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
Page | 24 http://www.masteracademy.com.my
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 )
Page | 25 http://www.masteracademy.com.my
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
Page | 27 http://www.masteracademy.com.my
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, maximum point if dx 2 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
Page | 28 http://www.masteracademy.com.my
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
Page | 29 http://www.masteracademy.com.my
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
Page | 30 http://www.masteracademy.com.my
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
IV Page | 31
http://www.masteracademy.com.my
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 .
Page | 32 http://www.masteracademy.com.my
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
Page | 33 http://www.masteracademy.com.my
