CBSE X MATHS SA2 Synopsis

8/10/2019 CBSE X MATHS SA2 Synopsis

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QUADRATIC EQUATIONS

1) Standard form of quadratic polynomial in

one real variable x is 2 0ax bx c a

where a, b, c are real numbers.

2) Standard form of quadratic equation in

one real variable x is 2

0 0ax bx c a where a, b, c are real numbers.

3) If is a root/solution of the equation

2 0 0ax bx c a , in other words if

satisfies 2 0 0ax bx c a , then

2 0a b c is a true statement.

4)

Zeros of polynomial 2 0ax bx c a are

nothing but the roots of quadratic

equation 2 0 0ax bx c a .

5)

A quadratic equation has at most (a

maximum) of two roots.

6) Let the roots of 2 0 0ax bx c a be

denoted by , . Then the roots are,

2 4

2

b b ac

a

and

2 4

2

b b ac

a

7)

Roots of a quadratic equation may or may

not be real.

8) 2 4b ac , denoted by or D is called

the discriminant of the quadratic

equation 2 0 0ax bx c a .

9)

Using discriminant of a quadratic

equation, we can find nature of its roots.

10)

Consider discriminant 2 4b ac D of

quadratic equation 2 0 0ax bx c a

a)

If 0 D , then the roots of

2 0 0ax bx c a are imaginary

(not real).

b) If 0 D , then the roots of

2 0 0ax bx c a are real and

equal and each root is equal to2

b

a

.

In this case, polynomial

2 0ax bx c a will be a perfect

square.

c)

If 0 D , then the roots of

2 0 0ax bx c a are real and

distinct.

d)

Roots of 2 0 0ax bx c a are real

if and only if 0 D .

11)

If we can resolve 2 0ax bx c a into a

product of two linear factors, say

1 1 2 2a x b a x b , then the roots of

2 0 0ax bx c a can be found by

solving linear equations1 1

0a x b and

2 2 0a x b for x .

12)

In the completion of square method we

express a part or whole of polynomial

2 0ax bx c a in the form 2

m n or

2

m n and we then solve equation

2 0 0ax bx c a for x .

ARITHMETIC PROGRESSION

1)

A progression is a sequence which follows

certain pattern.

2) Arithmetic progression (AP) is a sequence

in which every term except the first is

obtained by adding a fixed number to its

preceding term. These are some

important results on A.P.

a)

Standard form:

, , 2 ,...., 1a a d a d a n d

b)

nth term or general term or last term:

1na l a n d

c) Common difference:

1 1n nd a a k

d)

Sum to n terms: 2 12

n

nS a n d

2 2

n

n na l a a

e) na in terms of nS : 1n n na S S



3) Some convenient ways of assuming the

terms in AP

a) If 3n , then , ,a d a a d are the

terms with a common difference d .

b) If 4n , then the terms may be

3 , , 3a d a d a d a d with acommon difference 2d .

c) If 5n , then the terms may be

2 , , , 2a d a d a a d a d with

a common difference d and so on.

4)

Properties of AP

a)

If a constant is added to or subtracted

from each term of an AP, then the

resulting sequence is also an AP with

the same common difference.

b)

If each term of a given AP is multiplied

by a constant k , then the resulting

sequence is also an AP with common

difference kd , where d is the common

difference of the given AP.

c) If each term of a given AP is divided by

a non-zero constant k , then the

resulting sequence is also an A.P with

common difference

d

k , where d is the

common difference of the given AP.

d) In a finite AP the sum of the terms

equidistant from the beginning and

end is always same and is equal to the

sum of first and last term.

i.e. 1 2 1 3 2 ...n n na a a a a a

e)

Three numbers , ,a b c are in AP if and

only if 2b a c .

f)

If the terms of an A.P are chosen at

regular intervals, they form an AP.

g)

If the terms of an A.P whose common

difference is d are written in reverse

order then the common difference of

new A.P is d .

5) Let there be n terms in an AP. Then k th

term from the end is same as 1 th

n k

term from the beginning.

6) Sum to n terms of certain AP’s

a) Sum of first n natural numbers

(1, 2, 3, …, n ) is 1

2

n n .

b)

Sum of first n odd natural numbers

(1, 3, 5, …,

2 1n ) is 2

n .

c)

Sum of first n even natural numbers

(2, 4, 6, …, 2n ) is 1n n .

