1. NOTES AND FORMULAE ADDITIONAL MATHEMATICS FORM 5
1. PROGRESSIONS (iii)
(a) Arithmetic Progression b c c
Tn = a + (n – 1)d
n

a

f ( x )dx  f ( x )dx 
b
 f ( x)dx
a
Sn = [2a  ( n  1)d ]
2 (d) Area under a curve
n   
 
= [ a  Tn ] AC  AB  BC
2
(b) Geometric Progression
(b) A, B and C are collinear if
Tn = ar
n–1

 
n AB   BC where  is a constant.
Sn 
a (1  r ) 
 
1 r AB and PQ are parallel if
Sum to infinity
 
b b
PQ   AB where  is a constant.
a
S 
1 r
A=

a
ydx A=
 xdy
a
(c) Subtraction of Two Vectors
(c) General
Tn = Sn − Sn – 1
T1 = a = S1 (e) Volume of Revolution
2. INTEGRATION
x n 1   
  
(a)
 xn dx  c
n 1 AB  OB  OA
(ax  b) n 1 (d) Vectors in the Cartesian Plane
(b)
 ( ax  b) n dx  c
(n  1)a
(c) Rules of Integration:
b b b b
V   y 2 dx
 V   x 2 dy

(i)
 nf ( x)dx  n f ( x)dx
a a a a
a b
3. VECTORS 

(ii)
 f ( x)dx   f ( x)dx
b a
(a) Triangle Law of Vector Addition OA  xi  yj
 
Magnitude of
 
 
OA  OA  x 2  y 2
Prepared by Mr. Sim Kwang Yaw 1

2. (g) Double Angle Formulae
Unit vector in the direction of OA sin 2A = 2 sin A cos A
r xi  yj 2
cos 2A = cos A – sin A
2
r    
ˆ 2
= 2cos A – 1
 r x2  y 2 2
= 1 – 2sin A

4. TRIGONOMETRIC FUNCTIONS 2 tan A
tan 2A =
(iii) y = tan x 1  tan 2 A
(a) Sign of trigonometric functions in the four
5. PROBABILITY
quadrants.
(a) Probability of Event A
n( A)
Acronym: P(A) =
“Add Sugar To Coffee” n( S )
(b) Probability of Complementary Event
P(A) = 1 – P(A)
(c) Probability of Mutually Exclusive Events
(iv) y = a sin nx
(b) Definition and Relation P(A or B) = P(A  B) = P(A) + P(B)
sec x =
1 cosec x = 1 (d) Probability of Independent Events
cos x sin x
P(A and B) = P(A  B) = P(A) × P(B)
1 sin x
cot x = tan x =
tan x cos x 6. PROBABILTY DISTRIBUTION
(a) Binomial Distribution
(c) Supplementary Angles n
P(X = r) = Cr p q
r n r
o
sin (90 − x) = cos x a = amplitude
o
cot (90 – x) = tan x n = number of cycles n = number of trials
(e) Basic Identities p = probability of success
2 2
(d) Graphs of Trigonometric Function (i) sin x + cos x = 1 q = probability of failure
2 2
(i) y = sin x (ii) 1 + tan x = sec x Mean = np
2 2
(iii) 1 + cot x = cosec x
Standard deviation = npq
(f) Addition Formulae
(i) sin (A  B) (b) Normal Distribution
= sin A cos B  cos A sin B X 
Z=
(ii) cos (A  B) 
= cos A cos B  sin A sin B Z = Standard Score
(ii) y = cos x
(iii) tan (A  B) = tan A  tan B X = Normal Score
1  tan A tan B  = mean  = standard deviation
Prepared by Mr. Sim Kwang Yaw 2

3. (b) Condition and Implication:
(a) Normal Distribution Graph Condition Implication
Returns to O s=0
To the left of O s<0
To the right of O s>0
Maximum/Minimum ds = 0
displacement dt
Initial velocity v when t = 0
Uniform velocity a=0
Moves to the left v<0
Moves to the right v>0
Stops/change v=0
direction of motion
P(Z < k) = 1 – P(Z > P(Z < -k) = P(Z > k) Maximum/Minimum dv = 0
k) velocity dt
Initial acceleration a when t = 0
Increasing speed a>0
Decreasing speed a<0
(c) Total Distance Travelled in the Period
P(Z > -k) = 1 – P(Z < - P(a < Z < b) 0 ≤ t ≤ b Second
k) = 1 – P(Z > k) = P(Z > a) – P(Z > b) (i) If the particle does not stop in the
period of 0 ≤ t ≤ b seconds
Total distance travelled
= displacement at t = b second
(ii) If the particle stops in t = a second
when t = a is in the interval of 0 ≤ t ≤
P(-b < Z < -a) = P(a < P(- b < Z < a) b second,
Z < b) = P(Z > a) – = 1 – P(z > b) – P(Z > Total distance travelled in b second
P(Z > b) a) = Sa  S0  Sb  Sa
7. MOTION ALONG A STRAIGHT LINE
(a) Relation Between Displacement,
Velocity and Acceleration
 vdt  adt
Prepared by Mr. Sim Kwang Yaw 3