

Surface Area of Cube of side L 6L2

Lateral surface area of the cube 4L2

Diagonal of the cube radic3 Volume of a cube L3

Surface Area and Volume of Right circular cylinder

Radius The radius (r) of the circular base is called the radius of the

cylinder

Height The length of the axis of the cylinder is called the height (h) of the

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cylinder

Lateral

Surface

The curved surface joining the two base of a right circular cylinder is

called Lateral Surface

Type Measurement

Curved or lateral Surface Area of

cylinder

2πrh

Total surface area of cylinder 2πr (h+r)

Volume of Cylinder π r2h

Surface Area and Volume of Right circular cone

Radius The radius (r) of the circular base is called the radius of the

cone

Height The length of the line segment joining the vertex to the center of

base is called the height (h) of the cone

Slant

Height

The length of the segment joining the vertex to any point on the

circular edge of the base is called the slant height (L) of the cone

Lateral

surface

Area

The curved surface joining the base and uppermost point of a right

circular cone is called Lateral Surface

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Type Measurement

Curved or lateral Surface Area of

cone

πrL

Total surface area of cone πr (L+r)

Volume of Cone

Surface Area and Volume of sphere and hemisphere

Sphere A sphere can also be considered as a solid obtained on

rotating a circle About its diameter

Hemisphere A plane through the centre of the sphere divides the sphere into two

equal parts each of which is called a hemisphere

radius The radius of the circle by which it is formed

SphericalShell

The difference of two solid concentric spheres is called a sphericalshell

Lateral

Surface

Area for

Total surface area of the sphere

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Sphere

Lateral

Surface

area of

Hemisphere

It is the curved surface area leaving the circular base

Type Measurement

Surface area of Sphere 4πr2

Volume of Sphere 43 Curved Surface area of hemisphere 2πr2

Total Surface area of hemisphere 3πr2

Volume of hemisphere 23 Volume of the spherical shell whose

outer and inner radii and lsquoRrsquo and lsquorrsquorespectively

43

How the Surface area and Volume are determined

Area of Circle The circumference of a circle is 2πr

This is the definition of π (pi) Divide

the circle into many triangular

segments The area of the triangles

is 12 times the sum of their bases

2πr (the circumference) times their

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height r

12 2 Surface Area of cylinder This can be imagined as unwrapping the

surface into a rectangle

Surface area of cone This can be achieved by divide the

surface of the cone into its triangles or

the surface of the cone into many thin

triangles The area of the triangles is 12

times the sum of their bases p times

their height

12 2

Surface Area and Volume of frustum of cone

h = vertical height of the frustum

l = slant height of the frustum

r 1 and r 2 are radii of the two bases (ends) of the frustum

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Type Measurement

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Statistics

Sno Term Description

1 Statistics Statistics is a broad mathematical disciplinewhich studies ways to collect summarize and

draw conclusions from data

2 Data A systematic record of facts or different values of

a quantity is called data

Data is of two types - Primary data and Secondary

data

Primary Data The data collected by a researcher

with a specific purpose in mind is called primarydata

Secondary Data The data gathered from a

source where it already exists is called secondary

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data

3 Features of data bull Statistics deals with collectionpresentation analysis and interpretation ofnumerical data

bull Arranging data in an order to study theirsalient features is called presentation ofdata

bull Data arranged in ascending or descendingorder is called arrayed data or an array

bull Range of the data is the differencebetween the maximum and the minimum

values of the observationsbull Table that shows the frequency of different

values in the given data is called afrequency distribution table

bull A frequency distribution table that showsthe frequency of each individual value in thegiven data is called an ungrouped frequencydistribution table

bull A table that shows the frequency of groupsof values in the given data is called agrouped frequency distribution table

bull

The groupings used to group the values ingiven data are called classes or class-intervals The number of values that eachclass contains is called the class size orclass width The lower value in a class iscalled the lower class limit The higher valuein a class is called the upper class limit

bull Class mark of a class is the mid value ofthe two limits of that class

bull A frequency distribution in which the upperlimit of one class differs from the lower limitof the succeeding class is called an

Inclusive or discontinuous FrequencyDistribution

bull A frequency distribution in which the upperlimit of one class coincides from the lowerlimit of the succeeding class is called anexclusive or continuous Frequency

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Distribution bull

4 Bar graph A bar graph is a pictorial representation of data inwhich rectangular bars of uniform width are drawnwith equal spacing between them on one axisusually the x axis The value of the variable isshown on the other axis that is the y axis

5 Histogram A histogram is a set of adjacent rectangles whoseareas are proportional to the frequencies of agiven continuous frequency distribution

6 Mean The mean value of a variable is defined as thesum of all the values of the variable divided by thenumber of values

4 sum

7 Median The median of a set of data values is the middle

value of the data set when it has been arranged inascending order That is from the smallest valueto the highest value

Median is calculated as

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12 1 Where n is the number of values in the data

If the number of values in the data set is eventhen the median is the average of the two middlevalues

8 Mode Mode of a statistical data is the value of thatvariable which has the maximum frequency

Sno Term Description

1983117983141983137983150 983142983151983154

983125983150983143983154983151983157983152

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Mean is given by

sum sum

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983110983154983141983153983157983141983150983139983161 983156983137983138983148983141

In these distribution it is assumed that frequency of each classinterval is centered around its mid-point ie class marks

2

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sum

sum

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sum sum

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Where

a=gt Assumed mean

di =gt xi ndasha

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sum sum

Where

a=gt Assumed mean

ui =gt (xi ndasha)h

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Modal class The class interval having highest frequency is

called the modal class and Mode is obtained using the modal

class

2

983127983144983141983154983141

l = lower limit of the modal classh = size of the class interval (assuming all class sizes tobe equal)f 1 = frequency of the modal classf 0 = frequency of the class preceding the modal classf 2 = frequency of the class succeeding the modal class

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For the given data we need to have class interval frequency

distribution and cumulative frequency distribution

Median is calculated as

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2

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983144 983101 983139983148983137983155983155 983155983145983162983141 (983137983155983155983157983149983145983150983143 983139983148983137983155983155 983155983145983162983141 983156983151 983138983141 983141983153983157983137983148)

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3 Median=Mode +2 Mean

ProbabilitySn

o

Term Description

1 Empirical

probability

It is a probability of event which is calculated based on

experiments

Example

A coin is tossed 1000 times we get 499 times head and

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501 times tail

So empirical or experimental probability of getting head is

calculated as

4991000 499

Empirical probability depends on experiment and

different will get different values based on the

experiment

2 Important point

about events

If the event A B C covers the entire possible outcome inthe experiment Then

P (A) +P (B) +P(C)=1

3 impossible

event

The probability of an event (U) which is impossible tooccur is 0 Such an event is called an impossible event

P (U)=0

4 Sure or certain

event

The probability of an event (X) which is sure (or certain)to occur is 1 Such an event is called a sure event or acertain event

P(X) =1

5 Probability of

any event

Probability of any event can be as

0 1

Sn

o

Term Description

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1

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The theoretical probability or the classical probability of the event isdefined as

2 Elementary

events

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983113983141 983113983142 983159983141 983156983144983154983141983141 983141983148983141983149983141983150983156983137983154983161 983141983158983141983150983156 983105983106983107 983145983150 983156983144983141 983141983160983152983141983154983145983149983141983150983156 983156983144983141983150

983120(983105)+983120(983106) +983120(983107)9831011

3 Complementar

y events

983124983144983141 983141983158983141983150983156 Ᾱ 983154983141983152983154983141983155983141983150983156983145983150983143 991256983150983151983156 983105991257 983145983155 983139983137983148983148983141983140 983156983144983141 983139983151983149983152983148983141983149983141983150983156 983151983142 983156983144983141 983141983158983141983150983156

983105 983127983141 983137983148983155983151 983155983137983161 983156983144983137983156 Ᾱ 983137983150983140 983105 983137983154983141 983139983151983149983152983148983141983149983141983150983156983137983154983161 983141983158983141983150983156983155 983105983148983155983151

983120(983105) +983120(Ᾱ)9831011

4 Sure or certain

event

The probability of an event (X) which is sure (or certain)

to occur is 1 Such an event is called a sure event or acertain event

P(X) =1

5 Probability of

any event

Probability of any event can be as

0 1

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The coordinates of the points in the four quadrants will have sign according to thebelow table

Quadrant x-coordinate y-coordinate

Ist Quadrant + +

IInd quadrant - +

IIIrd quadrant - -

IVth quadrant + -

Sno Terms Descriptions

1 Distance formula Distance between the points AB is given by

Distance of Point A from Origin

2 Section Formula A point P(xy) which divide the line segment AB in

the ratio m1 and m2 is given by

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The midpoint P is given by

3 Area of Triangle Area of triangle ABC of coordinates A(x 1y 1 )

B(x 2 y 2 ) and C(x 3y 3 )

12 For point AB and C to be collinear The value of A

should be zero

LINEAR EQUATIONS IN TWO

VARIABLESAn equation of the form ax + by + c = 0 where a b and c are real numbers such

that a andb are not both zero is called a linear equation in two variablesImportant points to Note

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Sno Points

1 A linear equation in two variable has infinite solutions

2 The graph of every linear equation in two variable is a straight line

3 x = 0 is the equation of the y-axis and y = 0 is the equation of the x-axis

4 The graph x=a is a line parallel to y -axis

5 The graph y=b is a line parallel to x -axis

6 An equation of the type y = mx represents a line passing through theorigin

7 Every point on the graph of a linear equation in two variables is a solution

of the linear

equation Moreover every solution of the linear equation is a point on the

graph

Sno Type of equation Mathematical

representation

Solutions

1 Linear equation in one Variable ax+b=0 ane0

a and b are real

number

One solution

2 Linear equation in two Variable ax+by+c=0 ane0 and

bne0

a b and c are real

number

Infinite solution

possible

3 Linear equation in three Variable ax+by+cz+d=0 ane0

bne0 and cne0

Infinite solution

possible

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a b c d are realnumber

Simultaneous pair of Linear equation

A pair of Linear equation in two variables

a1x+b1y+c1=0

a2x +b2y+c2=0

Graphically it is represented by two straight lines on Cartesian plane

Simultaneous pair of

Linear equation

Condition Graphical

representation

Algebraic

interpretation

a1x+b1y+c1=0

a2x +b2y+c2=0

Example

x-4y+14=0

3x+2y-14=0

Intersecting lines The

intersecting point

coordinate is the only

solution

One unique solution

only

a1x+b1y+c1=0

a2x +b2y+c2=0

Coincident lines Theany coordinate on the

line is the solution

Infinite solution

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Example

2x+4y=16

3x+6y=24

a1x+b1y+c1=0

a2x +b2y+c2=0

Example

2x+4y=6

4x+8y=18

Parallel Lines No solution

The graphical solution can be obtained by drawing the lines on the Cartesian plane

Algebraic Solution of system of Linear equation

Sno Type of method Working of method

1 Method of elimination by

substitution

1) Suppose the equation are

a1x+b1y+c1=0

a2x +b2y+c2=0

2) Find the value of variable of either x or y in other

variable term in first equation

3) Substitute the value of that variable in second

equation

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4) Now this is a linear equation in one variable Findthe value of the variable

5) Substitute this value in first equation and get the

second variable

2 Method of elimination by

equating the coefficients

1) Suppose the equation are

a1x+b1y+c1=0

a2x +b2y+c2=0

2) Find the LCM of a1 and a2 Let it k

3) Multiple the first equation by the value ka1

4) Multiple the first equation by the value ka2

4) Subtract the equation obtained This way one

variable will be eliminated and we can solve to get the

value of variable y

5) Substitute this value in first equation and get the

second variable

3 Cross Multiplication method 1) Suppose the equation are

a1x+b1y+c1=0

a2x +b2y+c2=0

2) This can be written as

3) This can be written as

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4) Value of x and y can be find using the

x =gt first and last expression

y=gt second and last expression

Quadratic EquationsSno Terms Descriptions

1 Quadratic Polynomial P(x) = ax +bx+c where ane0

2 Quadratic equation ax +bx+c =0 where ane0

3 Solution or root of the

Quadratic equation

A real number α is called the root or solution of the

quadratic equation if

aα2 +bα+c=0

4 zeroes of the

polynomial p(x)

The root of the quadratic equation are called zeroes

5 Maximum roots of

quadratic equations

We know from chapter two that a polynomial of

degree can have max two zeroes So a quadratic

equation can have maximum two roots

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6 Condition for realroots A quadratic equation has real roots if b

2

- 4ac gt 0

How to Solve Quadratic equation

Sno Method Working

1 factorization This method we factorize the equation by splitting themiddle term b

In ax2+bx+c=0

Example

6x2-x-2=0

1) First we need to multiple the coefficient a and cIn this

case =6X-2=-12

2) Splitting the middle term so that multiplication is 12

and difference is the coefficient b

6x2 +3x-4x-2=0

3x( 2x+1) -2(2x+1)=0

(3x-2) (2x+1)=0

3) Roots of the equation can be find equating the factorsto zero

3x-2=0 =gt x=32

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2x+1=0 =gt x=-12

2 Square method In this method we create square on LHS and RHS and

then find the value

ax2 +bx+c=0

1) x2 +(ba) x+(ca)=0

2) ( x+b2a)2 ndash(b2a)2 +(ca)=0

3) ( x+b2a)2=(b2-4ac)4a2

4) radic

Example

x2 +4x-5=0

1) (x+2)2 -4-5=0

2) (x+2)2=9

3) Roots of the equation can be find using square root on

both the sides

x+2 =-3 =gt x=-5

x+2=3=gt x=1

3 Quadratic

method

For quadratic equation

ax2 +bx+c=0

roots are given by radic radic

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For b2 -4ac gt 0 Quadratic equation has two real roots of

different value

For b2-4ac =0 quadratic equation has one real root

For b2-4ac lt 0 no real roots for quadratic equation

Nature of roots of Quadratic equation

Sno Condition Nature of roots

1 b -4ac gt 0 Two distinct real roots

2 b -4ac =0 One real root

3 b

2

-4ac lt 0 No real roots

Triangles

Sno Terms Descriptions

1 Congruence Two Geometric figure are said to be congruence ifthey are exactly same size and shapeSymbol used is cong Two angles are congruent if they are equalTwo circle are congruent if they have equal radiiTwo squares are congruent if the sides are equal