7) Let ,a b are two numbers. Then2

a b is

called the arithmetic mean (AM) of a and

b such that , ,2

a ba b

are in A.P.

COORDINATE GEOMETRY

1)

Coordinate/Cartesian Plane: The

Cartesian plane consists of two

perpendicular axes that cross at a central

point called the origin. Positions or

coordinates are determined according to

the east/west and north/south

displacements from the origin. The

east/west axis is called the x axis, and

the north/south axis is called the y axis.

2)

A point in a co-ordinate plane is of the

form , x y . Here x is the distance of a

point from the y -axis and is called

x -coordinate or abscissa. Similarly y is

the distance of a point from the x -axis

and is called y -coordinate or ordinate.

3) General forms of some specific points in

the coordinate plane:

a) A point on X-axis is of form ,0a .

b) A point on Y-axis is of form

0,b .

c)

A point in first quadrant is , x y ,

where 0 x , 0 y .

d) A point in second quadrant is , x y ,

where 0 x , 0 y .

e)

A point in third quadrant is , x y ,

where 0 x , 0 y .



f) A point in fourth quadrant is , x y ,

where 0 x , 0 y .

4) Distance Formula

a)

Distance between two points 1 1, x y ,

2 2, x y is

2 2

2 1 2 1 x x y y

or

2 2

1 2 1 2 x x y y .

b) Distance between a point 1 1, x y and

the origin is 2 2

1 1 x y .

5)

Important hints for identifying the nature

of polygons

a) A triangle is equilateral if all of its

sides are equal.

b) A triangle is isosceles if any two of its

sides are equal.

c)

A triangle is a scalene triangle if no

two of its sides are equal.

d)

A triangle is a right triangle if the

square of its largest side is equal to

the sum of the squares of remaining

two sides.

e) A triangle is an isosceles right triangle

if two of its sides are equal and thesquare of the largest side is equal to

the sum of the squares of remaining

two sides.

f) A quadrilateral is a parallelogram if

both the pairs of opposite sides are

equal.

g)

A quadrilateral is a rectangle if both

the pairs of opposite sides are equal

and the diagonals are equal.

h)

A quadrilateral is a rhombus if all of

its sides are equal.

i) A quadrilateral is a square if all of its

sides are equal and the diagonals are

equal.

6)

Three points A, B, C are collinear if either

AB BC CA or BC CA AB or

CA AB BC .

7) Section Formulae

a) Let AB be a line segment joining the

points A and B. Let M be the point on

the line segment such that

: : AM MB m n . Then we say that M

divides AB into two sections (parts) in

the ratio :m n .

b)

Let us consider the line segment

joining the points 1 1 2 2, , ,P x y Q x y .

The point R which divides the join PQ

in the ratio :m n internally is

2 1 2 1,mx nx my ny

m n m n

.

c)

The point R which divides the join PQ

in the ratio 1:1 internally is

1 2 1 2,2 2

x x y y

. Here R is called the

midpoint or the point of bisection

(dividing into two equal parts) of PQ .

d) The points1

R ,2

R which divide the

join PQ in the ratio 1: 2 and 2 :1

respectively are 2 1 2 12 2,

3 3

x x y y

and 2 1 2 12 2

,3 3

x x y y

. Here1

R ,2

R

are called the points of trisection

(dividing into 3 equal parts) of PQ .





CIRCLES (TANGENTS)

1) A circle is a locus of points in a plane

equidistant from a fixed point. Here, the

fixed distance is the radius ( r ) and fixed

point is the centre (O) of the circle.

2) Secant, tangent, and chord of a circle

a) A line which intersects a circle in two

distinct points is called a secant.

b) A line which touches a circle in one

and only one point is called a tangent.

c)

The line segment that connects any

two points on the outline of a circle is

called a chord.

3)

Arc, segment and sector

a)

An arc is a segment of outline of a

circle.

b) Two points on the outline of a circle

determine two arcs of the circle. The

smaller of them is called the minor arc

and the bigger one is called the major

arc.

c) The region enclosed between an arc

and its corresponding chord, of a

circle is called a segment.

d)

Major segment is the region bounded

by major arc and its corresponding

chord.

e)

Minor

segment is

the regionbounded by

minor arc and its corresponding

chord.

f) The region enclosed between an arc

and radii at its end points, of a circle

is called a sector.

g) Major sector is the region bounded by

major arc and radii at its end points.

h) Minor sector is

the region

bounded by

minor arc and

radii at its end

points.

