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

                                                           01 Functions


Absolute Value Function                                             Inverse Function

                                                                          If   y = f ( x ) , then f −1 ( y ) = x
                             f ( x ), if f ( x ) ≥ 0
          f ( x)                                                    Remember:
                          − f ( x), if f ( x ) < 0                  Object = the value of x
                                                                    Image = the value of y or f(x)
                                                                    f(x) map onto itself means f(x) = x


                                                 02 Quadratic Equations


General Form                                                        Quadratic Formula

                       ax 2 + bx + c = 0
                                                                                    −b ± b 2 − 4ac
      where a, b, and c are constants and a ≠ 0.                                 x=
                                                                                         2a
*Note that the highest power of an unknown of a
quadratic equation is 2.                                                  When the equation can not be factorized.

Forming Quadratic Equation From its Roots:                          Nature of Roots

If α and β are the roots of a quadratic equation
                         b                             c
        α +β =−                             αβ =                    b 2 − 4ac   >0    ⇔ two real and different roots
                         a                             a
                                                                    b 2 − 4ac   =0    ⇔ two real and equal roots
The Quadratic Equation                                              b 2 − 4ac   <0    ⇔ no real roots
              x 2 − (α + β ) x + αβ = 0                             b 2 − 4ac   ≥0    ⇔ the roots are real
                          or
              x − ( SoR ) x + ( PoR ) = 0
                   2


          SoR = Sum of Roots
          PoR = Product of Roots




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                                             03 Quadratic Functions


General Form                                                                 Completing the square:

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

      where a, b, and c are constants and a ≠ 0.                (i)     the value of x, x = − p
                                                                (ii)    min./max. value = q
*Note that the highest power of an unknown of a                 (iii)   min./max. point = (− p, q)
quadratic function is 2.                                        (iv)    equation of axis of symmetry, x = − p

                                                             Alternative method:
a > 0 ⇒ minimum ⇒ ∪ (smiling face)
                                                                                f ( x) = ax 2 + bx + c
a < 0 ⇒ maximum ⇒ ∩ (sad face)
                                                                                                 b
                                                                (i)     the value of x, x = −
                                                                                                 2a
                                                                                                      b
                                                                (ii)    min./max. value = f (−           )
                                                                                                      2a
                                                                                                              b
                                                                (iii)   equation of axis of symmetry, x = −
                                                                                                              2a

Quadratic Inequalities                                       Nature of Roots

  a > 0 and f ( x) > 0            a > 0 and f ( x) < 0
                                                             b 2 − 4ac > 0 ⇔ intersects two different points
                                                                              at x-axis
  a            b                   a            b            b − 4ac = 0 ⇔ touch one point at x-axis
                                                               2

                                                             b 2 − 4ac < 0 ⇔ does not meet x-axis

  x < a or x > b                       a< x<b

                                            04 Simultaneous Equations

To find the intersection point ⇒ solves simultaneous equation.

Remember: substitute linear equation into non- linear equation.




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                                          05 Indices and Logarithm

Fundamental if Indices                                   Laws of Indices

     Zero Index,             a0 = 1                           a m × a n = a m+n
                                      1
     Negative Index,         a −1 =
                                      a                       a m ÷ a n = a m−n
                              a
                             ( ) −1 =
                                      b                       ( a m ) n = a m× n
                              b       a

                              1                               ( ab) n = a n b n
     Fractional Index        an   = a n



                              m                                a n an
                             an   = a n   m
                                                              ( ) = n
                                                               b   b
Fundamental of Logarithm                                 Law of Logarithm

     log a y = x ⇔ a x = y                                    log a mn = log a m + log a n

     log a a = 1                                              log a
                                                                      m
                                                                        = log a m − log a n
                                                                      n
     log a a x = x
                                                              log a mn = n log a m
     log a 1 = 0
                                                         Changing the Base

                                                                           log c b
                                                              log a b =
                                                                           log c a