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2 Triangle Congruencebull

Two triangles are congruent if three sidesand three angles of one triangle is

congruent to the corresponding sides and

angles of the other

bull Corresponding sides are equal

AB=DE BC=EF AC=DF

bull Corresponding angles are equalang angang angang ang

bull We write this as

cong

bull The above six equalities are between the

corresponding parts of the two congruent

triangles In short form this is called

CPCT

bull We should keep the letters in correct order

on both sides

3 Inequalities in

Triangles

1) In a triangle angle opposite to longer side is

larger

2) In a triangle side opposite to larger angle is

larger

3) The sum of any two sides of the triangle is

greater than the third side

983105

983107983106

983108

983109 983110

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In triangle ABC

AB +BC gt AC

Different Criterion for Congruence of the triangles

N Criterion Description Figures and

expression

1 Side angle

Side (SAS)

congruence

bull Two triangles are congruent if the

two sides and included angles of

one triangle is equal to the two

sides and included angle

bull It is an axiom as it cannot be

proved so it is an accepted truth

bull ASS and SSA type two triangles

may not be congruent alwaysIf following

condition

983105

983110983109

983108

983107983106

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AB=DE BC=EFang ang

Then

cong

2 Angle side

angle (ASA)

congruence

bull Two triangles are congruent if the

two angles and included side of

one triangle is equal to the

corresponding angles and side

bull It is a theorem and can be proved

If following

condition

BC=EF

ang angang ang

Then

cong

3 Angle angle

side( AAS)

congruence

bull Two triangles are congruent if the

any two pair of angles and any

side of one triangle is equal to the

corresponding angles and side

bull It is a theorem and can be proved

983110983109

983108

983107983106

983105

983110983109

983108

983107983106

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If followingcondition

BC=EF

ang angang ang

Then

cong

4 Side-Side-Side

(SSS)congruence

bull Two triangles are congruent if the

three sides of one triangle is equalto the three sides of the another

If following

condition

BC=EFAB=DEDF

=AC

Then

cong

5 Right angle ndash

hypotenuse-

side(RHS)

bull Two right triangles are congruent if

the hypotenuse and a side of the

one triangle are equal to

983110983109

983108

983107983106

983105

983105 983108

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congruence corresponding hypotenuse and sideof the another

If following

condition

AC=DFBC=EF

Then

cong

Some Important points on Triangles

Terms Description

Orthocenter Point of intersection of the three altitudeof the triangle

Equilateral triangle whose all sides are equal and allangles are equal to 600

Median A line Segment joining the corner of thetriangle to the midpoint of the oppositeside of the triangle

Altitude A line Segment from the corner of thetriangle and perpendicular to theopposite side of the triangle

Isosceles A triangle whose two sides are equalCentroid Point of intersection of the three median

of the triangle is called the centroid of

the triangleIn center All the angle bisector of the trianglepasses through same point

Circumcenter The perpendicular bisector of the sidesof the triangles passes through samepoint

983106 983107 983109 983110

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2983093

Scalene triangle Triangle having no equal angles and noequal sidesRight Triangle Right triangle has one angle equal to 900 Obtuse Triangle One angle is obtuse angle while other

two are acute anglesAcute Triangle All the angles are acute

Similarity of Triangles

Sno Points

1 Two figures having the same shape but not necessarily the same size are called similarfigures

2 All the congruent figures are similar but the converse is not true

3 If a line is drawn parallel to one side of a triangle to intersect the other two sides in

distinct points then the other two sides are divided in the same ratio

4 If a line divides any two sides of a triangle in the same ratio then the line is parallel tothe third side

Different Criterion for Similarity of the triangles

N Criterion Description Expression

1 Angle Angle

angle(AAA)

similarity

bull Two triangles are similar if

corresponding angle are equal

If following

condition

ang ang

ang ang

ang ang

Then

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983151983150983148983161

2983094

Then 983166

2 Angle angle

(AA) similarity

bull Two triangles are similar if the two

corresponding angles are equal as

by angle property third angle will

be also equal

If following

condition

ang ang

ang ang

Then

ang ang

Then

Then

983166

3 Side side

side(SSS)

Similarity

Two triangles are similar if the

sides of one triangle is

proportional to the sides of other

triangle

If following

condition

Then

ang ang

ang ang

ang ang

Then

cong

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2983095

4 Side-Angle-Side (SAS)

similarity

bull

Two triangles are similar if the oneangle of a triangle is equal to one

angle of other triangles and sides

including that angle is proportional

If followingcondition

And ang ang

Then

cong

Area of Similar triangles

If the two triangle ABC and DEF are similar

cong

Then

Pythagoras Theorem

Sno Points

1 If a perpendicular is drawn from the vertex of the right angle of a right triangle to thehypotenuse then the triangles on both sides of the perpendicular are similar to the

whole triangle and also to each other

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2983096

2 In a right triangle the square of the hypotenuse is equal to the sum of the squares of the

other two sides (Pythagoras Theorem)

(hyp)2 =(base)

2 + (perp)

2

3 If in a triangle square of one side is equal to the sum of the squares of the other twosides then the angle opposite the first side is a right angle

Arithmetic Progression

Sno Terms Descriptions

1 ArithmeticProgression

An arithmetic progression is a sequence ofnumbers such that the difference of any two

successive members is a constant

Examples

1) 1591317hellip

2) 12345hellip

2 common difference of

the AP

the difference between any successive members is a

constant and it is called the common difference of AP

1) If a1 a2a3a4a5 are the terms in AP then

D=a2 -a1 =a3 - a2 =a4 ndash a3=a5 ndasha4

2) We can represent the general form of AP inthe form

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2983097

aa+da+2da+3da+4d

Where a is first term and d is the commondifference

3 nth term of Arithmetic Progression

n term = a + (n - 1)d

4 Sum of nth item in ArithmeticProgression

Sn =(n2)[a + (n-1)d]

Or

Sn =(n2)[t1+ tn]

Trigonometry

Sno Terms Descriptions

1 What is

Trigonometry

Trigonometry from Greek trigotildenon triangle and

metron measure) is a branch of mathematics

that studies relationships involving lengths and

angles of triangles The field emerged during the

3rd century BC from applications of geometry to

astronomical studies

Trigonometry is most simply associated with

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planar right angle triangles (each of which is atwo-dimensional triangle with one angle equal to

90 degrees) The applicability to non-right-angle

triangles exists but since any non-right-angle

triangle (on a flat plane) can be bisected to

create two right-angle triangles most problems

can be reduced to calculations on right-angle

triangles Thus the majority of applications relate

to right-angle triangles

2 Trigonometric

Ratiorsquos

In a right angle triangle ABC where B=90deg

We can define following term for angle A

Base Side adjacent to angle

Perpendicular Side Opposite of angleHypotenuse Side opposite to right angle

We can define the trigonometric ratios for

angle A as

sin A= PerpendicularHypotenuse =BCAC

cosec A= HypotenusePerpendicular

=ACBC

cos A= BaseHypotenuse =ABAC

sec A= HypotenuseBase=ACAB

tan A= PerpendicularBase =BCAB

cot A= BasePerpendicular=ABBC

Notice that each ratio in the right-hand column is

the inverse or the reciprocal of the ratio in the

left-hand column

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9830911

3 Reciprocal offunctions The reciprocal of sin A is cosec A and vice-versa

The reciprocal of cos A is sec A

And the reciprocal of tan A is cot A

These are valid for acute angles

We can define tan A = sin Acos A

And Cot A =cos A Sin A

4 Value of of sin and

cos

Is always less 1

5 Trigonometric ration

from another angle

We can define the trigonometric ratios for angle Cas

sin C= PerpendicularHypotenuse =ABACcosec C= HypotenusePerpendicular =ACABBcos C= BaseHypotenuse =BCACsec C= HypotenuseBase=ACBCtan A= PerpendicularBase =ABBCcot A= BasePerpendicular=BCAB

6 Trigonometric ratios

of complimentary

Sin (90-A) =cos(A)

Cos(90-A) = sin A

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9830912

angles Tan(90-A) =cot A

Sec(90-A)= cosec A

Cosec (90-A) =sec A

Cot(90- A) =tan A7 Trigonometric

identities

Sin2 A + cos2 A =0

1 + tan2 A =sec2 A

1 + cot2 A =cosec2 A

Trigonometric Ratios of Common angles

We can find the values of trigonometric ratiorsquos various angle

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Area of Circles

Sno Points

1 A circle is a collection of all the points in a plane which are equidistant

from a fixed point in the plane

2 Equal chords of a circle (or of congruent circles) subtend equal angles at

the center

3 If the angles subtended by two chords of a circle (or of congruent circles)

at the center (corresponding center) are equal the chords are equal

4 The perpendicular from the center of a circle to a chord bisects the chord

5 The line drawn through the center of a circle to bisect a chord is

perpendicular to the chord

6 There is one and only one circle passing through three non-collinear points

7 Equal chords of a circle (or of congruent circles) are equidistant from the

center (or corresponding centers)

8 Chords equidistant from the center (or corresponding centers) of a circle

(or of congruent circles) are equal

9 If two arcs of a circle are congruent then their corresponding chords are

equal and conversely if two chords of a circle are equal then their

corresponding arcs (minor major) are congruent

10 Congruent arcs of a circle subtend equal angles at the center

11 The angle subtended by an arc at the center is double the angle subtended

by it at any point on the remaining part of the circle

12 Angles in the same segment of a circle are equal

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13 Angle in a semicircle is a right angle

14 If a line segment joining two points subtends equal angles at two other

points lying on the same side of the line containing the line segment the

four points lie on a circle

15 The sum of either pair of opposite angles of a cyclic quadrilateral is 180deg

16 If the sum of a pair of opposite angles of a quadrilateral is 180deg then the

quadrilateral is cyclic

Sno Terms Descriptions

1 Circumference of a circle 2 π r

2 Area of circle π r 2

3 Length of the arc of

the sector of angle

Length of the arc of the sector of angle θ 2

4 Area of the sector of

angle

Area of the sector of angle θ

5 Area of segment of a

circle

Area of the corresponding sector ndash Area of the

corresponding triangle

Surface Area and Volume

Sno Term Description

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1 Mensuration It is branch of mathematics which is concernedabout the measurement of length area and

Volume of plane and Solid figure

2 Perimeter a)The perimeter of plane figure is defined as thelength of the boundaryb)It units is same as that of length ie m cmkm

3 Area a)The area of the plane figure is the surfaceenclosed by its boundaryb) It unit is square of length unit ie m2 km2

4 Volume Volume is the measure of the amount of spaceinside of a solid figure like a cube ball cylinderor pyramid Its units are always cubic that isthe number of little element cubes that fit insidethe figure

Volume Unit conversion

1 cm3 1mL 1000 mm3

1 Litre 1000ml 1000 cm3

1 m3 106 cm3 1000 L

1 dm3 1000 cm3 1 L

Surface Area and Volume of Cube and Cuboid

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Type Measurement

Surface Area of Cuboid of Length L

Breadth B and Height H

2(LB + BH + LH)

Lateral surface area of the cuboids 2( L + B ) H

Diagonal of the cuboids Volume of a cuboids LBH

Length of all 12 edges of the cuboids 4 (L+B+H)

Surface Area of Cube of side L 6L2

Lateral surface area of the cube 4L2

Diagonal of the cube radic3 Volume of a cube L3

Surface Area and Volume of Right circular cylinder

Radius The radius (r) of the circular base is called the radius of the

cylinder

Height The length of the axis of the cylinder is called the height (h) of the

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cylinder

Lateral

Surface

The curved surface joining the two base of a right circular cylinder is

called Lateral Surface

Type Measurement

Curved or lateral Surface Area of

cylinder

2πrh

Total surface area of cylinder 2πr (h+r)

Volume of Cylinder π r2h

Surface Area and Volume of Right circular cone

Radius The radius (r) of the circular base is called the radius of the

cone

Height The length of the line segment joining the vertex to the center of

base is called the height (h) of the cone

Slant

Height

The length of the segment joining the vertex to any point on the

circular edge of the base is called the slant height (L) of the cone

Lateral

surface

Area

The curved surface joining the base and uppermost point of a right

circular cone is called Lateral Surface

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983091983096

Type Measurement

Curved or lateral Surface Area of

cone

πrL

Total surface area of cone πr (L+r)

