SPM AddMath Formula List NOT Given

SPM Additional Mathematics

Formula List and Important topics (for SPM Additional Mathematics) 1.

Functions (a) Composite function. (b) Inverse function. (c) Finding function

(i) or ( ii ) or

given function given function given function given function

f g g f

and and and and

fg , find function gf , find function fg , find function gf , find function

g. f. f. g.

(d) Graph sketching

2.

Quadratic Equations (a) ax 2 + bx + c = 0 , roots of the quadratic equation x = α , β

b a c S.O.P. = Product of Root = a

Hence, S.O.R. = Sum of Roots = −

(b) x 2 − ( New S .O . R ) x + ( New P .O . R ) = 0 (c) α 2 + β 2 = (α + β ) 2 − 2αβ (d) Factorisation, ax 2 + bx + c = 0 For a = 1 , given p > q

Sign for

b + − + −

c + + − −

( x + p)( x + q ) ( x − p )( x − q ) ( x + p )( x − q ) ( x − p )( x + q )

(e) ( i ) Two real and distinct/different roots ( ii ) Two real and equal/same roots ( iii ) Two real roots (special case) ( iv ) No real roots

means means means means

b2 b2 b2 b2

− 4ac > 0 − 4ac = 0 − 4ac ≥ 0 − 4ac < 0

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SPM Additional Mathematics 3.

Quadratic Functions y = a( x + p ) 2 + q

(a) Completing the square (b) Quadratic Inequalities ( i ) y = ax 2 + bx + c > 0

if a > 0 , the range of x : x < α or x > β . if a < 0 , the range of x : α < x < β .

if a > 0 , the range of x : α < x < β if a < 0 , the range of x : x < α or x > β . Two ways to solve quadratic inequalities i.e. Number line method and Graph sketching method. ( ii ) y = ax 2 + bx + c < 0

(c) Points of intersection between a straight and a curve. Simultaneous Equation – equalises the two equations to form a quadratic equation

ax 2 + bx + c = 0 ( i ) Intersects at two different points means b 2 − 4ac > 0 ( ii ) touches at one point @ tangent means b 2 − 4ac = 0 ( iii ) Does not intersect, always positive ( a > 0 ) @ always negative ( a < 0 ) means b 2 − 4ac < 0 4.

Simultaneous Equation (a) ax 2 + bx + c = kx + hy = m where a, b, c, k , h, m are constants. - Separate the equation into two equations ax 2 + bx + c = m & kx + hy = m - Always start from the linear equation - Substitute the linear equation into the non-linear equation and solve it. (b) Graph – finding the points of intersection between a straight line and a curve. - Always starts from the straight line equation - Substitute the straight line equation into the equation of the curve and solve it. (c) Daily problems - Form two equation base on the information given (one linear and one non-linear) Always start from the linear equation - Substitute the linear equation into the non-linear equation and solve it.

5.

Indices and Logarithm Indices N = a x , a > 0, N > 0 (a)

1 ax

(b)

a 0 = 1 , a1 = a 1 n

1 3

(c)

a −x =

(e)

( a m ) n = ( a n ) m = a m×n

(g)

If a ( Left _ Hand _ side ) = a ( Right _ Hand _ side ) , Then ( Left _ Hand _ side) = ( Right _ Hand _ side) (Compare the indices)

(d)

(f)

a = a

eg., a = 3 a

n

m n

1 m n

1 n m

(a ) = (a ) = (a )

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SPM Additional Mathematics Logarithm (a) log a N = x

⇔

N = a x (interchange form)

(b)

log a 1 = 0 ,

(d)

If log a ( Left _ Hand _ side ) = log a ( Right _ Hand _ side ) ,

(c)

log a a = 1

Then ( Left _ Hand _ side ) = ( Right _ Hand _ side ) (Compare the values) If ( Left _ Hand _ side ) > ( Right _ Hand _ side ) ,

(e)

Then log a ( Left _ Hand _ side ) > log a ( Right _ Hand _ side ) 6.

Coordinate Geometry (a) Finding area of quadrilateral.

A( x1 , y1 )

Area =

1 x1 2 y1

B( x2 , y 2 )

x2 y2

x3 y3

x4 y4

x1 y1

D( x4 , y 4 )

C ( x3 , y 3 ) Area =

1 ( x1 y2 + x2 y3 + x3 y4 + x4 y1 ) − ( y1 x2 + y2 x3 + x3 y4 + y4 x1 ) 2

(b) Method to find the equation of straight line. ( i ) Given the gradient of the straight line, m and 1 point A( x1 , y1 )

y − y1 = m( x − x1 ) ( ii ) Given 2 points A( x1 , y1 ) and B( x 2 , y 2 )

