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Add Maths Formulae List: Form 4 (Update 18/9/08)

01 Functions

Absolute Value Function Inverse Function

f (x ), iff (x)0 Ify =f (x), then f

1(y )=x

f (x) Remember:f(x),iff(x) 0 two real and different rootsa ab 4ac = 0 two real and equal roots

The Quadratic Equation b 4ac

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03 Quadratic Functions

General Form Completing the square:

f (x )=ax 2+bx +c (x)=a(x +p )2+q

where a, b, and care constants and a0. (i) the value ofx, x = p

*Note that the highest power of an unknown of a(ii) min./max. value = q(iii) min./max. point = (p, q)

quadratic function is 2. (iv) equation of axis of symmetry, x =p

Alternative method:

a >0minimum(smiling face)

f (x)=ax

2

+bx +ca 0and f (x)>0 a >0and f (x)0 intersects two different points

atx-axis

b24ac= 0 touch one point atx-axis

a b a bb

24ac

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05 Indices and Logarithm

Fundamental if Indices Laws of Indices

Zero Index, a0

=1 am

an

=am

+n

a

1 1

aman=amn

Negative Index, =a

a 1

=

b ( am

)n=a

mn

( )b a

Fractional Index

1

= na

( ab)n=a

nb

n

an

m (a)n= an

a = a

bn

b

Fundamental of Logarithm Law of Logarithm

log amn=log am+loganlogay=xax

=y

log aa=1 m

logaax

=x

log an=log amlogan

log amn= nlog am

log a1 =0Changing the Base

logab= logcb

logca

logab=1

logba

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06 Coordinate Geometry

Distance and GradientDistance Between Point A and C =

( x1x2)2

+( x1x2)2

Gradient of line AC, m =y2y1xx

Or2 1

Gradient of a line, m = y intercept

x intercept

Parallel Lines Perpendicular Lines

When 2 lines are parallel, When 2 lines are perpendicular to each other,

m =m2. m1m2= 11

m1= gradient of line 1

m2= gradient of line 2

Midpoint A point dividing a segment of a line

x +x y +y A point dividing a segment of a line

Midpoint, M =1 2

,1 2

nx +mx ny +my2 2 2 2P = 1 , 1

m +n m +n

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Area of triangle:

Area of Triangle

1

= 2

A = ( x1y2+x2y3 +x3y1)(x2y1+x3y2+x1y3)2

Form of Equation of Straight LineGeneral form Gradient form I ntercept form

ax +by +c =0 y =mx +c x+y =1a

m = gradientb

bc =y-intercepta =x-intercept m = a

b =y-intercept

Equation of Straight LineGradient (m) and 1 point (x1, y1) 2 points, (x1, y1) and (x2, y2) given x-intercept and y-intercept given

given

y y1=m(x x1) y y1 =y2y1 x+y =1x x a bx x

1 2 1

Equation of perpendicular bisectorgets midpoint and gradient of perpendicular line.

Information in a rhombus:

A B (i) same lengthAB=BC=CD=AD

(ii) parallel linesmAB=mCD or mAD=mBC

(iii) diagonals (perpendicular)mACmBD= 1

(iv) share same midpointmidpointAC= midpoint

DC

BD

(v) any pointsolve the simultaneous equations

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Remember:

y-interceptx=0

cuty-axisx=0

x-intercepty =0

cutx-axisy=0

**point lies on the linesatisfy the equationsubstitute the value ofxand ofyof the point into the

equation.