Volume of Cone

Surface Area and Volume of sphere and hemisphere

Sphere A sphere can also be considered as a solid obtained on

rotating a circle About its diameter

Hemisphere A plane through the centre of the sphere divides the sphere into two

equal parts each of which is called a hemisphere

radius The radius of the circle by which it is formed

SphericalShell

The difference of two solid concentric spheres is called a sphericalshell

Lateral

Surface

Area for

Total surface area of the sphere

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983091983097

Sphere

Lateral

Surface

area of

Hemisphere

It is the curved surface area leaving the circular base

Type Measurement

Surface area of Sphere 4πr2

Volume of Sphere 43 Curved Surface area of hemisphere 2πr2

Total Surface area of hemisphere 3πr2

Volume of hemisphere 23 Volume of the spherical shell whose

outer and inner radii and lsquoRrsquo and lsquorrsquorespectively

43

How the Surface area and Volume are determined

Area of Circle The circumference of a circle is 2πr

This is the definition of π (pi) Divide

the circle into many triangular

segments The area of the triangles

is 12 times the sum of their bases

2πr (the circumference) times their

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height r

12 2 Surface Area of cylinder This can be imagined as unwrapping the

surface into a rectangle

Surface area of cone This can be achieved by divide the

surface of the cone into its triangles or

the surface of the cone into many thin

triangles The area of the triangles is 12

times the sum of their bases p times

their height

12 2

Surface Area and Volume of frustum of cone

h = vertical height of the frustum

l = slant height of the frustum

r 1 and r 2 are radii of the two bases (ends) of the frustum

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983151983150983148983161

9830921

Type Measurement

983126983151983148983157983149983141 983151983142 983137 983142983154983157983155983156983157983149 983151983142 983137 983139983151983150983141 983123983148983137983150983156 983144983141983145983143983144983156 983151983142 983142983154983157983155983156983157983149 983151983142 983137 983139983151983150983141 983107983157983154983158983141983140 983155983157983154983142983137983139983141 983137983154983141983137 983151983142 983137 983142983154983157983155983156983157983149 983151983142 983137 983139983151983150983141 983124983151983156983137983148 983155983157983154983142983137983139983141 983137983154983141983137 983151983142 983142983154983157983155983156983157983149 983151983142 983137 983139983151983150983141

Statistics

Sno Term Description

1 Statistics Statistics is a broad mathematical disciplinewhich studies ways to collect summarize and

draw conclusions from data

2 Data A systematic record of facts or different values of

a quantity is called data

Data is of two types - Primary data and Secondary

data

Primary Data The data collected by a researcher

with a specific purpose in mind is called primarydata

Secondary Data The data gathered from a

source where it already exists is called secondary

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data

3 Features of data bull Statistics deals with collectionpresentation analysis and interpretation ofnumerical data

bull Arranging data in an order to study theirsalient features is called presentation ofdata

bull Data arranged in ascending or descendingorder is called arrayed data or an array

bull Range of the data is the differencebetween the maximum and the minimum

values of the observationsbull Table that shows the frequency of different

values in the given data is called afrequency distribution table

bull A frequency distribution table that showsthe frequency of each individual value in thegiven data is called an ungrouped frequencydistribution table

bull A table that shows the frequency of groupsof values in the given data is called agrouped frequency distribution table

bull

The groupings used to group the values ingiven data are called classes or class-intervals The number of values that eachclass contains is called the class size orclass width The lower value in a class iscalled the lower class limit The higher valuein a class is called the upper class limit

bull Class mark of a class is the mid value ofthe two limits of that class

bull A frequency distribution in which the upperlimit of one class differs from the lower limitof the succeeding class is called an

Inclusive or discontinuous FrequencyDistribution

bull A frequency distribution in which the upperlimit of one class coincides from the lowerlimit of the succeeding class is called anexclusive or continuous Frequency

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Distribution bull

4 Bar graph A bar graph is a pictorial representation of data inwhich rectangular bars of uniform width are drawnwith equal spacing between them on one axisusually the x axis The value of the variable isshown on the other axis that is the y axis

5 Histogram A histogram is a set of adjacent rectangles whoseareas are proportional to the frequencies of agiven continuous frequency distribution

6 Mean The mean value of a variable is defined as thesum of all the values of the variable divided by thenumber of values

4 sum

7 Median The median of a set of data values is the middle

value of the data set when it has been arranged inascending order That is from the smallest valueto the highest value

Median is calculated as

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12 1 Where n is the number of values in the data

If the number of values in the data set is eventhen the median is the average of the two middlevalues

8 Mode Mode of a statistical data is the value of thatvariable which has the maximum frequency

Sno Term Description

1983117983141983137983150 983142983151983154

983125983150983143983154983151983157983152

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Mean is given by

sum sum

2983117983141983137983150 983142983151983154 983143983154983151983157983152

983110983154983141983153983157983141983150983139983161 983156983137983138983148983141

In these distribution it is assumed that frequency of each classinterval is centered around its mid-point ie class marks

2

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983137983081 983108983145983154983141983139983156 983149983141983156983144983151983140

sum

sum

983138983081 983105983155983155983157983149983141983140 983149983141983137983150 983149983141983156983144983151983140

sum sum

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983151983150983148983161

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Where

a=gt Assumed mean

di =gt xi ndasha

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sum sum

Where

a=gt Assumed mean

ui =gt (xi ndasha)h

3983117983151983140983141 983142983151983154

983143983154983151983157983152983141983140

983142983154983141983153983157983141983150983139983161 983156983137983138983148983141

Modal class The class interval having highest frequency is

called the modal class and Mode is obtained using the modal

class

2

983127983144983141983154983141

l = lower limit of the modal classh = size of the class interval (assuming all class sizes tobe equal)f 1 = frequency of the modal classf 0 = frequency of the class preceding the modal classf 2 = frequency of the class succeeding the modal class

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For the given data we need to have class interval frequency

distribution and cumulative frequency distribution

Median is calculated as

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2

983127983144983141983154983141

983148 983101 983148983151983159983141983154 983148983145983149983145983156 983151983142 983149983141983140983145983137983150 983139983148983137983155983155

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983139983142 983101 983139983157983149983157983148983137983156983145983158983141 983142983154983141983153983157983141983150983139983161 983151983142 983139983148983137983155983155 983152983154983141983139983141983140983145983150983143 983156983144983141 983149983141983140983145983137983150 983139983148983137983155983155

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983144 983101 983139983148983137983155983155 983155983145983162983141 (983137983155983155983157983149983145983150983143 983139983148983137983155983155 983155983145983162983141 983156983151 983138983141 983141983153983157983137983148)

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3 Median=Mode +2 Mean

ProbabilitySn

o

Term Description

1 Empirical

probability

It is a probability of event which is calculated based on

experiments

Example

A coin is tossed 1000 times we get 499 times head and

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501 times tail

So empirical or experimental probability of getting head is

calculated as

4991000 499

Empirical probability depends on experiment and

different will get different values based on the

experiment

2 Important point

about events

If the event A B C covers the entire possible outcome inthe experiment Then

P (A) +P (B) +P(C)=1

3 impossible

event

The probability of an event (U) which is impossible tooccur is 0 Such an event is called an impossible event

P (U)=0

4 Sure or certain

event

The probability of an event (X) which is sure (or certain)to occur is 1 Such an event is called a sure event or acertain event

P(X) =1

5 Probability of

any event

Probability of any event can be as

0 1

Sn

o

Term Description

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1

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The theoretical probability or the classical probability of the event isdefined as

2 Elementary

events

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983120(983105)+983120(983106) +983120(983107)9831011

3 Complementar

y events

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983120(983105) +983120(Ᾱ)9831011

4 Sure or certain

event

The probability of an event (X) which is sure (or certain)

to occur is 1 Such an event is called a sure event or acertain event

P(X) =1

5 Probability of

any event

Probability of any event can be as

0 1

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11

The midpoint P is given by

3 Area of Triangle Area of triangle ABC of coordinates A(x 1y 1 )

B(x 2 y 2 ) and C(x 3y 3 )

12 For point AB and C to be collinear The value of A

should be zero

LINEAR EQUATIONS IN TWO

VARIABLESAn equation of the form ax + by + c = 0 where a b and c are real numbers such

that a andb are not both zero is called a linear equation in two variablesImportant points to Note

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12

Sno Points

1 A linear equation in two variable has infinite solutions

2 The graph of every linear equation in two variable is a straight line

3 x = 0 is the equation of the y-axis and y = 0 is the equation of the x-axis

4 The graph x=a is a line parallel to y -axis

5 The graph y=b is a line parallel to x -axis

6 An equation of the type y = mx represents a line passing through theorigin

7 Every point on the graph of a linear equation in two variables is a solution

of the linear

equation Moreover every solution of the linear equation is a point on the

graph

Sno Type of equation Mathematical

representation

Solutions

1 Linear equation in one Variable ax+b=0 ane0

a and b are real

number

One solution

2 Linear equation in two Variable ax+by+c=0 ane0 and

bne0

a b and c are real

number

Infinite solution

possible

3 Linear equation in three Variable ax+by+cz+d=0 ane0

bne0 and cne0

Infinite solution

possible

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a b c d are realnumber

Simultaneous pair of Linear equation

A pair of Linear equation in two variables

a1x+b1y+c1=0

a2x +b2y+c2=0

Graphically it is represented by two straight lines on Cartesian plane

Simultaneous pair of

Linear equation

Condition Graphical

representation

Algebraic

interpretation

a1x+b1y+c1=0

a2x +b2y+c2=0

Example

x-4y+14=0

3x+2y-14=0

Intersecting lines The

intersecting point

coordinate is the only

solution

One unique solution

only

a1x+b1y+c1=0

a2x +b2y+c2=0

Coincident lines Theany coordinate on the

line is the solution

Infinite solution

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Example

2x+4y=16

3x+6y=24

a1x+b1y+c1=0

a2x +b2y+c2=0

Example

2x+4y=6

4x+8y=18

Parallel Lines No solution

The graphical solution can be obtained by drawing the lines on the Cartesian plane

Algebraic Solution of system of Linear equation

Sno Type of method Working of method

1 Method of elimination by

substitution

1) Suppose the equation are

a1x+b1y+c1=0

a2x +b2y+c2=0

2) Find the value of variable of either x or y in other

variable term in first equation

3) Substitute the value of that variable in second

equation

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1983093

4) Now this is a linear equation in one variable Findthe value of the variable

5) Substitute this value in first equation and get the

second variable

2 Method of elimination by

equating the coefficients

1) Suppose the equation are

a1x+b1y+c1=0

a2x +b2y+c2=0

2) Find the LCM of a1 and a2 Let it k

3) Multiple the first equation by the value ka1

4) Multiple the first equation by the value ka2

4) Subtract the equation obtained This way one

variable will be eliminated and we can solve to get the

value of variable y

5) Substitute this value in first equation and get the

second variable

3 Cross Multiplication method 1) Suppose the equation are

a1x+b1y+c1=0

a2x +b2y+c2=0

2) This can be written as

3) This can be written as

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4) Value of x and y can be find using the

x =gt first and last expression

y=gt second and last expression

Quadratic EquationsSno Terms Descriptions

1 Quadratic Polynomial P(x) = ax +bx+c where ane0

2 Quadratic equation ax +bx+c =0 where ane0

3 Solution or root of the

Quadratic equation

A real number α is called the root or solution of the

quadratic equation if

aα2 +bα+c=0

4 zeroes of the

polynomial p(x)

The root of the quadratic equation are called zeroes

5 Maximum roots of

quadratic equations

We know from chapter two that a polynomial of

degree can have max two zeroes So a quadratic

equation can have maximum two roots

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6 Condition for realroots A quadratic equation has real roots if b

2

- 4ac gt 0

How to Solve Quadratic equation

Sno Method Working

1 factorization This method we factorize the equation by splitting themiddle term b

In ax2+bx+c=0

Example

6x2-x-2=0

1) First we need to multiple the coefficient a and cIn this

case =6X-2=-12

2) Splitting the middle term so that multiplication is 12

and difference is the coefficient b

6x2 +3x-4x-2=0

3x( 2x+1) -2(2x+1)=0

(3x-2) (2x+1)=0

3) Roots of the equation can be find equating the factorsto zero

3x-2=0 =gt x=32

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2x+1=0 =gt x=-12

2 Square method In this method we create square on LHS and RHS and

then find the value

ax2 +bx+c=0

1) x2 +(ba) x+(ca)=0

2) ( x+b2a)2 ndash(b2a)2 +(ca)=0

3) ( x+b2a)2=(b2-4ac)4a2

4) radic

Example

x2 +4x-5=0

1) (x+2)2 -4-5=0

2) (x+2)2=9

3) Roots of the equation can be find using square root on

both the sides

x+2 =-3 =gt x=-5

x+2=3=gt x=1

3 Quadratic

method

For quadratic equation

ax2 +bx+c=0

roots are given by radic radic

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For b2 -4ac gt 0 Quadratic equation has two real roots of

different value

For b2-4ac =0 quadratic equation has one real root

For b2-4ac lt 0 no real roots for quadratic equation

Nature of roots of Quadratic equation

Sno Condition Nature of roots

1 b -4ac gt 0 Two distinct real roots

2 b -4ac =0 One real root

3 b

2

-4ac lt 0 No real roots

Triangles

Sno Terms Descriptions

1 Congruence Two Geometric figure are said to be congruence ifthey are exactly same size and shapeSymbol used is cong Two angles are congruent if they are equalTwo circle are congruent if they have equal radiiTwo squares are congruent if the sides are equal

