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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