( iii ) Given x − intercept = b and y − intercept = c

y − y1 y2 − y1 = x − x1 x2 − x1 x y + =1 b c

(c) The equation of straight line can be written in three forms ( i ) y = mx + c ( ii ) ax + by + c = 0 ( iii )

x y + =1 b c

(d) If two straight lines are parallel, then m1 = m2 (e) If two straight lines are perpendicular to each other, then m1 × m2 = −1 Page | 26 http://www.masteracademy.com.my

SPM Additional Mathematics (f) Locus of point P( x, y ) The general form of answer for locus is

ax 2 + by 2 + cx + dy + e = 0

where a , b, c, d , e = constant

( i ) Distance from point A( x1 , y1 ) is always k units. ∴ AP = k

⇒ ( x − x1 ) 2 + ( y − y1 ) 2 = k ( ii ) Equidistance from two fixed points A( x1 , y1 ) and B( x2 , y 2 ) ∴ AP = BP

⇒ ( x − x1 ) 2 + ( y − y1 ) 2 = ( x − x2 ) 2 + ( y − y2 ) 2 ( iii ) Distance from two points A( x1 , y1 ) and B( x2 , y 2 ) always in the ratio of m : n

∴

AP m = ⇒ nAP = mBP BP n

⇒ n ( x − x1 ) 2 + ( y − y1 ) 2 = m ( x − x 2 ) 2 + ( y − y 2 ) 2 Square both sides,

⇒ n 2 [( x − x1 ) 2 + ( y − y1 ) 2 ] = m 2 [( x − x 2 ) 2 + ( y − y 2 ) 2 ] 7.

Statistics (a) Median,

1 N −F m = L+( 2 )C fm

L - lower boundary of median class N - total frequency, ∑ f F - cumulative frequency before median class f m - frequency of median class C - width of median class (b) Find the mode from a histogram x − axis - the lower boundaries and upper boundaries of all the classes y − axis - the frequency of each class 5. (c) Cumulative Frequency curve or Ogive x − axis - upper boundaries of classes including the class before the first class. y − axis - cumulative frequencies of classes (the cumulative frequency of the class before the first class is ZERO) 10. (d) The effects on mean and variance when all the data changed uniformly A new set of data v = ku ± h

v = k × (mean of u ) ± h standard deviation of v = k × (standard deviation of u ) 2 variance of v = k × (variance of u )

Then, mean of

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SPM Additional Mathematics 8.

Circular Measure (a) Length of chord AB = 2r sin

(b) Area of triangle OAB =

θ

1 2 r sin θ , θ in unit ( O ) 2

(c) Area of the segment ACB = 9.

2

, θ in unit ( O )

B

θ

O

1 2 j (θ − sin θ ) 2

C

r A

Differentiation dy = n ax n−1 dx

(a) If y = ax n , then

(b) If y = ( ax + b ) n , then

dy = n (ax + b) n −1 • a dx

(c) For graph of a curve, the gradient of tangent to the curve at the point A( x1 , y1 ) ,

dy = f ' ( x1 ) dx dy = m1 when x = x1 , dx m1 =

The gradient of the normal to curve at point A( x1 , y1 ) , m2 = −

1 because m1

m1 × m2 = −1 (d) Maximum and minimum point

dy = 0 , the value of x is the x − coordinate for dx d2y 0. minimum point if dx 2

When -

(e) Rate of change

dy dy dx = × dt dx dt

Example, volume of sphere, V =

dV dV dr 4 3 = × πr . then, 3 dt dr dt

(f) Small changes and approximations

δ y≈

dy ×δ x dx

Where δ x = xnew − xinitial and the value of

ynew = yinitial + δ y

dy is when x = xinitial dx

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SPM Additional Mathematics 10. Solution of Triangles (a) Ambiguous Case

B ∠BC ' C = ∠BCC ' BC = BC ' ∠BAC = constant

A C'

C

11. Index Number (a) Finding weighs If a circle is given, the weightages are the simplest ratio of the angles. Example,

A B

60 o

C

100

o

xo

D

Items

Angle

Weightage

A

100 o

10

B

60 o

6

C

90 o

9

D

110 o

11

x o = 360o − (100o + 60o + 90o ) = 110o (b) Information given ( i ) The price increased by 30% from year 2003 to year 2006 means Price index, I =

P2006 ×100 = 130 P2003

( ii ) The price decreased by 20% from year 2003 to year 2006 means Price index, I =

P2006 × 100 = 80 P2003

(c) Change of base time If given I 1 =

P2006 P × 100 = 120 and I 2 = 2004 × 100 = 90 P2003 P2003

Price Index for year 2006 based on year 2004,

I=

P2006 P P 120 100 × 100 = 2006 × 2003 × 100 = × × 100 = 133.3 P2004 P2003 P2004 100 90

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SPM Additional Mathematics 12. Progressions (a) Arithmetic Progression (A.P.). ( i ) Method to prove a series of terms are Arithmetic Progression where exists a common difference, Tn +1 − Tn = Tn − Tn −1 example, T3 − T2 = T2 − T1 (b) Geometry Progression (G.P.) ( i ) Method to prove a series of terms are Geometry Progression where exists a common ratio,

Tn +1 T T T = n example, 3 = 2 Tn Tn −1 T2 T1 (c) A.P. and G.P. ( i ) S n − S n −1 = Tn ( ii ) The sum of the first 4th terms to the first 13th terms.