4) Some important results on secant and

tangent of a circle

a)

At any point on a circle there can be

one and only one tangent.

b) A circle can have infinitely many

tangents.

c) No tangent is possible from a point

interior to the circle.

d) Utmost two tangents can be drawn

from a point exterior to the circle.

e) The point of intersection of tangent

and circle is called point of contact.

f)

The tangent at

any point of a

circle is

perpendicular to the radius through

the point of contact.

g) The line containing the radius

through the point of contact is called

normal to the circle at the point.

h) Tangents drawn at the end points of a

diameter of a circle are parallel.

i) The perpendicular at the point of

contact to the tangent to a circle

passes through the centre.

j) A line drawn through the end point of

a radius and perpendicular to it is a

tangent to the

circle.

k)

The lengths ofthe tangents

drawn from a point exterior to the

circle are equal in length. In the

figure, AB AC

l) Tangents drawn

from exterior point

to a circle subtend

equal angles at the centre.

In the figure, AOB AOC



m) Line joining

the centre of a

circle and the

point exterior

to the circle

from where tangents are drawn,

bisects the angle between the

tangents. In the fig., OAB OAC

5) Some important results on arc, and chord

of a circle

a)

Line passing through the centre of a

circle and perpendicular to its chord

bisects the chord.

b)

Line passing through the centre of a

circle and midpoint of its chord is

perpendicular to the chord.

c)

Perpendicular bisector of the chord ofa circle passes through the centre.

d)

The perpendicular bisectors of two

non-parallel chords of a circle

intersect at the centre of the circle.

e)

If two chords are equal, they are

equidistant from centre of the circle.

f) If two chords of a circle are equidistant

from the centre, they are equal.

g) Equal chords of a circle lie equidistant

from the centre.

h)

Equal chords of a circle subtend equal

angles at the centre.

i) If two chords of a circle subtend equal

angles at the centre, then their

lengths are equal.

j)

In a circle (or in congruent circles)

equal arcs form equal chords.

k) Diameter is the longest chord of a

circle and it passes through the

centre. Length of diameter is twice

the length of radius, i.e. 2d r .

l) Diameter divides a circle into two

equal segments called semi-circular

regions.

m) Chord other than the diameter divides

a circle into two unequal segments-

major segment and minor segment.

n) Angle in the major segment is acute,

whereas

angle in

the minor

segment

is obtuse.

o) Angle in the semi-circle is a right

angle.

p) Angle subtended by major arc at the

centre is 180 , whereas the angle

subtended

by minor

arc at the

centre is

180 .

q) Angle subtended by semicircular arc

at the centre is 180 .

r) Angles in the same

segment of a circle

are equal. In the

adjacent figure,

ACB ADB .

s)

In equal circles (or in the same circle),

if two arcs subtend equal angles at

the corresponding centres (or center),

then the arcs are equal in length.

t) In equal circles (or in the same circle),

if two arcs are equal, they subtend

equal angles at their corresponding

centres (or center).

u) The angle subtended by an arc of a

circle at the centre is twice the angle it

subtends at any point on the

remaining circle.

In the above figure,

2 2 AOB ACB ADB .



6) Cyclic Quadrilateral

a) If all the four

vertices of a

quadrilateral lie

on a circle, then the quadrilateral is

called cyclic quadrilateral.

b)

In a cyclic quadrilateral any pair ofopposite angles is supplementary.

c) In a cyclic quadrilateral any of the

exterior angle is equal to its interior

opposite angle. DAB BCN

AREAS RELATED TO CIRCLES

1)

Formulae related to circle

a) Diameter, 2d r

b) Circumference, 2C r

c)

Area,2

A r

2) Formulae related to sector of a circle

a)

Length of minor arc,

1 2360

xl r

b)

Length of major arc,

2

3602

360

xl r

c)

Area of minor sector,

2

1360

x A r

1

1

2l r

d) Area of major sector,

22 360

360

x A r

212

l r

e) Perimeter of minor sector,1 1

2P l r

f)

Perimeter of major sector,2 2

2P l r

g) Semi-circle is a sector whose sector

angle is equal to 180 .

h)

Quadrant of a circle is a sector whose

sector angle is equal to 90 .

