Prepared by Mr. Sim Kwang Yaw
1
NOTES AND FORMULAE ADDITIONAL MATHEMATICS FORM 5
1. PROGRESSIONS
(a) Arithmetic Progression Tn = a + (n – 1)d
Sn =
[2
(
1) ]
2
n
a
n
d
=
[
]
2
nn
a T
(b) Geometric Progression Tn = arn – 1
(1
)
1
n n
a
r
S
r
Sum to infinity1
a
S
r
(c) General Tn = Sn− Sn – 1
T1 = a = S1
2. INTEGRATION
(a)
1
1
nn
x
x dx
c
n
(b)
1
(
)
(
)
(
1)
nn
ax
b
ax
b dx
c
n
a
(c) Rules of Integration:
(i)
( )
( )
b b
a a
nf x dx
n f x dx
(ii)
( )
( )
a b
b a
f x dx
f x dx
(iii)
( )
( )
( )
b c c
a b a
f x dx
f x dx
f x dx
(d) Area under a curve
A = b
a
ydx
A =b
a
xdy
(e) Volume of Revolution
2
b
a
V
y dx
2b
a
V
x dy
3. VECTORS
(a) Triangle Law of Vector Addition
AC
AB
BC
(b) A, B and C are collinear if
AB
BC
where
is a constant.AB
and
PQ
are parallel ifPQ
AB
where
is a constant.(c) Subtraction of Two Vectors
AB
OB
OA
(d) Vectors in the Cartesian Plane
OA
xi
yj
Magnitude of
2 2
OA
OA
x
y
Prepared by Mr. Sim Kwang Yaw
2
Unit vector in the direction ofOA
2 2
ˆ
r
xi
yj
r
r
x
y
4. TRIGONOMETRIC FUNCTIONS
(a) Sign of trigonometric functions in the four quadrants.
(b) Definition and Relation
sec x = 1 cos x
cosec x = 1 sin x
cot x =
1
tan x
tan x =sin
cos
x
x
(c) Supplementary Angles sin (90o− x) = cos x cot (90o – x) = tan x
(d) Graphs of Trigonometric Function (i) y = sin x
(ii) y = cos x
(iii) y = tan x
(iv) y = a sin nx
a = amplitude n = number of cycles (e) Basic Identities
(i) sin2 x + cos2 x = 1 (ii) 1 + tan2 x = sec2 x (iii) 1 + cot2 x = cosec2 x
(f) 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(iii) tan (A
B) = tan tan 1 tan tanA B
A B
(g) Double Angle Formulae sin 2A = 2 sin A cos A cos 2A = cos2 A – sin2 A
= 2cos2 A – 1 = 1 – 2sin2 A
tan 2A =
2
2 tan
1 tan A
A 5. PROBABILITY
(a) Probability of Event A
P(A) =
( )
( )
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. PROBABILTY DISTRIBUTION
(a) Binomial Distribution
P(X = r) = n
C p q
r r n rn = number of trials p = probability of success q = probability of failure Mean = np
Standard deviation =
npq
(b) Normal Distribution
Z =
X
Z = Standard Score X = Normal Score
= mean
= standard deviation Acronym:Prepared by Mr. Sim Kwang Yaw
3
(a) Normal Distribution GraphP(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. MOTION ALONG A STRAIGHT LINE (a) Relation Between Displacement,
Velocity and Acceleration
(b) Condition and Implication:
Condition Implication
Returns to O To the left of O To the right of O Maximum/Minimum displacement
s = 0 s < 0 s > 0
ds dt
= 0
Initial velocity Uniform velocity Moves to the left Moves to the right Stops/change direction of motion Maximum/Minimum velocity
v when t = 0 a = 0 v < 0 v > 0 v = 0
dv dt
= 0
Initial acceleration Increasing speed Decreasing speed
a when t = 0 a > 0 a < 0
(c) Total Distance Travelled in the Period 0 ≤ t ≤ b Second
(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 ≤ b second,
Total distance travelled in b second =
S
a
S
0
S
b
S
avdt