                                                                             1
                                                              log a b =
                                                                           logb a




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                                        06 Coordinate Geometry

Distance and Gradient
                                                                Distance Between Point A and C =
                                                                       (x1 − x2 )2 + (x1 − x2 )2

                                                                                              y2 − y1
                                                               Gradient of line AC, m =
                                                                                              x2 − x1
                                                                                 Or
                                                                                     ⎛ y − int ercept ⎞
                                                           Gradient of a line, m = − ⎜                ⎟
                                                                                     ⎝ x − int ercept ⎠


                   Parallel Lines                                     Perpendicular Lines




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

                      m1 = m2 .                                            m1 × m2 = −1

                                                                     m1 = gradient of line 1
                                                                     m2 = gradient of line 2


Midpoint                                              A point dividing a segment of a line




                         ⎛ x1 + x2 y1 + y2 ⎞                    A point dividing a segment of a line
           Midpoint, M = ⎜        ,        ⎟                          ⎛ nx + mx2 ny1 + my2 ⎞
                         ⎝ 2          2 ⎠                          P =⎜ 1       ,          ⎟
                                                                      ⎝ m+n        m+n ⎠




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Area of triangle:


Area of Triangle

     1
 =
     2

     1
A=
     2
       ( x1 y2 + x2 y3 + x3 y1 ) − ( x2 y1 + x3 y2 + x1 y3 )



Form of Equation of Straight Line
         General form                                      Gradient form                            Intercept form

             ax + by + c = 0                                    y = mx + c                              x y
                                                                                                         + =1
                                                                                                        a b
                                                            m = gradient
                                                           c = y-intercept                                                     b
                                                                                                     a = x-intercept    m=−
                                                                                                     b = y-intercept           a



Equation of Straight Line
Gradient (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 y2 − y1                                             x y
                                                    =                                                    + =1
                                             x − x1 x2 − x1                                             a b

Equation of perpendicular bisector ⇒ gets midpoint and gradient of perpendicular line.

Information in a rhombus:
              A                                B
                                                               (i)         same length ⇒ AB = BC = CD = AD
                                                               (ii)        parallel lines ⇒ mAB = mCD or mAD = mBC
                                                               (iii)       diagonals (perpendicular) ⇒ mAC × mBD = −1
                                                               (iv)        share same midpoint ⇒ midpoint AC = midpoint
         D                                                                 BD
                                       C                       (v)         any point ⇒ solve the simultaneous equations




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Remember:

y-intercept ⇒ x = 0
cut y-axis ⇒ x = 0
x-intercept ⇒ y = 0
cut x-axis ⇒ y = 0
**point lies on the line ⇒ satisfy the equation ⇒ substitute the value of x and of y of 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 point P ( 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 points A and B
moving point P ( x, y ) which       from        two    fixed        points   is the perpendicular bisector of the
is always at a constant             A ( x1 , y1 ) and B ( x2 , y 2 ) with    straight line AB.
distance (r) from a fixed point     a ratio m : n is
A ( x1 , y1 ) is                                                                                 PA = PB
                                                  PA m
                                                                                      2
                                                                             ( x − x1 ) + ( y − y1 ) 2 = ( x − x2 ) 2 + ( y − y2 ) 2
                                                    =
            PA = r                                PB n
( x − x1 ) 2 + ( y − y1 ) 2 = r 2   ( x − x1 ) 2 + ( y − y1 ) 2 m 2
                                                               =
                                    ( x − x2 ) + ( y − y 2 ) 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 at




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                                                                  07 Statistics
Measure of Central Tendency