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20

2 Triangle Congruencebull

Two triangles are congruent if three sidesand three angles of one triangle is

congruent to the corresponding sides and

angles of the other

bull Corresponding sides are equal

AB=DE BC=EF AC=DF

bull Corresponding angles are equalang angang angang ang

bull We write this as

cong

bull The above six equalities are between the

corresponding parts of the two congruent

triangles In short form this is called

CPCT

bull We should keep the letters in correct order

on both sides

3 Inequalities in

Triangles

1) In a triangle angle opposite to longer side is

larger

2) In a triangle side opposite to larger angle is

larger

3) The sum of any two sides of the triangle is

greater than the third side

983105

983107983106

983108

983109 983110

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21

In triangle ABC

AB +BC gt AC

Different Criterion for Congruence of the triangles

N Criterion Description Figures and

expression

1 Side angle

Side (SAS)

congruence

bull Two triangles are congruent if the

two sides and included angles of

one triangle is equal to the two

sides and included angle

bull It is an axiom as it cannot be

proved so it is an accepted truth

bull ASS and SSA type two triangles

may not be congruent alwaysIf following

condition

983105

983110983109

983108

983107983106

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22

AB=DE BC=EFang ang

Then

cong

2 Angle side

angle (ASA)

congruence

bull Two triangles are congruent if the

two angles and included side of

one triangle is equal to the

corresponding angles and side

bull It is a theorem and can be proved

If following

condition

BC=EF

ang angang ang

Then

cong

3 Angle angle

side( AAS)

congruence

bull Two triangles are congruent if the

any two pair of angles and any

side of one triangle is equal to the

corresponding angles and side

bull It is a theorem and can be proved

983110983109

983108

983107983106

983105

983110983109

983108

983107983106

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If followingcondition

BC=EF

ang angang ang

Then

cong

4 Side-Side-Side

(SSS)congruence

bull Two triangles are congruent if the

three sides of one triangle is equalto the three sides of the another

If following

condition

BC=EFAB=DEDF

=AC

Then

cong

5 Right angle ndash

hypotenuse-

side(RHS)

bull Two right triangles are congruent if

the hypotenuse and a side of the

one triangle are equal to

983110983109

983108

983107983106

983105

983105 983108

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congruence corresponding hypotenuse and sideof the another

If following

condition

AC=DFBC=EF

Then

cong

Some Important points on Triangles

Terms Description

Orthocenter Point of intersection of the three altitudeof the triangle

Equilateral triangle whose all sides are equal and allangles are equal to 600

Median A line Segment joining the corner of thetriangle to the midpoint of the oppositeside of the triangle

Altitude A line Segment from the corner of thetriangle and perpendicular to theopposite side of the triangle

Isosceles A triangle whose two sides are equalCentroid Point of intersection of the three median

of the triangle is called the centroid of

the triangleIn center All the angle bisector of the trianglepasses through same point

Circumcenter The perpendicular bisector of the sidesof the triangles passes through samepoint

983106 983107 983109 983110

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Scalene triangle Triangle having no equal angles and noequal sidesRight Triangle Right triangle has one angle equal to 900 Obtuse Triangle One angle is obtuse angle while other

two are acute anglesAcute Triangle All the angles are acute

Similarity of Triangles

Sno Points

1 Two figures having the same shape but not necessarily the same size are called similarfigures

2 All the congruent figures are similar but the converse is not true

3 If a line is drawn parallel to one side of a triangle to intersect the other two sides in

distinct points then the other two sides are divided in the same ratio

4 If a line divides any two sides of a triangle in the same ratio then the line is parallel tothe third side

Different Criterion for Similarity of the triangles

N Criterion Description Expression

1 Angle Angle

angle(AAA)

similarity

bull Two triangles are similar if

corresponding angle are equal

If following

condition

ang ang

ang ang

ang ang

Then

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Then 983166

2 Angle angle

(AA) similarity

bull Two triangles are similar if the two

corresponding angles are equal as

by angle property third angle will

be also equal

If following

condition

ang ang

ang ang

Then

ang ang

Then

Then

983166

3 Side side

side(SSS)

Similarity

Two triangles are similar if the

sides of one triangle is

proportional to the sides of other

triangle

If following

condition

Then

ang ang

ang ang

ang ang

Then

cong

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4 Side-Angle-Side (SAS)

similarity

bull

Two triangles are similar if the oneangle of a triangle is equal to one

angle of other triangles and sides

including that angle is proportional

If followingcondition

And ang ang

Then

cong

Area of Similar triangles

If the two triangle ABC and DEF are similar

cong

Then

Pythagoras Theorem

Sno Points

1 If a perpendicular is drawn from the vertex of the right angle of a right triangle to thehypotenuse then the triangles on both sides of the perpendicular are similar to the

whole triangle and also to each other

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2 In a right triangle the square of the hypotenuse is equal to the sum of the squares of the

other two sides (Pythagoras Theorem)

(hyp)2 =(base)

2 + (perp)

2

3 If in a triangle square of one side is equal to the sum of the squares of the other twosides then the angle opposite the first side is a right angle

Arithmetic Progression

Sno Terms Descriptions

1 ArithmeticProgression

An arithmetic progression is a sequence ofnumbers such that the difference of any two

successive members is a constant

Examples

1) 1591317hellip

2) 12345hellip

2 common difference of

the AP

the difference between any successive members is a

constant and it is called the common difference of AP

1) If a1 a2a3a4a5 are the terms in AP then

D=a2 -a1 =a3 - a2 =a4 ndash a3=a5 ndasha4

2) We can represent the general form of AP inthe form

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aa+da+2da+3da+4d

Where a is first term and d is the commondifference

3 nth term of Arithmetic Progression

n term = a + (n - 1)d

4 Sum of nth item in ArithmeticProgression

Sn =(n2)[a + (n-1)d]

Or

Sn =(n2)[t1+ tn]

Trigonometry

Sno Terms Descriptions

1 What is

Trigonometry

Trigonometry from Greek trigotildenon triangle and

metron measure) is a branch of mathematics

that studies relationships involving lengths and

angles of triangles The field emerged during the

3rd century BC from applications of geometry to

astronomical studies

Trigonometry is most simply associated with

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planar right angle triangles (each of which is atwo-dimensional triangle with one angle equal to

90 degrees) The applicability to non-right-angle

triangles exists but since any non-right-angle

triangle (on a flat plane) can be bisected to

create two right-angle triangles most problems

can be reduced to calculations on right-angle

triangles Thus the majority of applications relate

to right-angle triangles

2 Trigonometric

Ratiorsquos

In a right angle triangle ABC where B=90deg

We can define following term for angle A

Base Side adjacent to angle

Perpendicular Side Opposite of angleHypotenuse Side opposite to right angle

We can define the trigonometric ratios for

angle A as

sin A= PerpendicularHypotenuse =BCAC

cosec A= HypotenusePerpendicular

=ACBC

cos A= BaseHypotenuse =ABAC

sec A= HypotenuseBase=ACAB

tan A= PerpendicularBase =BCAB

cot A= BasePerpendicular=ABBC

Notice that each ratio in the right-hand column is

the inverse or the reciprocal of the ratio in the

left-hand column

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3 Reciprocal offunctions The reciprocal of sin A is cosec A and vice-versa

The reciprocal of cos A is sec A

And the reciprocal of tan A is cot A

These are valid for acute angles

We can define tan A = sin Acos A

And Cot A =cos A Sin A

4 Value of of sin and

cos

Is always less 1

5 Trigonometric ration

from another angle

We can define the trigonometric ratios for angle Cas

sin C= PerpendicularHypotenuse =ABACcosec C= HypotenusePerpendicular =ACABBcos C= BaseHypotenuse =BCACsec C= HypotenuseBase=ACBCtan A= PerpendicularBase =ABBCcot A= BasePerpendicular=BCAB

6 Trigonometric ratios

of complimentary

Sin (90-A) =cos(A)

Cos(90-A) = sin A

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angles Tan(90-A) =cot A

Sec(90-A)= cosec A

Cosec (90-A) =sec A

Cot(90- A) =tan A7 Trigonometric

identities

Sin2 A + cos2 A =0

1 + tan2 A =sec2 A

1 + cot2 A =cosec2 A

Trigonometric Ratios of Common angles

We can find the values of trigonometric ratiorsquos various angle

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Area of Circles

Sno Points

1 A circle is a collection of all the points in a plane which are equidistant

from a fixed point in the plane

2 Equal chords of a circle (or of congruent circles) subtend equal angles at

the center

3 If the angles subtended by two chords of a circle (or of congruent circles)

at the center (corresponding center) are equal the chords are equal

4 The perpendicular from the center of a circle to a chord bisects the chord

5 The line drawn through the center of a circle to bisect a chord is

perpendicular to the chord

6 There is one and only one circle passing through three non-collinear points

7 Equal chords of a circle (or of congruent circles) are equidistant from the

center (or corresponding centers)

8 Chords equidistant from the center (or corresponding centers) of a circle

(or of congruent circles) are equal

9 If two arcs of a circle are congruent then their corresponding chords are

equal and conversely if two chords of a circle are equal then their

corresponding arcs (minor major) are congruent

10 Congruent arcs of a circle subtend equal angles at the center

11 The angle subtended by an arc at the center is double the angle subtended

by it at any point on the remaining part of the circle

12 Angles in the same segment of a circle are equal

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13 Angle in a semicircle is a right angle

14 If a line segment joining two points subtends equal angles at two other

points lying on the same side of the line containing the line segment the

four points lie on a circle

15 The sum of either pair of opposite angles of a cyclic quadrilateral is 180deg

16 If the sum of a pair of opposite angles of a quadrilateral is 180deg then the

quadrilateral is cyclic

Sno Terms Descriptions

1 Circumference of a circle 2 π r

2 Area of circle π r 2

3 Length of the arc of

the sector of angle

Length of the arc of the sector of angle θ 2

4 Area of the sector of

angle

Area of the sector of angle θ

5 Area of segment of a

circle

Area of the corresponding sector ndash Area of the

corresponding triangle

Surface Area and Volume

Sno Term Description

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1 Mensuration It is branch of mathematics which is concernedabout the measurement of length area and

Volume of plane and Solid figure

2 Perimeter a)The perimeter of plane figure is defined as thelength of the boundaryb)It units is same as that of length ie m cmkm

3 Area a)The area of the plane figure is the surfaceenclosed by its boundaryb) It unit is square of length unit ie m2 km2

4 Volume Volume is the measure of the amount of spaceinside of a solid figure like a cube ball cylinderor pyramid Its units are always cubic that isthe number of little element cubes that fit insidethe figure

Volume Unit conversion

1 cm3 1mL 1000 mm3

1 Litre 1000ml 1000 cm3

1 m3 106 cm3 1000 L

1 dm3 1000 cm3 1 L

Surface Area and Volume of Cube and Cuboid

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Type Measurement

Surface Area of Cuboid of Length L

Breadth B and Height H

2(LB + BH + LH)

Lateral surface area of the cuboids 2( L + B ) H

Diagonal of the cuboids Volume of a cuboids LBH

Length of all 12 edges of the cuboids 4 (L+B+H)

Surface Area of Cube of side L 6L2

Lateral surface area of the cube 4L2

Diagonal of the cube radic3 Volume of a cube L3

Surface Area and Volume of Right circular cylinder

Radius The radius (r) of the circular base is called the radius of the

cylinder

Height The length of the axis of the cylinder is called the height (h) of the

983107983157983138983141 983107983157983138983151983145983140

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cylinder

Lateral

Surface

The curved surface joining the two base of a right circular cylinder is

called Lateral Surface

Type Measurement

Curved or lateral Surface Area of

cylinder

2πrh

Total surface area of cylinder 2πr (h+r)

Volume of Cylinder π r2h

Surface Area and Volume of Right circular cone

Radius The radius (r) of the circular base is called the radius of the

cone

Height The length of the line segment joining the vertex to the center of

base is called the height (h) of the cone

Slant

Height

The length of the segment joining the vertex to any point on the

circular edge of the base is called the slant height (L) of the cone

Lateral

surface

Area

The curved surface joining the base and uppermost point of a right

circular cone is called Lateral Surface

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Type Measurement

Curved or lateral Surface Area of

cone

πrL

Total surface area of cone πr (L+r)

Volume of Cone

Surface Area and Volume of sphere and hemisphere

Sphere A sphere can also be considered as a solid obtained on

rotating a circle About its diameter

Hemisphere A plane through the centre of the sphere divides the sphere into two

equal parts each of which is called a hemisphere

radius The radius of the circle by which it is formed

SphericalShell

The difference of two solid concentric spheres is called a sphericalshell

Lateral

Surface

Area for

Total surface area of the sphere

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Sphere

Lateral

Surface

area of

Hemisphere

It is the curved surface area leaving the circular base

Type Measurement

Surface area of Sphere 4πr2

Volume of Sphere 43 Curved Surface area of hemisphere 2πr2

Total Surface area of hemisphere 3πr2

Volume of hemisphere 23 Volume of the spherical shell whose

outer and inner radii and lsquoRrsquo and lsquorrsquorespectively

43

How the Surface area and Volume are determined

Area of Circle The circumference of a circle is 2πr

This is the definition of π (pi) Divide

the circle into many triangular

segments The area of the triangles

is 12 times the sum of their bases

2πr (the circumference) times their

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height r

12 2 Surface Area of cylinder This can be imagined as unwrapping the

surface into a rectangle

Surface area of cone This can be achieved by divide the

surface of the cone into its triangles or

the surface of the cone into many thin

triangles The area of the triangles is 12

times the sum of their bases p times

their height

12 2

Surface Area and Volume of frustum of cone

h = vertical height of the frustum

l = slant height of the frustum

r 1 and r 2 are radii of the two bases (ends) of the frustum

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Type Measurement

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Statistics

Sno Term Description

1 Statistics Statistics is a broad mathematical disciplinewhich studies ways to collect summarize and

draw conclusions from data

2 Data A systematic record of facts or different values of

a quantity is called data

Data is of two types - Primary data and Secondary

data

Primary Data The data collected by a researcher

with a specific purpose in mind is called primarydata

Secondary Data The data gathered from a

source where it already exists is called secondary

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data

3 Features of data bull Statistics deals with collectionpresentation analysis and interpretation ofnumerical data

bull Arranging data in an order to study theirsalient features is called presentation ofdata

bull Data arranged in ascending or descendingorder is called arrayed data or an array

bull Range of the data is the differencebetween the maximum and the minimum

values of the observationsbull Table that shows the frequency of different

values in the given data is called afrequency distribution table

bull A frequency distribution table that showsthe frequency of each individual value in thegiven data is called an ungrouped frequencydistribution table

bull A table that shows the frequency of groupsof values in the given data is called agrouped frequency distribution table

bull

The groupings used to group the values ingiven data are called classes or class-intervals The number of values that eachclass contains is called the class size orclass width The lower value in a class iscalled the lower class limit The higher valuein a class is called the upper class limit

bull Class mark of a class is the mid value ofthe two limits of that class

bull A frequency distribution in which the upperlimit of one class differs from the lower limitof the succeeding class is called an