Equation of Locus

( use the formula of The equation of the locus of a The equation of the locus of a moving

distance) moving pointP(x,y) which is point P(x,y) which is always

The equation of the locus of a always at a constant distance equidistant from two fixed pointsAandB

moving pointP(x,y)which from two fixed points is the perpendicular bisector of theis always at a constant A (x,y)andB (x

2,y2) with straight lineAB.

distance (r) from a fixed point1 1

a ratio m:nisA (x1,y1)is

(xx)2

PA =PB+ (yy 2

PA m+ (yy) = (xx)

2PA =r =

1 1 2PB n

(xx1)2+(yy1)

2=r

2 (xx1)2+ (yy1)2 =

m 2

(xx2) +(yy2)2 n

2

More Formulae and Equation List:

SPM Form 4 Physics - Formulae List

SPM Form 5 Physics - Formulae List

SPM Form 4 Chemistry - List of Chemical Reactions

SPM Form 5 Chemistry - List of Chemical Reactions

All atOne-School.net

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Measure of Central Tendency

07 Statistics

Ungrouped DataGrouped Data

Without Class Interval With Class IntervalMean

x =x

x=fx

x =fx

N f f

x =mean x =mean x =mean

x= sum ofx x= sum ofx f =frequency

x =value of the data f =frequency x =class mark

N = total number of the x =value of the data(lower limit+upper limit)data =

2

Medianm =TN+1 m =TN+1 1 N F

22 2 m =L +

fmC

WhenNis an odd number. WhenNis an odd number.

m = medianT N +TN T N +TN L = Lower boundary of median class

m = +1 m = +1 N = Number of data2 2 2 22 2 F = Total frequency before median class

WhenNis an even WhenN is an even number. fm= Total frequency in median class

number. c = Size class

= (Upper boundary

lower boundary)

Measure of Dispersion

Ungrouped DataGrouped Data

Without Class Interval With Class Interval

variance

2

=

x2 2

2

=

fx2x

2

2

=

fx 2 2

N x f f x

= variance = variance = variance

Standardxx

2 (xx)2

f(xx)2

= =DeviationN

=N f

x 2 2 x2 2 2 = N x = N x = fx x

f

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The varianceis a measure of the meanfor the square of the deviationsfrom the mean.

The standard deviati onrefers to the square rootfor the variance.

Effects of data changes on Measures of Central Tendency and Measures of dispersion

Data are changed uniformly with

+ k k k kMeasures of Mean, median, mode + k k k k

Central Tendency

Measures ofRange , Interquartile Range No changes k k

Standard Deviation No changes k kdispersion

Variance No changes k k

08 Circular MeasuresTerminology

Convert degree to radian:

Convert radian to degree:

x

o=(x )radians

180D 180

x radians=(x

180 )degrees

radians degrees

180

D

Remember:

180 =rad 1.2 radO

???0.7 rad ???360 =2rad

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Length and Area

r = radius

A = area

s = arc length = angle

l = length of chord

Arc Length: Length of chord: Area of Sector: Area of Triangle: Area of Segment:

s =r 1 2 1 2 1 2

l =2r sin2 A = 2r A = 2 sin A =2r (sin )

09 Differentiation

Gradient of a tangent of a line (curve or

straight)

dy= lim (ydx x0x

Differentiation of Algebraic Function

Differentiation of a Constant

y =a a is a constant

dy

dx

= 0

Example

y =2

dydx= 0

Differentiation of a Function I

y =xn

dydx= nx

n1

Example

y =x3

dydx= 3x

2

Differentiation of a Function II

y =ax

dydx= ax

11= ax

0= a

Example

y =3x

dy

dx= 3

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Differentiation of a Function III

y =axn

dydx= anx

n1

Example

y =2x3

dydx= 2(3)x

2= 6x

2

Differentiation of a Fractional Function

y =x

1n

Rewrite

y =xn

dydx= nx

n1=x

n

n+1

Example

y =1x

y =x1

dy

dx= 1x2=

x

12

Law of Differentiation

Sum and Difference Rule

y =u v u andv are functions inx

dydx=

dudx

dvdx

Example

y =2x3+5x

2

dydx= 2(3)x

2+ 5(2)x= 6x

2+10x

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Chain Rule

y =un u andv are functions inx

dy =dydudx du dx

Example

y =(2x2+3)

5

u =2x2+3, therefore =4x

dx

y =u5, therefore dy =5u4

dy =dydudu

dx du dx

= 5u44x

= 5(2x2+3)

44x=20x(2x

2+3)