                                                                                                  Grouped Data
               Ungrouped Data
                                                        Without Class Interval                             With Class Interval
Mean
                             Σx                                        Σ fx                                               Σ fx
                       x=                                         x=                                                 x=
                             N                                         Σf                                                 Σf

            x = mean                                   x = mean                                     x = mean
            Σx = sum of x                              Σx = sum of x                                f = frequency
            x = value of the data                      f = frequency                                x = class mark
            N = total number of the                    x = value of the data
            data                                                                                         (lower limit+upper limit)
                                                                                                     =
                                                                                                                     2
Median
                       m = TN +1                                  m = TN +1                                          ⎛ 1N −F⎞
                                  2                                          2                                m = L+ ⎜ 2
                                                                                                                     ⎜ f    ⎟C
                                                                                                                            ⎟
            When N is an odd number.                    When N is an odd number.                                     ⎝    m ⎠

                                                                                                   m = median
                        TN + TN                                    TN + T N                        L = Lower boundary of median class
                                          +1                                         +1
                 m=      2            2
                                                             m=        2         2                 N = Number of data
                             2                                              2                      F = Total frequency before median class
               When N is an even                       When N is an even number.                   fm = Total frequency in median class
                  number.                                                                          c = Size class
                                                                                                     = (Upper boundary – lower boundary)


Measure of Dispersion

                                                                                Grouped Data
                 Ungrouped Data
                                                         Without Class Interval          With Class Interval

variance
                σ =2∑ x2                   −x
                                                   2
                                                            σ =
                                                              2 ∑ fx 2                −x
                                                                                              2
                                                                                                               σ =
                                                                                                                 2 ∑ fx 2        −x
                                                                                                                                      2

                             N                                  ∑f                                                 ∑f
                 σ = variance                                σ = variance                                       σ = variance

                                                                       Σ(x − x )
                                                                                          2
                    Σ(x − x )
                                               2
Standard                                                                                                           Σ f (x − x)
                                                                                                                                  2
                 σ=                                          σ=
Deviation              N                                                        N                               σ=
                                                                                                                       Σf
                             Σx 2                                          Σx 2
                  σ=              − x2                        σ=                − x2                                   Σ fx 2
                              N                                             N                                    σ=           − x2
                                                                                                                        Σf




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 The variance is a measure of the mean for the square of the deviations from the mean.

 The standard deviation refers to the square root for the variance.

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

                                                                 Data are changed uniformly with
                                                                +k        −k         ×k        ÷k
  Measures of    Mean, median, mode                             +k        −k         ×k        ÷k
Central Tendency
                 Range , Interquartile Range                          No changes                ×k      ÷k
  Measures of
                 Standard Deviation                                   No changes                ×k      ÷k
   dispersion
                 Variance                                             No changes                × k2    ÷ k2

                                         08 Circular Measures
Terminology




Convert degree to radian:
Convert radian to degree:
                                                                                      π
                                                                         xo = ( x ×      )radians
                         180                                                        180
                     ×
                          π                                                             180
                                                                      x radians = ( x ×     ) degrees
                                                                                            π
       radians                 degrees


                          π
                     ×
                         180

Remember:

180 = π rad                                                                       1.2 rad
                                                                            O
                                   ???    0.7 rad                           ???
360 = 2π rad




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Length and Area


                                                                                    r = radius
                                                                                    A = area
                                                                                    s = arc length
                                                                                    θ = angle
                                                                                    l = length of chord




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

      s = rθ                          θ             1 2                     1 2                 1 2
                         l = 2r sin            A=     rθ           A=         r sin θ      A=     r (θ − sin θ )
                                      2             2                       2                   2


                                          09 Differentiation

                                                        Differentiation of a Function I
 Gradient of a tangent of a line (curve or
 straight)                                                    y = xn
                  dy        δy                                dy
                     = lim ( )                                   = nx n−1
                  dx δ x →0 δ x                               dx

                                                             Example
                                                             y = x3
Differentiation of Algebraic Function
                                                              dy
Differentiation of a Constant                                    = 3x 2
                                                              dx
      y=a          a is a constant
      dy
         =0                                             Differentiation of a Function II
      dx
                                                              y = ax
     Example                                                  dy
     y=2                                                         = ax1−1 = ax 0 = a
                                                              dx
     dy
        =0
     dx                                                      Example
                                                             y = 3x
                                                             dy
                                                                =3
                                                             dx