Inclusive or discontinuous FrequencyDistribution

bull A frequency distribution in which the upperlimit of one class coincides from the lowerlimit of the succeeding class is called anexclusive or continuous Frequency

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Distribution bull

4 Bar graph A bar graph is a pictorial representation of data inwhich rectangular bars of uniform width are drawnwith equal spacing between them on one axisusually the x axis The value of the variable isshown on the other axis that is the y axis

5 Histogram A histogram is a set of adjacent rectangles whoseareas are proportional to the frequencies of agiven continuous frequency distribution

6 Mean The mean value of a variable is defined as thesum of all the values of the variable divided by thenumber of values

4 sum

7 Median The median of a set of data values is the middle

value of the data set when it has been arranged inascending order That is from the smallest valueto the highest value

Median is calculated as

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12 1 Where n is the number of values in the data

If the number of values in the data set is eventhen the median is the average of the two middlevalues

8 Mode Mode of a statistical data is the value of thatvariable which has the maximum frequency

Sno Term Description

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983125983150983143983154983151983157983152

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Mean is given by

sum sum

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In these distribution it is assumed that frequency of each classinterval is centered around its mid-point ie class marks

2

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sum

sum

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sum sum

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Where

a=gt Assumed mean

di =gt xi ndasha

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sum sum

Where

a=gt Assumed mean

ui =gt (xi ndasha)h

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Modal class The class interval having highest frequency is

called the modal class and Mode is obtained using the modal

class

2

983127983144983141983154983141

l = lower limit of the modal classh = size of the class interval (assuming all class sizes tobe equal)f 1 = frequency of the modal classf 0 = frequency of the class preceding the modal classf 2 = frequency of the class succeeding the modal class

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983143983154983151983157983152983141983140 983140983137983156983137

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For the given data we need to have class interval frequency

distribution and cumulative frequency distribution

Median is calculated as

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2

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983148 983101 983148983151983159983141983154 983148983145983149983145983156 983151983142 983149983141983140983145983137983150 983139983148983137983155983155

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983139983142 983101 983139983157983149983157983148983137983156983145983158983141 983142983154983141983153983157983141983150983139983161 983151983142 983139983148983137983155983155 983152983154983141983139983141983140983145983150983143 983156983144983141 983149983141983140983145983137983150 983139983148983137983155983155

983142 983101 983142983154983141983153983157983141983150983139983161 983151983142 983149983141983140983145983137983150 983139983148983137983155983155

983144 983101 983139983148983137983155983155 983155983145983162983141 (983137983155983155983157983149983145983150983143 983139983148983137983155983155 983155983145983162983141 983156983151 983138983141 983141983153983157983137983148)

5983109983149983152983145983154983145983139983137983148

983110983151983154983149983157983148983137 983138983141983156983159983141983141983150

983117983151983140983141983084 983117983141983137983150 983137983150983140

983117983141983140983145983137983150

3 Median=Mode +2 Mean

ProbabilitySn

o

Term Description

1 Empirical

probability

It is a probability of event which is calculated based on

experiments

Example

A coin is tossed 1000 times we get 499 times head and

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983151983150983148983161

983092983095

501 times tail

So empirical or experimental probability of getting head is

calculated as

4991000 499

Empirical probability depends on experiment and

different will get different values based on the

experiment

2 Important point

about events

If the event A B C covers the entire possible outcome inthe experiment Then

P (A) +P (B) +P(C)=1

3 impossible

event

The probability of an event (U) which is impossible tooccur is 0 Such an event is called an impossible event

P (U)=0

4 Sure or certain

event

The probability of an event (X) which is sure (or certain)to occur is 1 Such an event is called a sure event or acertain event

P(X) =1

5 Probability of

any event

Probability of any event can be as

0 1

Sn

o

Term Description

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983092983096

1

983124983144983141983151983154983141983156983145983139983137983148

983120983154983151983138983137983138983145983148983145983156983161

The theoretical probability or the classical probability of the event isdefined as

2 Elementary

events

983105983150 983141983158983141983150983156 983144983137983158983145983150983143 983151983150983148983161 983151983150983141 983151983157983156983139983151983149983141 983151983142 983156983144983141 983141983160983152983141983154983145983149983141983150983156 983145983155 983139983137983148983148983141983140 983137983150

983141983148983141983149983141983150983156983137983154983161 983141983158983141983150983156

991260983124983144983141 983155983157983149 983151983142 983156983144983141 983152983154983151983138983137983138983145983148983145983156983145983141983155 983151983142 983137983148983148 983156983144983141 983141983148983141983149983141983150983156983137983154983161 983141983158983141983150983156983155 983151983142983137983150 983141983160983152983141983154983145983149983141983150983156 983145983155 1991261

983113983141 983113983142 983159983141 983156983144983154983141983141 983141983148983141983149983141983150983156983137983154983161 983141983158983141983150983156 983105983106983107 983145983150 983156983144983141 983141983160983152983141983154983145983149983141983150983156 983156983144983141983150

983120(983105)+983120(983106) +983120(983107)9831011

3 Complementar

y events

983124983144983141 983141983158983141983150983156 Ᾱ 983154983141983152983154983141983155983141983150983156983145983150983143 991256983150983151983156 983105991257 983145983155 983139983137983148983148983141983140 983156983144983141 983139983151983149983152983148983141983149983141983150983156 983151983142 983156983144983141 983141983158983141983150983156

983105 983127983141 983137983148983155983151 983155983137983161 983156983144983137983156 Ᾱ 983137983150983140 983105 983137983154983141 983139983151983149983152983148983141983149983141983150983156983137983154983161 983141983158983141983150983156983155 983105983148983155983151

983120(983105) +983120(Ᾱ)9831011

4 Sure or certain

event

The probability of an event (X) which is sure (or certain)

to occur is 1 Such an event is called a sure event or acertain event

P(X) =1

5 Probability of

any event

Probability of any event can be as

0 1

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12

Sno Points

1 A linear equation in two variable has infinite solutions

2 The graph of every linear equation in two variable is a straight line

3 x = 0 is the equation of the y-axis and y = 0 is the equation of the x-axis

4 The graph x=a is a line parallel to y -axis

5 The graph y=b is a line parallel to x -axis

6 An equation of the type y = mx represents a line passing through theorigin

7 Every point on the graph of a linear equation in two variables is a solution

of the linear

equation Moreover every solution of the linear equation is a point on the

graph

Sno Type of equation Mathematical

representation

Solutions

1 Linear equation in one Variable ax+b=0 ane0

a and b are real

number

One solution

2 Linear equation in two Variable ax+by+c=0 ane0 and

bne0

a b and c are real

number

Infinite solution

possible

3 Linear equation in three Variable ax+by+cz+d=0 ane0

bne0 and cne0

Infinite solution

possible

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a b c d are realnumber

Simultaneous pair of Linear equation

A pair of Linear equation in two variables

a1x+b1y+c1=0

a2x +b2y+c2=0

Graphically it is represented by two straight lines on Cartesian plane

Simultaneous pair of

Linear equation

Condition Graphical

representation

Algebraic

interpretation

a1x+b1y+c1=0

a2x +b2y+c2=0

Example

x-4y+14=0

3x+2y-14=0

Intersecting lines The

intersecting point

coordinate is the only

solution

One unique solution

only

a1x+b1y+c1=0

a2x +b2y+c2=0

Coincident lines Theany coordinate on the

line is the solution

Infinite solution

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Example

2x+4y=16

3x+6y=24

a1x+b1y+c1=0

a2x +b2y+c2=0

Example

2x+4y=6

4x+8y=18

Parallel Lines No solution

The graphical solution can be obtained by drawing the lines on the Cartesian plane

Algebraic Solution of system of Linear equation

Sno Type of method Working of method

1 Method of elimination by

substitution

1) Suppose the equation are

a1x+b1y+c1=0

a2x +b2y+c2=0

2) Find the value of variable of either x or y in other

variable term in first equation

3) Substitute the value of that variable in second

equation

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4) Now this is a linear equation in one variable Findthe value of the variable

5) Substitute this value in first equation and get the

second variable

2 Method of elimination by

equating the coefficients

1) Suppose the equation are

a1x+b1y+c1=0

a2x +b2y+c2=0

2) Find the LCM of a1 and a2 Let it k

3) Multiple the first equation by the value ka1

4) Multiple the first equation by the value ka2

4) Subtract the equation obtained This way one

variable will be eliminated and we can solve to get the

value of variable y

5) Substitute this value in first equation and get the

second variable

3 Cross Multiplication method 1) Suppose the equation are

a1x+b1y+c1=0

a2x +b2y+c2=0

2) This can be written as

3) This can be written as

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4) Value of x and y can be find using the

x =gt first and last expression

y=gt second and last expression

Quadratic EquationsSno Terms Descriptions

1 Quadratic Polynomial P(x) = ax +bx+c where ane0

2 Quadratic equation ax +bx+c =0 where ane0

3 Solution or root of the

Quadratic equation

A real number α is called the root or solution of the

quadratic equation if

aα2 +bα+c=0

4 zeroes of the

polynomial p(x)

The root of the quadratic equation are called zeroes

5 Maximum roots of

quadratic equations

We know from chapter two that a polynomial of

degree can have max two zeroes So a quadratic

equation can have maximum two roots

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6 Condition for realroots A quadratic equation has real roots if b

2

- 4ac gt 0

How to Solve Quadratic equation

Sno Method Working

1 factorization This method we factorize the equation by splitting themiddle term b

In ax2+bx+c=0

Example

6x2-x-2=0

1) First we need to multiple the coefficient a and cIn this

case =6X-2=-12

2) Splitting the middle term so that multiplication is 12

and difference is the coefficient b

6x2 +3x-4x-2=0

3x( 2x+1) -2(2x+1)=0

(3x-2) (2x+1)=0

3) Roots of the equation can be find equating the factorsto zero

3x-2=0 =gt x=32

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2x+1=0 =gt x=-12

2 Square method In this method we create square on LHS and RHS and

then find the value

ax2 +bx+c=0

1) x2 +(ba) x+(ca)=0

2) ( x+b2a)2 ndash(b2a)2 +(ca)=0

3) ( x+b2a)2=(b2-4ac)4a2

4) radic

Example

x2 +4x-5=0

1) (x+2)2 -4-5=0

2) (x+2)2=9

3) Roots of the equation can be find using square root on

both the sides

x+2 =-3 =gt x=-5

x+2=3=gt x=1

3 Quadratic

method

For quadratic equation

ax2 +bx+c=0

roots are given by radic radic

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For b2 -4ac gt 0 Quadratic equation has two real roots of

different value

For b2-4ac =0 quadratic equation has one real root

For b2-4ac lt 0 no real roots for quadratic equation

Nature of roots of Quadratic equation

Sno Condition Nature of roots

1 b -4ac gt 0 Two distinct real roots

2 b -4ac =0 One real root

3 b

2

-4ac lt 0 No real roots

Triangles

Sno Terms Descriptions

1 Congruence Two Geometric figure are said to be congruence ifthey are exactly same size and shapeSymbol used is cong Two angles are congruent if they are equalTwo circle are congruent if they have equal radiiTwo squares are congruent if the sides are equal

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2 Triangle Congruencebull

Two triangles are congruent if three sidesand three angles of one triangle is

congruent to the corresponding sides and

angles of the other

bull Corresponding sides are equal

AB=DE BC=EF AC=DF

bull Corresponding angles are equalang angang angang ang

bull We write this as

cong

bull The above six equalities are between the

corresponding parts of the two congruent

triangles In short form this is called

CPCT

bull We should keep the letters in correct order

on both sides

3 Inequalities in

Triangles

1) In a triangle angle opposite to longer side is

larger

2) In a triangle side opposite to larger angle is

larger

3) The sum of any two sides of the triangle is

greater than the third side

983105

983107983106

983108

983109 983110

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In triangle ABC

AB +BC gt AC

Different Criterion for Congruence of the triangles

N Criterion Description Figures and

expression

1 Side angle

Side (SAS)

congruence

bull Two triangles are congruent if the

two sides and included angles of

one triangle is equal to the two

sides and included angle

bull It is an axiom as it cannot be

proved so it is an accepted truth

bull ASS and SSA type two triangles

may not be congruent alwaysIf following

condition

983105

983110983109

983108

983107983106

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22

AB=DE BC=EFang ang

Then

cong

2 Angle side

angle (ASA)

congruence

bull Two triangles are congruent if the

two angles and included side of

one triangle is equal to the

corresponding angles and side

bull It is a theorem and can be proved

If following

condition

BC=EF

ang angang ang

Then

cong

3 Angle angle

side( AAS)

congruence

bull Two triangles are congruent if the

any two pair of angles and any

side of one triangle is equal to the

corresponding angles and side

bull It is a theorem and can be proved

983110983109

983108

983107983106

983105

983110983109

983108

983107983106

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If followingcondition

BC=EF

ang angang ang

Then

cong

4 Side-Side-Side

(SSS)congruence

bull Two triangles are congruent if the

three sides of one triangle is equalto the three sides of the another

If following

condition

BC=EFAB=DEDF

=AC

Then

cong

5 Right angle ndash

hypotenuse-

side(RHS)

bull Two right triangles are congruent if

the hypotenuse and a side of the

one triangle are equal to

983110983109

983108

983107983106

983105

983105 983108

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983151983150983148983161

2983092

congruence corresponding hypotenuse and sideof the another

If following

condition

AC=DFBC=EF

Then

cong

Some Important points on Triangles

Terms Description

Orthocenter Point of intersection of the three altitudeof the triangle

Equilateral triangle whose all sides are equal and allangles are equal to 600

Median A line Segment joining the corner of thetriangle to the midpoint of the oppositeside of the triangle