T4 + T5 + T6 + ... + T13 = S13 − S 3 13. Linear Law Change the non-linear equation to linear form

where

Y = mX + c

Y − axis − y new X − axis − x new m − gradient of graph c − Y − intercept 14. Integration

dy = f ( x ) , then dx dy y = ∫ ( ) dx = ∫ f ( x ) dx dx

(a) If

(b) ∫ ( ax + b) n dx =

( ax + b) n +1 +c a( n + 1)

(c) Graph– equation of a curve and gradient function dy = f ( x) , dx dy Then the equation of the curve, y = ∫ ( ) dx = ∫ f ( x ) dx dx

If gradient function of a curve,

(d) Additional formulae

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SPM Additional Mathematics (i)

b

a

∫a f ( x)dx = −∫b f ( x)dx b

c

c

∫a b a ( iii ) ∫ a • f (x )dx = a ∫ f (x )dx example, ∫ 3x dx = 3∫ x dx

( ii )

f ( x )dx + ∫ f ( x )dx = ∫ f ( x )dx

15. Vector

(a) If a parallel to b , then a = k b ~

~

~

~

where k is a constant.

(b) If AB = k BC , then A, B and C are collinear. (c) AB = OB − OA (d) If AB : BC = m : n , then AB =

A

m BC . n

B n C

m

If AB : AC = m : m + n , then AB =

⎛ x⎞

m AC . m+n

(e) r = x i + y j = ⎜⎜ ⎟⎟ ~ ~ ~ ⎝ y⎠ ⎛ x1 ⎞ ⎛x ⎞ ⎛x +x ⎞ ⎛x −x ⎞ ⎟⎟ and v = ⎜⎜ 2 ⎟⎟ , then u + v = ⎜⎜ 1 2 ⎟⎟ , u − v = ⎜⎜ 1 2 ⎟⎟ and ~ ~ ~ ~ ⎝ y1 ⎠ ⎝ y2 ⎠ ⎝ y1 + y2 ⎠ ~ ~ ⎝ y1 − y2 ⎠ ⎛ x ⎞ ⎛ kx ⎞ ku = k ⎜⎜ 1 ⎟⎟ = ⎜⎜ 1 ⎟⎟ ~ ⎝ y1 ⎠ ⎝ ky1 ⎠

(f) If u = ⎜⎜

16. Trigonometric

Functions

(a) Quadrants

A

S

II III T

I IV C

II θ 2 = 180 o − θ θ 3 = 180 o + θ

I θ

θ 4 = 360 o − θ

III

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SPM Additional Mathematics

(b) Graph sketching of trigonometric functions sin θ , kosθ and tan θ . (c) Number of solutions 17. Permutation and Combination (a) Permutation – Choose with arrangement which means arrangement does affect the number of choices (b) Combination – Choose without involving arrangement which means arrangement does not affect the number of choices 18. Probability

(a) Concept of Complement

P( A) = 1 − P( A' ) n( A' ) and n( A' ) = n ( S ) − n ( A) where P( A' ) = n( S )

(b) Tree diagram – Total probability of all the branches is 1 19. Distribution

of Probability (a) Binomial distribution ( i ) Concept of Complement

P( X ≥ 3) = 1 − P( X < 3) = 1 − P( X = 2) − P( X = 1) − P( X = 0)

( ii ) P( X ≥ 1) = 1 − P( X = 0) and n C 0 = 1 20. Motion

(a)

on a Straight Line ( i ) Displacement , s = ∫ v dt

( ii ) Velocity, v =

ds dt

( iii ) Acceleration, a =

∫

; v = a dt

dv dt

(b) Hidden Information ( i ) Stop for a while, turn, change direction of motion ⇒ v = 0 ( ii ) Maximum displacement, ⇒ displacement when v = 0 ( ( iii ) Pass through the origin again ⇒ s = 0 ( iv ) Always move to the right ⇒ v > 0 ( v ) On the left side of point O , ⇒ s < 0 ( vi ) Particle P and particle Q meet ⇒ s P = s Q

ds = 0) dt

( vii ) Maximum velocity ⇒ velocity when a = 0 .

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SPM Additional Mathematics 21. Linear

Programming

Conditions

Inequalities

y not more than x

y≤x

y not less than x

y≥x

y at least k times of x

y ≥ kx

y at most k times of x

y ≤ kx

The Sum of x

and y not less than k

Minimum of y

is k

Maximum of y

Ratio of y

y≥k y≤k

is k

Value of y more than x

x+ y≥k

at least k

to x is k or more

y−x≥k y ≥k x

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