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Differentiation of a Function III                Chain Rule

      y = ax n                                      y = un       u and v are functions in x
      dy                                            dy dy du
         = anx n−1                                    =  ×
      dx                                            dx du dx

     Example                                        Example
     y = 2 x3                                       y = (2 x 2 + 3)5
      dy                                                                           du
         = 2(3) x 2 = 6 x 2                         u = 2 x 2 + 3,     therefore      = 4x
      dx                                                                           dx
                                                                             dy
                                                    y = u5 ,    therefore       = 5u 4
                                                                             du
Differentiation of a Fractional Function
                                                    dy dy du
                                                      =     ×
         1                                          dx du dx
      y=
         xn                                           = 5u 4 × 4 x
      Rewrite                                           = 5(2 x 2 + 3) 4 × 4 x = 20 x(2 x 2 + 3) 4
      y = x−n
      dy                −n                          Or differentiate directly
         = − nx − n−1 = n+1                         y = (ax + b) n
      dx               x
                                                    dy
                                                       = n.a.(ax + b) n −1
     Example                                        dx
         1
     y=
         x                                          y = (2 x 2 + 3)5
     y = x −1                                       dy
                                                       = 5(2 x 2 + 3) 4 × 4 x = 20 x(2 x 2 + 3) 4
     dy           −1                                dx
        = −1x −2 = 2
     dx           x



Law of Differentiation

Sum and Difference Rule

    y =u±v   u and v are functions in x
    dy du dv
       =   ±
    dx dx dx

   Example
   y = 2 x3 + 5 x 2
    dy
       = 2(3) x 2 + 5(2) x = 6 x 2 + 10 x
    dx




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Product Rule                                                      Quotient Rule

    y = uv    u and v are functions in x                                   u
                                                                      y=             u and v are functions in x
    dy     du    dv                                                        v
       = v +u
    dx     dx    dx                                                                du    dv
                                                                               v      −u
                                                                      dy           dx    dx
                                                                         =
   Example                                                            dx              v2
   y = (2 x + 3)(3 x 3 − 2 x 2 − x)
                                                                     Example
    u = 2x + 3           v = 3x3 − 2 x 2 − x
                                                                            x2
    du                 dv                                            y=
        =2                = 9 x2 − 4 x − 1                               2x +1
    dx                 dx
                                                                     u = x2           v = 2x +1
    dy     du     dv
       =v      +u                                                    du                  dv
    dx     dx     dx                                                     = 2x                =2
                                                                     dx                  dx
       =(3 x − 2 x − x)(2) + (2 x + 3)(9 x 2 − 4 x − 1)
            3     2
                                                                             du      dv
                                                                           v    −u
                                                                     dy
                                                                        = dx 2 dx
   Or differentiate directly                                         dx         v
   y = (2 x + 3)(3x3 − 2 x 2 − x)                                    dy (2 x + 1)(2 x) − x 2 (2)
                                                                        =
    dy                                                               dx          (2 x + 1) 2
       = (3x3 − 2 x 2 − x)(2) + (2 x + 3)(9 x 2 − 4 x − 1)
    dx                                                                    4 x2 + 2 x − 2 x2 2 x2 + 2 x
                                                                        =                  =
                                                                             (2 x + 1) 2     (2 x + 1) 2

                                                                     Or differentiate directly
                                                                           x2
                                                                     y=
                                                                         2x +1
                                                                     dy (2 x + 1)(2 x) − x 2 (2)
                                                                        =
                                                                     dx         (2 x + 1) 2
                                                                          4 x2 + 2 x − 2 x2 2 x2 + 2 x
                                                                        =                  =
                                                                             (2 x + 1) 2     (2 x + 1) 2




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Gradients of tangents, Equation of tangent and Normal

                                                           Gradient of tangent at A(x1, y1):