Altitude A line Segment from the corner of thetriangle and perpendicular to theopposite side of the triangle

Isosceles A triangle whose two sides are equalCentroid Point of intersection of the three median

of the triangle is called the centroid of

the triangleIn center All the angle bisector of the trianglepasses through same point

Circumcenter The perpendicular bisector of the sidesof the triangles passes through samepoint

983106 983107 983109 983110

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983151983150983148983161

2983093

Scalene triangle Triangle having no equal angles and noequal sidesRight Triangle Right triangle has one angle equal to 900 Obtuse Triangle One angle is obtuse angle while other

two are acute anglesAcute Triangle All the angles are acute

Similarity of Triangles

Sno Points

1 Two figures having the same shape but not necessarily the same size are called similarfigures

2 All the congruent figures are similar but the converse is not true

3 If a line is drawn parallel to one side of a triangle to intersect the other two sides in

distinct points then the other two sides are divided in the same ratio

4 If a line divides any two sides of a triangle in the same ratio then the line is parallel tothe third side

Different Criterion for Similarity of the triangles

N Criterion Description Expression

1 Angle Angle

angle(AAA)

similarity

bull Two triangles are similar if

corresponding angle are equal

If following

condition

ang ang

ang ang

ang ang

Then

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983151983150983148983161

2983094

Then 983166

2 Angle angle

(AA) similarity

bull Two triangles are similar if the two

corresponding angles are equal as

by angle property third angle will

be also equal

If following

condition

ang ang

ang ang

Then

ang ang

Then

Then

983166

3 Side side

side(SSS)

Similarity

Two triangles are similar if the

sides of one triangle is

proportional to the sides of other

triangle

If following

condition

Then

ang ang

ang ang

ang ang

Then

cong

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983151983150983148983161

2983095

4 Side-Angle-Side (SAS)

similarity

bull

Two triangles are similar if the oneangle of a triangle is equal to one

angle of other triangles and sides

including that angle is proportional

If followingcondition

And ang ang

Then

cong

Area of Similar triangles

If the two triangle ABC and DEF are similar

cong

Then

Pythagoras Theorem

Sno Points

1 If a perpendicular is drawn from the vertex of the right angle of a right triangle to thehypotenuse then the triangles on both sides of the perpendicular are similar to the

whole triangle and also to each other

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983151983150983148983161

2983096

2 In a right triangle the square of the hypotenuse is equal to the sum of the squares of the

other two sides (Pythagoras Theorem)

(hyp)2 =(base)

2 + (perp)

2

3 If in a triangle square of one side is equal to the sum of the squares of the other twosides then the angle opposite the first side is a right angle

Arithmetic Progression

Sno Terms Descriptions

1 ArithmeticProgression

An arithmetic progression is a sequence ofnumbers such that the difference of any two

successive members is a constant

Examples

1) 1591317hellip

2) 12345hellip

2 common difference of

the AP

the difference between any successive members is a

constant and it is called the common difference of AP

1) If a1 a2a3a4a5 are the terms in AP then

D=a2 -a1 =a3 - a2 =a4 ndash a3=a5 ndasha4

2) We can represent the general form of AP inthe form

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983151983150983148983161

2983097

aa+da+2da+3da+4d

Where a is first term and d is the commondifference

3 nth term of Arithmetic Progression

n term = a + (n - 1)d

4 Sum of nth item in ArithmeticProgression

Sn =(n2)[a + (n-1)d]

Or

Sn =(n2)[t1+ tn]

Trigonometry

Sno Terms Descriptions

1 What is

Trigonometry

Trigonometry from Greek trigotildenon triangle and

metron measure) is a branch of mathematics

that studies relationships involving lengths and

angles of triangles The field emerged during the

3rd century BC from applications of geometry to

astronomical studies

Trigonometry is most simply associated with

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983151983150983148983161

9830910

planar right angle triangles (each of which is atwo-dimensional triangle with one angle equal to

90 degrees) The applicability to non-right-angle

triangles exists but since any non-right-angle

triangle (on a flat plane) can be bisected to

create two right-angle triangles most problems

can be reduced to calculations on right-angle

triangles Thus the majority of applications relate

to right-angle triangles

2 Trigonometric

Ratiorsquos

In a right angle triangle ABC where B=90deg

We can define following term for angle A

Base Side adjacent to angle

Perpendicular Side Opposite of angleHypotenuse Side opposite to right angle

We can define the trigonometric ratios for

angle A as

sin A= PerpendicularHypotenuse =BCAC

cosec A= HypotenusePerpendicular

=ACBC

cos A= BaseHypotenuse =ABAC

sec A= HypotenuseBase=ACAB

tan A= PerpendicularBase =BCAB

cot A= BasePerpendicular=ABBC

Notice that each ratio in the right-hand column is

the inverse or the reciprocal of the ratio in the

left-hand column

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9830911

3 Reciprocal offunctions The reciprocal of sin A is cosec A and vice-versa

The reciprocal of cos A is sec A

And the reciprocal of tan A is cot A

These are valid for acute angles

We can define tan A = sin Acos A

And Cot A =cos A Sin A

4 Value of of sin and

cos

Is always less 1

5 Trigonometric ration

from another angle

We can define the trigonometric ratios for angle Cas

sin C= PerpendicularHypotenuse =ABACcosec C= HypotenusePerpendicular =ACABBcos C= BaseHypotenuse =BCACsec C= HypotenuseBase=ACBCtan A= PerpendicularBase =ABBCcot A= BasePerpendicular=BCAB

6 Trigonometric ratios

of complimentary

Sin (90-A) =cos(A)

Cos(90-A) = sin A

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983151983150983148983161

9830912

angles Tan(90-A) =cot A

Sec(90-A)= cosec A

Cosec (90-A) =sec A

Cot(90- A) =tan A7 Trigonometric

identities

Sin2 A + cos2 A =0

1 + tan2 A =sec2 A

1 + cot2 A =cosec2 A

Trigonometric Ratios of Common angles

We can find the values of trigonometric ratiorsquos various angle

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983091983091

Area of Circles

Sno Points

1 A circle is a collection of all the points in a plane which are equidistant

from a fixed point in the plane

2 Equal chords of a circle (or of congruent circles) subtend equal angles at

the center

3 If the angles subtended by two chords of a circle (or of congruent circles)

at the center (corresponding center) are equal the chords are equal

4 The perpendicular from the center of a circle to a chord bisects the chord

5 The line drawn through the center of a circle to bisect a chord is

perpendicular to the chord

6 There is one and only one circle passing through three non-collinear points

7 Equal chords of a circle (or of congruent circles) are equidistant from the

center (or corresponding centers)

8 Chords equidistant from the center (or corresponding centers) of a circle

(or of congruent circles) are equal

9 If two arcs of a circle are congruent then their corresponding chords are

equal and conversely if two chords of a circle are equal then their

corresponding arcs (minor major) are congruent

10 Congruent arcs of a circle subtend equal angles at the center

11 The angle subtended by an arc at the center is double the angle subtended

by it at any point on the remaining part of the circle

12 Angles in the same segment of a circle are equal

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983151983150983148983161

983091983092

13 Angle in a semicircle is a right angle

14 If a line segment joining two points subtends equal angles at two other

points lying on the same side of the line containing the line segment the

four points lie on a circle

15 The sum of either pair of opposite angles of a cyclic quadrilateral is 180deg

16 If the sum of a pair of opposite angles of a quadrilateral is 180deg then the

quadrilateral is cyclic

Sno Terms Descriptions

1 Circumference of a circle 2 π r

2 Area of circle π r 2

3 Length of the arc of

the sector of angle

Length of the arc of the sector of angle θ 2

4 Area of the sector of

angle

Area of the sector of angle θ

5 Area of segment of a

circle

Area of the corresponding sector ndash Area of the

corresponding triangle

Surface Area and Volume

Sno Term Description

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983091983093

1 Mensuration It is branch of mathematics which is concernedabout the measurement of length area and

Volume of plane and Solid figure

2 Perimeter a)The perimeter of plane figure is defined as thelength of the boundaryb)It units is same as that of length ie m cmkm

3 Area a)The area of the plane figure is the surfaceenclosed by its boundaryb) It unit is square of length unit ie m2 km2

4 Volume Volume is the measure of the amount of spaceinside of a solid figure like a cube ball cylinderor pyramid Its units are always cubic that isthe number of little element cubes that fit insidethe figure

Volume Unit conversion

1 cm3 1mL 1000 mm3

1 Litre 1000ml 1000 cm3

1 m3 106 cm3 1000 L

1 dm3 1000 cm3 1 L

Surface Area and Volume of Cube and Cuboid

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Type Measurement

Surface Area of Cuboid of Length L

Breadth B and Height H

2(LB + BH + LH)

Lateral surface area of the cuboids 2( L + B ) H

Diagonal of the cuboids Volume of a cuboids LBH

Length of all 12 edges of the cuboids 4 (L+B+H)

Surface Area of Cube of side L 6L2

Lateral surface area of the cube 4L2

Diagonal of the cube radic3 Volume of a cube L3

Surface Area and Volume of Right circular cylinder

Radius The radius (r) of the circular base is called the radius of the

cylinder

Height The length of the axis of the cylinder is called the height (h) of the

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983091983095

cylinder

Lateral

Surface

The curved surface joining the two base of a right circular cylinder is

called Lateral Surface

Type Measurement

Curved or lateral Surface Area of

cylinder

2πrh

Total surface area of cylinder 2πr (h+r)

Volume of Cylinder π r2h

Surface Area and Volume of Right circular cone

Radius The radius (r) of the circular base is called the radius of the

cone

Height The length of the line segment joining the vertex to the center of

base is called the height (h) of the cone

Slant

Height

The length of the segment joining the vertex to any point on the

circular edge of the base is called the slant height (L) of the cone

Lateral

surface

Area

The curved surface joining the base and uppermost point of a right

circular cone is called Lateral Surface

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983091983096

Type Measurement

Curved or lateral Surface Area of

cone

πrL

Total surface area of cone πr (L+r)

Volume of Cone

Surface Area and Volume of sphere and hemisphere

Sphere A sphere can also be considered as a solid obtained on

rotating a circle About its diameter

Hemisphere A plane through the centre of the sphere divides the sphere into two

equal parts each of which is called a hemisphere

radius The radius of the circle by which it is formed

SphericalShell

The difference of two solid concentric spheres is called a sphericalshell

Lateral

Surface

Area for

Total surface area of the sphere

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983091983097

Sphere

Lateral

Surface

area of

Hemisphere

It is the curved surface area leaving the circular base

Type Measurement

Surface area of Sphere 4πr2

Volume of Sphere 43 Curved Surface area of hemisphere 2πr2

Total Surface area of hemisphere 3πr2

Volume of hemisphere 23 Volume of the spherical shell whose

outer and inner radii and lsquoRrsquo and lsquorrsquorespectively

43

How the Surface area and Volume are determined

Area of Circle The circumference of a circle is 2πr

This is the definition of π (pi) Divide

the circle into many triangular

segments The area of the triangles

is 12 times the sum of their bases

2πr (the circumference) times their

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height r

12 2 Surface Area of cylinder This can be imagined as unwrapping the

surface into a rectangle

Surface area of cone This can be achieved by divide the

surface of the cone into its triangles or

the surface of the cone into many thin

triangles The area of the triangles is 12

times the sum of their bases p times

their height

12 2

Surface Area and Volume of frustum of cone

h = vertical height of the frustum

l = slant height of the frustum

r 1 and r 2 are radii of the two bases (ends) of the frustum

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Type Measurement

983126983151983148983157983149983141 983151983142 983137 983142983154983157983155983156983157983149 983151983142 983137 983139983151983150983141 983123983148983137983150983156 983144983141983145983143983144983156 983151983142 983142983154983157983155983156983157983149 983151983142 983137 983139983151983150983141 983107983157983154983158983141983140 983155983157983154983142983137983139983141 983137983154983141983137 983151983142 983137 983142983154983157983155983156983157983149 983151983142 983137 983139983151983150983141 983124983151983156983137983148 983155983157983154983142983137983139983141 983137983154983141983137 983151983142 983142983154983157983155983156983157983149 983151983142 983137 983139983151983150983141

Statistics

Sno Term Description

1 Statistics Statistics is a broad mathematical disciplinewhich studies ways to collect summarize and

draw conclusions from data

2 Data A systematic record of facts or different values of

a quantity is called data

Data is of two types - Primary data and Secondary

data

Primary Data The data collected by a researcher

with a specific purpose in mind is called primarydata

Secondary Data The data gathered from a

source where it already exists is called secondary

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data

3 Features of data bull Statistics deals with collectionpresentation analysis and interpretation ofnumerical data

bull Arranging data in an order to study theirsalient features is called presentation ofdata

bull Data arranged in ascending or descendingorder is called arrayed data or an array

bull Range of the data is the differencebetween the maximum and the minimum

values of the observationsbull Table that shows the frequency of different

values in the given data is called afrequency distribution table

bull A frequency distribution table that showsthe frequency of each individual value in thegiven data is called an ungrouped frequencydistribution table

bull A table that shows the frequency of groupsof values in the given data is called agrouped frequency distribution table

bull

The groupings used to group the values ingiven data are called classes or class-intervals The number of values that eachclass contains is called the class size orclass width The lower value in a class iscalled the lower class limit The higher valuein a class is called the upper class limit

bull Class mark of a class is the mid value ofthe two limits of that class

bull A frequency distribution in which the upperlimit of one class differs from the lower limitof the succeeding class is called an