3) Formulae related to a ring

a) Width of the

ring, w R r

b) 2 2 A R r R r R r

R r w

4) Formulae related to segments of a circle

a)

Area of minor segment is 3 1 3

b) Area of major segment is 2 3

SURFACE AREAS AND VOLUMES

1) Formulae related to cuboid

a)

Total number of faces 6

b)

Number of lateral faces 4

c) Number of bases 2

d) Total number of

edges 12

e)

Sum of lengths of

edges 4 l b h

f)

Length of diagonal, 2 2 2d l b h

g)

Perimeter of base, 2baseP l b

h)

Area of base,base A l b

i)

Lateral surface area, LSA

2baseP h h l b

j) Total surface area, TSA

LSA 2 2base A lb bh hl

k)

Volume,baseV A h l b h



2) Formulae related to cube

In the formulae related

to cuboid, replace l , b ,

h with a to get the

formulae of cube.

3) Formulae related to right circular cylinder

Let the radius of the circular bases be r

and the height of the cylinder is h . Then-

a) Circumference of base, 2base

C r

b)

Area of a base, 2base A r

c) Curved surface area, CSA

2baseC h rh

d) Total surface area, TSA

=CSA 2 base A

22 2 2rh r r h r

e) Volume, 2

baseV A h r h

4)

Formulae related to right circular hollowcylinder

a)

Area of a base (or circular ring), base A R r R r R r w

b) Curved surface area, CSA

2 R r h

c) Total surface area, TSA

2 R r h w

d)

Volume, baseV A h R r wh

5) Formulae related to right circular cone

a) Slant height, l is given by the relation2 2 2l h r

b) Area of a base, 2

base A r

c)

Curved surface area, CSA

rl

d) Total surface area, TSA

2CSAbase A r rl r r l

e)

Volume, 21

3V r h

6) Formulae related to frustum of a right

circular cone

a)

Curved surface area,

CSA

R r l

b) Total surface area, TSA

2 2 R r l R r

c) Volume, 2 21

3V h R Rr r

7) Formulae related to sphere

a) Surface area, SA24 r

b)

Volume, 34

3V r

8) Formulae related to hemi-sphere

a) Curved surface area,

CSA 22 r

b) Total surface area, TSA 23 r

c)

Volume, 32

3V r

9)

Formulae related to spherical shell

a) Volume, 3 34

3V R r



10) Formulae related to hemi-spherical shell

a) Area of the base (ring), A R r w

b) Surface area of outer hemisphere 22 R

c) Surface area of inner hemisphere22 r

d)

Total surface area, TSA 2 23 R r

e) Volume, 3 32

3V R r

PROBABILITY

1) An event for an experiment is the

collection of some outcomes of the

experiment.

2)

A trial is an action which results in one

or several outcomes.

3) The empirical (or experimental)

probability P(E) of an event E is given by

No.of trials in which E has happened

Total no. of trials

P E

4) The Probability of an event lies between 0

and 1 (inclusive of 0, 1), i.e. 0 1P E

5) When we speak of a coin, we assume it to

be fair, that is, it is symmetrical so that

there is no reason for it to come down

more often on one side than the other.

We call this property of the coin as being

unbiased.

6) The outcomes of an unbiased coin or dice

are assumed to be equally likely.

7)

The theoretical probability (or classical

probability) of an event E, written as P(E),

is defined as

No.of outcomes favourable to E

No. of all possible outcomes of EP E

8) In theoretical probability we assume that

outcomes of an experiment are equally

likely.

9) Theoretical probability is referred to as

probability.

10)

An event having only one outcome of the

experiment is called an elementary event.

11) The sum of the probabilities of all the

elementary events of an experiment is 1.

12)

For an event E, 1P E P E where E

and E are complementary events.

13) Probability of an impossible event is zero.

14)

Probability of a sure or certain event is 1.

IMPORTANT RESULTS

1)

2 22 22 4a b a ab b a b ab

2)

2 22 22 4a b a ab b a b ab

3)

2 22 2 2 2a b a b ab a b ab

4)

2 2 2 22a b a b a b

5)

2 24a b a b ab

6) 2 2a b a b a b

7) 3 3 33a b a ab a b b

8) 3 3 33a b a ab a b b

9) 33 3 3a b a b ab a b

2 2a b a ab b

10) 33 3 3a b a b ab a b

2 2

a b a ab b

11) 2 2 2 2 2 2 2a b c a b c ab bc ca

12) 2 x a x b x a b x ab

13) 0 0 or 0ab a b

This rule is called ‘Zero Product Rule’

14)

2 2 0 A B 0 and 0 A B

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CBSE X MATHS SA2 Synopsis