                                                                         dy
                                                                            = gradient of tangent
                                                                         dx

                                                                 Equation of tangent: y − y1 = m( x − x1 )

                                                           Gradient of normal at A(x1, y1):

                                                                                               1
                                                                              mnormal = −
                                                                                            mtangent
If A(x1, y1) is a point on a line y = f(x), the gradient                  1
of the line (for a straight line) or the gradient of the                        = gradient of normal
                                                                       − dy
                                                     dy                    dx
tangent of the line (for a curve) is the value of
                                                     dx          Equation of normal : y − y1 = m( x − x1 )
when x = x1.


Maximum and Minimum Point




                                                               dy
                                           Turning point ⇒        =0
                                                               dx

At maximum point,                                          At minimum point ,
                         2
     dy                 d y                                       dy                d2y
        =0                   <0                                      =0                  >0
     dx                 dx 2                                      dx                dx 2




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Rates of Change                                  Small Changes and Approximation
                                                 Small Change:
                           dA dA dr
             Chain rule      =  ×
                           dt dr dt                            δ y dy        dy
                                                                  ≈   ⇒ δ y ≈ ×δ x
                                                               δ x dx        dx
                                dx
If x changes at the rate of 5 cms -1 ⇒
                                   =5            Approximation:
                                dt
Decreases/leaks/reduces ⇒ NEGATIVES values!!!                 ynew = yoriginal + δ y
                                                                                     dy
                                                                     = yoriginal +      ×δ x
                                                                                     dx

                                                          δ x = small changes in x
                                                          δ y = small changes in y
                                                      If x becomes smaller ⇒ δ x = NEGATIVE




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                                       10 Solution of Triangle




Sine Rule:                         Cosine Rule:                            Area of triangle:
                                                                                    a
           a     b     c                    a2 = b2 + c2 – 2bc cosA
              =     =                       b2 = a2 + c2 – 2ac cosB
         sin A sin B sin C                                                        C
                                            c2 = a2 + b2 – 2ab cosC
                                                                                            b
 Use, when given
   2 sides and 1 non included                          b2 + c2 − a 2
                                             cos A =
   angle                                                   2bc                             1
   2 angles and 1 side                                                                A=     a b sin C
                                                                                           2
                                   Use, when given
                                    2 sides and 1 included angle           C is the included angle of sides a
     a               a    B         3 sides                                and b.
                          A
             A                          a                     a        c
         b         180 – (A+B)
                                                A
                                            b                     b




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               Case of AMBIGUITY                           If ∠C, the length AC and length AB remain unchanged,
                                                           the point B can also be at point B′ where ∠ABC = acute
                                     A
                                                           and ∠A B′ C = obtuse.
                                                           If ∠ABC = θ, thus ∠AB′C = 180 – θ .
                           180 - θ
                                     θ                                Remember : sinθ = sin (180° – θ)
         C                B′             B
Case 1: When a < b sin A                                   Case 2: When a = b sin A
CB is too short to reach the side opposite to C.           CB just touch the side opposite to C




Outcome:                                                   Outcome:
No solution                                                1 solution
Case 3: When a > b sin A but a < b.                        Case 4: When a > b sin A and a > 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
                   c                     problems.
     b                                               (i) Phythagoras Theorem: c = a 2 + b2
                       θ
               a                                           Trigonometry ratio:
                                                    (ii)
                                                           sin θ = b , cos θ = a , tan θ =
                                                                   c           c
                                                                                             b
                                                                                             a

                                                    (iii) Area = ½ (base)(height)




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                                               11 Index Number
Price Index                                                   Composite index

                            P                                                          Σ Wi I i
                      I =    1
                               × 100                                              I=
                            P0                                                          Σ Wi

              I = Price index / Index number                                  I = Composite Index
               P0 = Price at the base time                                      W = Weightage
               P1 = Price at a specific time
                                                                                 I = Price index

                                               I A, B × I B ,C = I A,C ×100




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