Inclusive or discontinuous FrequencyDistribution

bull A frequency distribution in which the upperlimit of one class coincides from the lowerlimit of the succeeding class is called anexclusive or continuous Frequency

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Distribution bull

4 Bar graph A bar graph is a pictorial representation of data inwhich rectangular bars of uniform width are drawnwith equal spacing between them on one axisusually the x axis The value of the variable isshown on the other axis that is the y axis

5 Histogram A histogram is a set of adjacent rectangles whoseareas are proportional to the frequencies of agiven continuous frequency distribution

6 Mean The mean value of a variable is defined as thesum of all the values of the variable divided by thenumber of values

4 sum

7 Median The median of a set of data values is the middle

value of the data set when it has been arranged inascending order That is from the smallest valueto the highest value

Median is calculated as

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12 1 Where n is the number of values in the data

If the number of values in the data set is eventhen the median is the average of the two middlevalues

8 Mode Mode of a statistical data is the value of thatvariable which has the maximum frequency

Sno Term Description

1983117983141983137983150 983142983151983154

983125983150983143983154983151983157983152

983110983154983141983153983157983141983150983139983161 983156983137983138983148983141

Mean is given by

sum sum

2983117983141983137983150 983142983151983154 983143983154983151983157983152

983110983154983141983153983157983141983150983139983161 983156983137983138983148983141

In these distribution it is assumed that frequency of each classinterval is centered around its mid-point ie class marks

2

983117983141983137983150 983139983137983150 983138983141 983139983137983148983139983157983148983137983156983141983140 983157983155983145983150983143 983156983144983154983141983141 983149983141983156983144983151983140

983137983081 983108983145983154983141983139983156 983149983141983156983144983151983140

sum

sum

983138983081 983105983155983155983157983149983141983140 983149983141983137983150 983149983141983156983144983151983140

sum sum

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Where

a=gt Assumed mean

di =gt xi ndasha

983139983081 983123983156983141983152 983140983141983158983145983137983156983145983151983150 983117983141983156983144983151983140

sum sum

Where

a=gt Assumed mean

ui =gt (xi ndasha)h

3983117983151983140983141 983142983151983154

983143983154983151983157983152983141983140

983142983154983141983153983157983141983150983139983161 983156983137983138983148983141

Modal class The class interval having highest frequency is

called the modal class and Mode is obtained using the modal

class

2

983127983144983141983154983141

l = lower limit of the modal classh = size of the class interval (assuming all class sizes tobe equal)f 1 = frequency of the modal classf 0 = frequency of the class preceding the modal classf 2 = frequency of the class succeeding the modal class

4983117983141983140983145983137983150 983151983142 983137

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For the given data we need to have class interval frequency

distribution and cumulative frequency distribution

Median is calculated as

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2

983127983144983141983154983141

983148 983101 983148983151983159983141983154 983148983145983149983145983156 983151983142 983149983141983140983145983137983150 983139983148983137983155983155

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983139983142 983101 983139983157983149983157983148983137983156983145983158983141 983142983154983141983153983157983141983150983139983161 983151983142 983139983148983137983155983155 983152983154983141983139983141983140983145983150983143 983156983144983141 983149983141983140983145983137983150 983139983148983137983155983155

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983144 983101 983139983148983137983155983155 983155983145983162983141 (983137983155983155983157983149983145983150983143 983139983148983137983155983155 983155983145983162983141 983156983151 983138983141 983141983153983157983137983148)

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3 Median=Mode +2 Mean

ProbabilitySn

o

Term Description

1 Empirical

probability

It is a probability of event which is calculated based on

experiments

Example

A coin is tossed 1000 times we get 499 times head and

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501 times tail

So empirical or experimental probability of getting head is

calculated as

4991000 499

Empirical probability depends on experiment and

different will get different values based on the

experiment

2 Important point

about events

If the event A B C covers the entire possible outcome inthe experiment Then

P (A) +P (B) +P(C)=1

3 impossible

event

The probability of an event (U) which is impossible tooccur is 0 Such an event is called an impossible event

P (U)=0

4 Sure or certain

event

The probability of an event (X) which is sure (or certain)to occur is 1 Such an event is called a sure event or acertain event

P(X) =1

5 Probability of

any event

Probability of any event can be as

0 1

Sn

o

Term Description

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1

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The theoretical probability or the classical probability of the event isdefined as

2 Elementary

events

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983120(983105)+983120(983106) +983120(983107)9831011

3 Complementar

y events

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983120(983105) +983120(Ᾱ)9831011

4 Sure or certain

event

The probability of an event (X) which is sure (or certain)

to occur is 1 Such an event is called a sure event or acertain event

P(X) =1

5 Probability of

any event

Probability of any event can be as

0 1

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a b c d are realnumber

Simultaneous pair of Linear equation

A pair of Linear equation in two variables

a1x+b1y+c1=0

a2x +b2y+c2=0

Graphically it is represented by two straight lines on Cartesian plane

Simultaneous pair of

Linear equation

Condition Graphical

representation

Algebraic

interpretation

a1x+b1y+c1=0

a2x +b2y+c2=0

Example

x-4y+14=0

3x+2y-14=0

Intersecting lines The

intersecting point

coordinate is the only

solution

One unique solution

only

a1x+b1y+c1=0

a2x +b2y+c2=0

Coincident lines Theany coordinate on the

line is the solution

Infinite solution

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Example

2x+4y=16

3x+6y=24

a1x+b1y+c1=0

a2x +b2y+c2=0

Example

2x+4y=6

4x+8y=18

Parallel Lines No solution

The graphical solution can be obtained by drawing the lines on the Cartesian plane

Algebraic Solution of system of Linear equation

Sno Type of method Working of method

1 Method of elimination by

substitution

1) Suppose the equation are

a1x+b1y+c1=0

a2x +b2y+c2=0

2) Find the value of variable of either x or y in other

variable term in first equation

3) Substitute the value of that variable in second

equation

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4) Now this is a linear equation in one variable Findthe value of the variable

5) Substitute this value in first equation and get the

second variable

2 Method of elimination by

equating the coefficients

1) Suppose the equation are

a1x+b1y+c1=0

a2x +b2y+c2=0

2) Find the LCM of a1 and a2 Let it k

3) Multiple the first equation by the value ka1

4) Multiple the first equation by the value ka2

4) Subtract the equation obtained This way one

variable will be eliminated and we can solve to get the

value of variable y

5) Substitute this value in first equation and get the

second variable

3 Cross Multiplication method 1) Suppose the equation are

a1x+b1y+c1=0

a2x +b2y+c2=0

2) This can be written as

3) This can be written as

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1983094

4) Value of x and y can be find using the

x =gt first and last expression

y=gt second and last expression

Quadratic EquationsSno Terms Descriptions

1 Quadratic Polynomial P(x) = ax +bx+c where ane0

2 Quadratic equation ax +bx+c =0 where ane0

3 Solution or root of the

Quadratic equation

A real number α is called the root or solution of the

quadratic equation if

aα2 +bα+c=0

4 zeroes of the

polynomial p(x)

The root of the quadratic equation are called zeroes

5 Maximum roots of

quadratic equations

We know from chapter two that a polynomial of

degree can have max two zeroes So a quadratic

equation can have maximum two roots

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1983095

6 Condition for realroots A quadratic equation has real roots if b

2

- 4ac gt 0

How to Solve Quadratic equation

Sno Method Working

1 factorization This method we factorize the equation by splitting themiddle term b

In ax2+bx+c=0

Example

6x2-x-2=0

1) First we need to multiple the coefficient a and cIn this

case =6X-2=-12

2) Splitting the middle term so that multiplication is 12

and difference is the coefficient b

6x2 +3x-4x-2=0

3x( 2x+1) -2(2x+1)=0

(3x-2) (2x+1)=0

3) Roots of the equation can be find equating the factorsto zero

3x-2=0 =gt x=32

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1983096

2x+1=0 =gt x=-12

2 Square method In this method we create square on LHS and RHS and

then find the value

ax2 +bx+c=0

1) x2 +(ba) x+(ca)=0

2) ( x+b2a)2 ndash(b2a)2 +(ca)=0

3) ( x+b2a)2=(b2-4ac)4a2

4) radic

Example

x2 +4x-5=0

1) (x+2)2 -4-5=0

2) (x+2)2=9

3) Roots of the equation can be find using square root on

both the sides

x+2 =-3 =gt x=-5

x+2=3=gt x=1

3 Quadratic

method

For quadratic equation

ax2 +bx+c=0

roots are given by radic radic

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For b2 -4ac gt 0 Quadratic equation has two real roots of

different value

For b2-4ac =0 quadratic equation has one real root

For b2-4ac lt 0 no real roots for quadratic equation

Nature of roots of Quadratic equation

Sno Condition Nature of roots

1 b -4ac gt 0 Two distinct real roots

2 b -4ac =0 One real root

3 b

2

-4ac lt 0 No real roots

Triangles

Sno Terms Descriptions

1 Congruence Two Geometric figure are said to be congruence ifthey are exactly same size and shapeSymbol used is cong Two angles are congruent if they are equalTwo circle are congruent if they have equal radiiTwo squares are congruent if the sides are equal

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983151983150983148983161

20

2 Triangle Congruencebull

Two triangles are congruent if three sidesand three angles of one triangle is

congruent to the corresponding sides and

angles of the other

bull Corresponding sides are equal

AB=DE BC=EF AC=DF

bull Corresponding angles are equalang angang angang ang

bull We write this as

cong

bull The above six equalities are between the

corresponding parts of the two congruent

triangles In short form this is called

CPCT

bull We should keep the letters in correct order

on both sides

3 Inequalities in

Triangles

1) In a triangle angle opposite to longer side is

larger

2) In a triangle side opposite to larger angle is

larger

3) The sum of any two sides of the triangle is

greater than the third side

983105

983107983106

983108

983109 983110

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983151983150983148983161

21

In triangle ABC

AB +BC gt AC

Different Criterion for Congruence of the triangles

N Criterion Description Figures and

expression

1 Side angle

Side (SAS)

congruence

bull Two triangles are congruent if the

two sides and included angles of

one triangle is equal to the two

sides and included angle

bull It is an axiom as it cannot be

proved so it is an accepted truth

bull ASS and SSA type two triangles

may not be congruent alwaysIf following

condition

983105

983110983109

983108

983107983106

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983151983150983148983161

22

AB=DE BC=EFang ang

Then

cong

2 Angle side

angle (ASA)

congruence

bull Two triangles are congruent if the

two angles and included side of

one triangle is equal to the

corresponding angles and side

bull It is a theorem and can be proved

If following

condition

BC=EF

ang angang ang

Then

cong

3 Angle angle

side( AAS)

congruence

bull Two triangles are congruent if the

any two pair of angles and any

side of one triangle is equal to the

corresponding angles and side

bull It is a theorem and can be proved

983110983109

983108

983107983106

983105

983110983109

983108

983107983106

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983151983150983148983161

2983091

If followingcondition

BC=EF

ang angang ang

Then

cong

4 Side-Side-Side

(SSS)congruence

bull Two triangles are congruent if the

three sides of one triangle is equalto the three sides of the another

If following

condition

BC=EFAB=DEDF

=AC

Then

cong

5 Right angle ndash

hypotenuse-

side(RHS)

bull Two right triangles are congruent if

the hypotenuse and a side of the

one triangle are equal to

983110983109

983108

983107983106

983105

983105 983108

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2983092

congruence corresponding hypotenuse and sideof the another

If following

condition

AC=DFBC=EF

Then

cong

Some Important points on Triangles

Terms Description

Orthocenter Point of intersection of the three altitudeof the triangle

Equilateral triangle whose all sides are equal and allangles are equal to 600

Median A line Segment joining the corner of thetriangle to the midpoint of the oppositeside of the triangle

Altitude A line Segment from the corner of thetriangle and perpendicular to theopposite side of the triangle

Isosceles A triangle whose two sides are equalCentroid Point of intersection of the three median

of the triangle is called the centroid of

the triangleIn center All the angle bisector of the trianglepasses through same point

Circumcenter The perpendicular bisector of the sidesof the triangles passes through samepoint

983106 983107 983109 983110

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2983093

Scalene triangle Triangle having no equal angles and noequal sidesRight Triangle Right triangle has one angle equal to 900 Obtuse Triangle One angle is obtuse angle while other

two are acute anglesAcute Triangle All the angles are acute

Similarity of Triangles

Sno Points

1 Two figures having the same shape but not necessarily the same size are called similarfigures

2 All the congruent figures are similar but the converse is not true

3 If a line is drawn parallel to one side of a triangle to intersect the other two sides in

distinct points then the other two sides are divided in the same ratio

4 If a line divides any two sides of a triangle in the same ratio then the line is parallel tothe third side

Different Criterion for Similarity of the triangles

N Criterion Description Expression

1 Angle Angle

angle(AAA)

similarity

bull Two triangles are similar if

corresponding angle are equal

If following

condition

ang ang

ang ang

ang ang

Then

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2983094

Then 983166

2 Angle angle

(AA) similarity

bull Two triangles are similar if the two

corresponding angles are equal as

by angle property third angle will

be also equal

If following

condition

ang ang

ang ang

Then

ang ang

Then

Then

983166

3 Side side

side(SSS)

