form 5 formulae and note

NOTES AND FORMULAE ADDITIONAL MATHEMATICS FORM 5
1.



PROGRESSIONS
(a) Arithmetic Progression
Tn = a + (n – 1)d

b

n
Sn = [2a  ( n  1)d ]
2
n
= [ a  Tn ]
2

(b)

f ( x )dx  f ( x )dx 

c

a

(d)

b

a

Area under a curve

  
AC  AB  BC

a (1  r )
1 r

S 

A=

A, B and C are collinear if

(e)

 xdy

(c)

Subtraction of Two Vectors

b

ydx

A=

a

General
Tn = Sn − Sn – 1
T1 = a = S1




b

a
1 r



AB   BC where  is a constant.


AB and PQ are parallel if



PQ   AB where  is a constant.

(b)

n

Sum to infinity

2.

 f ( x)dx
c

Geometric Progression
Tn = arn – 1

Sn 

(c)



(iii)

a

Volume of Revolution

INTEGRATION
(a)
(b)
(c)



x n 1
c
n 1
(ax  b) n 1
c

( ax  b) n dx 
(n  1)a
xn dx 

(d)



 nf ( x)dx  n f ( x)dx

Rules of Integration:
b

(i)

 f ( x)dx   f ( x)dx
a
a

(ii)

b

Prepared by Mr. Sim Kwang Yaw

V   y 2 dx
b

b

a

a
b

a

3.

  

AB  OB  OA

Vectors in the Cartesian Plane



V   x 2 dy
b

a

VECTORS
(a) Triangle Law of Vector Addition


OA  xi  yj
 
Magnitude of
 
OA  OA  x 2  y 2

1

xi  yj
r
rˆ     
 r
x2  y 2


Unit vector in the direction of

(g)

OA

4.

TRIGONOMETRIC FUNCTIONS

(a)

Sign of trigonometric functions in the four
quadrants.

tan 2A =
(iii)

y = tan x
5.
(a)

(iv) y = a sin nx

Definition and Relation
1
cosec x = 1
sec x =
cos x
sin x

cot x =

1
tan x

tan x =

sin x
cos x

(c)

Supplementary Angles
o
sin (90 − x) = cos x
cot (90o – x) = tan x

(d)

Graphs of Trigonometric Function
(i) y = sin x

(e)

(f)

(ii)

y = cos x

(iii)

Prepared by Mr. Sim Kwang Yaw

tan (A

 B) = tan A  tan B

1  tan A tan B

n( A)
n( S )

(b)

Probability of Complementary Event
P(A) = 1 – P(A)

(c)

Probability of Mutually Exclusive Events
P(A or B) = P(A  B) = P(A) + P(B)

(d)

Probability of Independent Events
P(A and B) = P(A  B) = P(A) × P(B)

6.
(a)

PROBABILTY DISTRIBUTION
Binomial Distribution
n

r

n r

P(X = r) = Cr p q
n = number of trials
p = probability of success
q = probability of failure
Mean = np

a = amplitude
n = number of cycles
Basic Identities
2
2
(i) sin x + cos x = 1
2
(ii) 1 + tan x = sec2 x
(iii) 1 + cot2 x = cosec2 x
Addition Formulae
(i) sin (A  B)
= sin A cos B  cos A sin B
(ii) cos (A  B)
= cos A cos B  sin A sin B

2 tan A
1  tan 2 A

PROBABILITY
Probability of Event A
P(A) =

Acronym:
“Add Sugar To Coffee”

(b)

Double Angle Formulae
sin 2A = 2 sin A cos A
2
2
cos 2A = cos A – sin A
2
= 2cos A – 1
= 1 – 2sin2 A

Standard deviation =
(b)

npq

X 


Normal Distribution
Z=

Z = Standard Score
X = Normal Score
 = mean
 = standard deviation

2

(a)

Normal Distribution Graph

P(Z < k) = 1 – P(Z >
k)

P(Z < -k) = P(Z > k)

P(Z > -k) = 1 – P(Z < k) = 1 – P(Z > k)

P(a < Z < b)
= P(Z > a) – P(Z > b)

P(-b < Z < -a) = P(a <
Z < b) = P(Z > a) –
P(Z > b)

P(- b < Z < a)
= 1 – P(z > b) – P(Z >
a)

7.
(a)

(b) Condition and Implication:
Condition
Implication
Returns to O
s=0
To the left of O
s0
Maximum/Minimum
ds = 0
displacement
dt
v when t = 0
Initial velocity
a=0
Uniform velocity
v0
Moves to the right
v=0
Stops/change
direction of motion
Maximum/Minimum
dv = 0
velocity
dt
Initial acceleration
a when t = 0
Increasing speed
a>0
Decreasing speed
a