4

Or differentiate directly

y =(ax +b)n

dydx= n.a.(ax+b)

n1

y =(2x2+3)

5

dydx= 5(2x

2+ 3)

4 4x= 20x(2x

2+3)

4

10

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Product Rule

y =uv u andv are functions inx

dy=vdu+udvdx dx dx

Example

y =(2x +3)(3x32x

2x)

u =2x +3 v =3x32x

2x

du= 2 = 9x2 4x1

dx dx

dy= v +u

dx dx dx

=(3x32x

2x)(2) +(2x+3)(9x

24x1)

Or differentiate directly

y =(2x +3)(3x32x

2x)

dy = (3x32x

2x)(2)+ (2x+ 3)(9x

2 4x1)

dx

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Quotient Rule

y = u andv are functions inx

vdu dv

dy= dx

dxdx v

Example

y =x

2

2x+1

u =x2 v =2x +1

du = 2x dv = 2dx dx

du dvdy

=v

dx udx

dx v2

dy=

(2x+1)(2x) x2(2)

dx (2x+1)2

=4x

2+2x2x

2=

2x2+ 2x

(2x+1)2 (2x+1)

2

Or differentiate directly

y = x22x+1

dy=

(2x+1)(2x) x2(2)

dx (2x+1)2

=4x

2+2x2x

2=

2x2+2x

(2x+1)2 (2x+1)

2

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Gradients of tangents, Equation of tangent and Normal

If A(x1,y1) is a point on a liney = f(x), the gradient

of the line (for a straight line) or the gradient of the

tangent of the line (for a curve) is the value ofdy

dx

when x = x1.

Gradient of tangent at A( x1, y1):

dy

dx= gradient of tangent

Equation of tangent: yy1=m(xx1)

Gradient of normal at A( x1, y1):

mnormal

= m

1

tangent1 = gradient of normal

dy dxEquation of normal : yy1=m(xx1)

Maximum and Minimum Point

Turning point dy= 0dx

At maximum point, At minimum point ,

dy=

0

d y 0dx dx dx dx

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Rates of Change Small Changes and Approximation

Small Change:

Chain rule dA =dA dry dy dydt dr dt

x dxydxx

Ifxchanges at the rate of 5 cms-1

dx

= 5 Approximation:dt

Decreases/leaks/reduces NEGATIVES values!!! new original

=yoriginal

+ x

dx

x=small changes inx

y=small changes iny

Ifxbecomes smallerx=NEGATIVE

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10 Solution of Triangle

Sine Rule:

sina

A=

sinb

B=

sinc

C

Use, when given

2 sides and 1 non includedangle

2 angles and 1 side

a a B

AA

b 180 (A+B)

Cosine Rule:

a2= b

2+ c

22bc cosA

b2= a

2+ c

22ac cosB

c2

= a2

+ b2

2ab cosC

cosA=b2+c

2

a22bc

Use, when given

2 sides and 1 included angle 3 sides

a a c

bA

b

Area of triangle:

a

C

b

A =12a b sinC

C is the included angle of sidesa

and b.

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Case of AMBIGUITY

A

180 -

C B B

If C, the length ACand length AB remain unchanged,

the point Bcan also be at point Bwhere ABC= acute

andA BC= obtuse.IfABC= , thusABC= 180.

Remember : sin= sin (180)

Case 1: WhenabsinAbuta < b. Case 4: When a>bsinAanda > b.

CB cuts the side opposite to C at 2 points CB cuts the side opposite to C at 1 points

Outcome: Outcome:

2 solution 1 solution

Useful information:

In a right angled triangle, you may use the following to solve the

bc problems.

(i) Phythagoras Theorem: c= a2+b

2

a(ii)

Trigonometry ratio:

sin= b, cos= a, tan= bac c

(iii) Area = (base)(height)

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11 Index Number

Price Index Composite index

I = 1 100 WiIiI =P W

0 i

I =Price index/ Index number = Composite IndexIP0= Price at the base time W =Weightage

P1= Price at a specific time I =Price index

IA,BIB,C=IA,C100