Similarity

Two triangles are similar if the

sides of one triangle is

proportional to the sides of other

triangle

If following

condition

Then

ang ang

ang ang

ang ang

Then

cong

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4 Side-Angle-Side (SAS)

similarity

bull

Two triangles are similar if the oneangle of a triangle is equal to one

angle of other triangles and sides

including that angle is proportional

If followingcondition

And ang ang

Then

cong

Area of Similar triangles

If the two triangle ABC and DEF are similar

cong

Then

Pythagoras Theorem

Sno Points

1 If a perpendicular is drawn from the vertex of the right angle of a right triangle to thehypotenuse then the triangles on both sides of the perpendicular are similar to the

whole triangle and also to each other

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2 In a right triangle the square of the hypotenuse is equal to the sum of the squares of the

other two sides (Pythagoras Theorem)

(hyp)2 =(base)

2 + (perp)

2

3 If in a triangle square of one side is equal to the sum of the squares of the other twosides then the angle opposite the first side is a right angle

Arithmetic Progression

Sno Terms Descriptions

1 ArithmeticProgression

An arithmetic progression is a sequence ofnumbers such that the difference of any two

successive members is a constant

Examples

1) 1591317hellip

2) 12345hellip

2 common difference of

the AP

the difference between any successive members is a

constant and it is called the common difference of AP

1) If a1 a2a3a4a5 are the terms in AP then

D=a2 -a1 =a3 - a2 =a4 ndash a3=a5 ndasha4

2) We can represent the general form of AP inthe form

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aa+da+2da+3da+4d

Where a is first term and d is the commondifference

3 nth term of Arithmetic Progression

n term = a + (n - 1)d

4 Sum of nth item in ArithmeticProgression

Sn =(n2)[a + (n-1)d]

Or

Sn =(n2)[t1+ tn]

Trigonometry

Sno Terms Descriptions

1 What is

Trigonometry

Trigonometry from Greek trigotildenon triangle and

metron measure) is a branch of mathematics

that studies relationships involving lengths and

angles of triangles The field emerged during the

3rd century BC from applications of geometry to

astronomical studies

Trigonometry is most simply associated with

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planar right angle triangles (each of which is atwo-dimensional triangle with one angle equal to

90 degrees) The applicability to non-right-angle

triangles exists but since any non-right-angle

triangle (on a flat plane) can be bisected to

create two right-angle triangles most problems

can be reduced to calculations on right-angle

triangles Thus the majority of applications relate

to right-angle triangles

2 Trigonometric

Ratiorsquos

In a right angle triangle ABC where B=90deg

We can define following term for angle A

Base Side adjacent to angle

Perpendicular Side Opposite of angleHypotenuse Side opposite to right angle

We can define the trigonometric ratios for

angle A as

sin A= PerpendicularHypotenuse =BCAC

cosec A= HypotenusePerpendicular

=ACBC

cos A= BaseHypotenuse =ABAC

sec A= HypotenuseBase=ACAB

tan A= PerpendicularBase =BCAB

cot A= BasePerpendicular=ABBC

Notice that each ratio in the right-hand column is

the inverse or the reciprocal of the ratio in the

left-hand column

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3 Reciprocal offunctions The reciprocal of sin A is cosec A and vice-versa

The reciprocal of cos A is sec A

And the reciprocal of tan A is cot A

These are valid for acute angles

We can define tan A = sin Acos A

And Cot A =cos A Sin A

4 Value of of sin and

cos

Is always less 1

5 Trigonometric ration

from another angle

We can define the trigonometric ratios for angle Cas

sin C= PerpendicularHypotenuse =ABACcosec C= HypotenusePerpendicular =ACABBcos C= BaseHypotenuse =BCACsec C= HypotenuseBase=ACBCtan A= PerpendicularBase =ABBCcot A= BasePerpendicular=BCAB

6 Trigonometric ratios

of complimentary

Sin (90-A) =cos(A)

Cos(90-A) = sin A

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9830912

angles Tan(90-A) =cot A

Sec(90-A)= cosec A

Cosec (90-A) =sec A

Cot(90- A) =tan A7 Trigonometric

identities

Sin2 A + cos2 A =0

1 + tan2 A =sec2 A

1 + cot2 A =cosec2 A

Trigonometric Ratios of Common angles

We can find the values of trigonometric ratiorsquos various angle

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Area of Circles

Sno Points

1 A circle is a collection of all the points in a plane which are equidistant

from a fixed point in the plane

2 Equal chords of a circle (or of congruent circles) subtend equal angles at

the center

3 If the angles subtended by two chords of a circle (or of congruent circles)

at the center (corresponding center) are equal the chords are equal

4 The perpendicular from the center of a circle to a chord bisects the chord

5 The line drawn through the center of a circle to bisect a chord is

perpendicular to the chord

6 There is one and only one circle passing through three non-collinear points

7 Equal chords of a circle (or of congruent circles) are equidistant from the

center (or corresponding centers)

8 Chords equidistant from the center (or corresponding centers) of a circle

(or of congruent circles) are equal

9 If two arcs of a circle are congruent then their corresponding chords are

equal and conversely if two chords of a circle are equal then their

corresponding arcs (minor major) are congruent

10 Congruent arcs of a circle subtend equal angles at the center

11 The angle subtended by an arc at the center is double the angle subtended

by it at any point on the remaining part of the circle

12 Angles in the same segment of a circle are equal

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13 Angle in a semicircle is a right angle

14 If a line segment joining two points subtends equal angles at two other

points lying on the same side of the line containing the line segment the

four points lie on a circle

15 The sum of either pair of opposite angles of a cyclic quadrilateral is 180deg

16 If the sum of a pair of opposite angles of a quadrilateral is 180deg then the

quadrilateral is cyclic

Sno Terms Descriptions

1 Circumference of a circle 2 π r

2 Area of circle π r 2

3 Length of the arc of

the sector of angle

Length of the arc of the sector of angle θ 2

4 Area of the sector of

angle

Area of the sector of angle θ

5 Area of segment of a

circle

Area of the corresponding sector ndash Area of the

corresponding triangle

Surface Area and Volume

Sno Term Description

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1 Mensuration It is branch of mathematics which is concernedabout the measurement of length area and

Volume of plane and Solid figure

2 Perimeter a)The perimeter of plane figure is defined as thelength of the boundaryb)It units is same as that of length ie m cmkm

3 Area a)The area of the plane figure is the surfaceenclosed by its boundaryb) It unit is square of length unit ie m2 km2

4 Volume Volume is the measure of the amount of spaceinside of a solid figure like a cube ball cylinderor pyramid Its units are always cubic that isthe number of little element cubes that fit insidethe figure

Volume Unit conversion

1 cm3 1mL 1000 mm3

1 Litre 1000ml 1000 cm3

1 m3 106 cm3 1000 L

1 dm3 1000 cm3 1 L

Surface Area and Volume of Cube and Cuboid

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Type Measurement

Surface Area of Cuboid of Length L

Breadth B and Height H

2(LB + BH + LH)

Lateral surface area of the cuboids 2( L + B ) H

Diagonal of the cuboids Volume of a cuboids LBH

Length of all 12 edges of the cuboids 4 (L+B+H)

Surface Area of Cube of side L 6L2

Lateral surface area of the cube 4L2

Diagonal of the cube radic3 Volume of a cube L3

Surface Area and Volume of Right circular cylinder

Radius The radius (r) of the circular base is called the radius of the

cylinder

Height The length of the axis of the cylinder is called the height (h) of the

983107983157983138983141 983107983157983138983151983145983140

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983091983095

cylinder

Lateral

Surface

The curved surface joining the two base of a right circular cylinder is

called Lateral Surface

Type Measurement

Curved or lateral Surface Area of

cylinder

2πrh

Total surface area of cylinder 2πr (h+r)

Volume of Cylinder π r2h

Surface Area and Volume of Right circular cone

Radius The radius (r) of the circular base is called the radius of the

cone

Height The length of the line segment joining the vertex to the center of

base is called the height (h) of the cone

Slant

Height

The length of the segment joining the vertex to any point on the

circular edge of the base is called the slant height (L) of the cone

Lateral

surface

Area

The curved surface joining the base and uppermost point of a right

circular cone is called Lateral Surface

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Type Measurement

Curved or lateral Surface Area of

cone

πrL

Total surface area of cone πr (L+r)

Volume of Cone

Surface Area and Volume of sphere and hemisphere

Sphere A sphere can also be considered as a solid obtained on

rotating a circle About its diameter

Hemisphere A plane through the centre of the sphere divides the sphere into two

equal parts each of which is called a hemisphere

radius The radius of the circle by which it is formed

SphericalShell

The difference of two solid concentric spheres is called a sphericalshell

Lateral

Surface

Area for

Total surface area of the sphere

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Sphere

Lateral

Surface

area of

Hemisphere

It is the curved surface area leaving the circular base

Type Measurement

Surface area of Sphere 4πr2

Volume of Sphere 43 Curved Surface area of hemisphere 2πr2

Total Surface area of hemisphere 3πr2

Volume of hemisphere 23 Volume of the spherical shell whose

outer and inner radii and lsquoRrsquo and lsquorrsquorespectively

43

How the Surface area and Volume are determined

Area of Circle The circumference of a circle is 2πr

This is the definition of π (pi) Divide

the circle into many triangular

segments The area of the triangles

is 12 times the sum of their bases

2πr (the circumference) times their

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height r

12 2 Surface Area of cylinder This can be imagined as unwrapping the

surface into a rectangle

Surface area of cone This can be achieved by divide the

surface of the cone into its triangles or

the surface of the cone into many thin

triangles The area of the triangles is 12

times the sum of their bases p times

their height

12 2

Surface Area and Volume of frustum of cone

h = vertical height of the frustum

l = slant height of the frustum

r 1 and r 2 are radii of the two bases (ends) of the frustum

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Type Measurement

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Statistics

Sno Term Description

1 Statistics Statistics is a broad mathematical disciplinewhich studies ways to collect summarize and

draw conclusions from data

2 Data A systematic record of facts or different values of

a quantity is called data

Data is of two types - Primary data and Secondary

data

Primary Data The data collected by a researcher

with a specific purpose in mind is called primarydata

Secondary Data The data gathered from a

source where it already exists is called secondary

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data

3 Features of data bull Statistics deals with collectionpresentation analysis and interpretation ofnumerical data

bull Arranging data in an order to study theirsalient features is called presentation ofdata

bull Data arranged in ascending or descendingorder is called arrayed data or an array

bull Range of the data is the differencebetween the maximum and the minimum

values of the observationsbull Table that shows the frequency of different

values in the given data is called afrequency distribution table

bull A frequency distribution table that showsthe frequency of each individual value in thegiven data is called an ungrouped frequencydistribution table

bull A table that shows the frequency of groupsof values in the given data is called agrouped frequency distribution table

bull

The groupings used to group the values ingiven data are called classes or class-intervals The number of values that eachclass contains is called the class size orclass width The lower value in a class iscalled the lower class limit The higher valuein a class is called the upper class limit

bull Class mark of a class is the mid value ofthe two limits of that class

bull A frequency distribution in which the upperlimit of one class differs from the lower limitof the succeeding class is called an

Inclusive or discontinuous FrequencyDistribution

bull A frequency distribution in which the upperlimit of one class coincides from the lowerlimit of the succeeding class is called anexclusive or continuous Frequency

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Distribution bull

4 Bar graph A bar graph is a pictorial representation of data inwhich rectangular bars of uniform width are drawnwith equal spacing between them on one axisusually the x axis The value of the variable isshown on the other axis that is the y axis

5 Histogram A histogram is a set of adjacent rectangles whoseareas are proportional to the frequencies of agiven continuous frequency distribution

6 Mean The mean value of a variable is defined as thesum of all the values of the variable divided by thenumber of values

4 sum

7 Median The median of a set of data values is the middle

value of the data set when it has been arranged inascending order That is from the smallest valueto the highest value

Median is calculated as

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983092983092

12 1 Where n is the number of values in the data

If the number of values in the data set is eventhen the median is the average of the two middlevalues

8 Mode Mode of a statistical data is the value of thatvariable which has the maximum frequency

Sno Term Description

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Mean is given by

sum sum

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In these distribution it is assumed that frequency of each classinterval is centered around its mid-point ie class marks

2

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sum

sum

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sum sum

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983151983150983148983161

983092983093

Where

a=gt Assumed mean

di =gt xi ndasha

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sum sum

Where

a=gt Assumed mean

ui =gt (xi ndasha)h

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Modal class The class interval having highest frequency is

called the modal class and Mode is obtained using the modal

class

2

983127983144983141983154983141

l = lower limit of the modal classh = size of the class interval (assuming all class sizes tobe equal)f 1 = frequency of the modal classf 0 = frequency of the class preceding the modal classf 2 = frequency of the class succeeding the modal class

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For the given data we need to have class interval frequency

distribution and cumulative frequency distribution

Median is calculated as

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2

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3 Median=Mode +2 Mean

ProbabilitySn

o

Term Description

1 Empirical

probability

It is a probability of event which is calculated based on

experiments

Example

A coin is tossed 1000 times we get 499 times head and

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501 times tail

So empirical or experimental probability of getting head is

calculated as

4991000 499

Empirical probability depends on experiment and

different will get different values based on the

experiment

2 Important point

about events

If the event A B C covers the entire possible outcome inthe experiment Then

P (A) +P (B) +P(C)=1

3 impossible

event

The probability of an event (U) which is impossible tooccur is 0 Such an event is called an impossible event

P (U)=0

4 Sure or certain

event

The probability of an event (X) which is sure (or certain)to occur is 1 Such an event is called a sure event or acertain event

P(X) =1

5 Probability of

any event

Probability of any event can be as

0 1

Sn

o

Term Description

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The theoretical probability or the classical probab