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NOTES AND FORMULAE
SPM MATHEMATICS
FORM 1 – 3 NOTES
1
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CIRCLE THEOREM

Sector

Area of sector =  
r2
Length of arc =



360

 2r

Angle at the centre
= 2 × angle at the
circumference
x = 2y

360
Cylinder
Curve surface area
= 2rh

Angles in the same
segment are equal
x=y

Sphere
Curve surface area =
4r2
(b)

Solid and Volume
Cube:
V = x  x  x = x3

Cuboid:
V=lbh
= lbh

Cylinder

Angle in a
semicircle
ACB = 90o
Sum of opposite
angles of a cyclic
quadrilateral = 180o
a + b = 180o
The exterior angle
of a cyclic
quadrilateral is
equal to the interior
opposite angle
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x=y
If PT and PS are
tangents to a circle,
PT = PS
TPO = SPO
TOP = SOP
3
...

(a)

LAW OF INDICES
xm  x n = xm + n

(b)

xm  xn = xm – n

(c)

(xm)n = x m  n

(d)

x-n =

(e)

xn 

(f)

x n  (n x ) m

(g)

x0 = 1

7
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Solution:

Regular hexagon

Each exterior angle = 60o
(f)

1
xn

0

Each exterior angle = 72o
Each interior angle = 108o
(e)

(x + 3)2 = x2 + 2 × 3 × x + 32
= x2 + 6x + 9
(x – y)(x + y) = x2 + xy – xy – y2 = x2 – y2

1 10  k 1 3k  (10  k )


2k
6k 2
6k 2
= 3k  10  k  4k  10  2( k  5)  k  5
6k 2
6k 2
6k 2
3k 2
8
...

Solution:

1
(3n + 2) = n – 2
5
1
5 × (3n + 2) = 5(n – 2)
5

Each exterior angle = 45o
Each interior angle = 135o
4
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EXPANSION OF ALGERBRAIC
EXPRESSIONS
(a)

3n + 2 = 5n – 10
2 + 10 = 5n – 3n
9
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LINEAR INEQUALITIES
1
...

Solution:
3x – 2 > 10
3x > 10 + 2
3x > 12
x>4
2
...

Solve the simultaneous linear inequalities

Solution:
4p – 3 ≤ p
p≤1
p+2

1
p
2

4p – p ≤ 3

1
p
2

3p ≤ 3

× 2, 2p + 4  p

2p – p  −4
p  −4
 The solution is −4 ≤ p ≤ 1
...

Mode is the data with the highest frequency
Median is the middle data which is arranged in
ascending/descending order
...

3, 3, 4, 6, 8
Mean =

2
...
8
5

Mode = 3
Median = 4
4, 5, 6, 8, 9, 10, there is no middle number,
the median is the mean of the two middle
numbers
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4
...


k 2
4

4p – 3 ≤ p and p + 2 

12
...
The length or the height of
each bar represents the frequency of each data
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Solution:
k – (m + 2) = 3m
k – m – 2 = 3m
k – 2 = 3m + m = 4m
m=

11
...


68
=7
2

A pictograph uses symbols to represent a set of
data
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January
February
March
Represents 50 books

zefry@sas
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A pie chart showing the favourite drinks of a group
of students
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SIGNIFICANT FIGURES AND STANDARD
FORM
Significant Figures
1
...

Example: 3045 (4 significant figures)
2
...

Example: 4560 (3 significant figures)
3
...

Example: 0
...

Zero behind decimal numbers are significant
...
140 (4 significant figures)
Standard Form
Standard form are numbers written in the form A ×
10n, where 1 ≤ A < 10 and n are integers
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4 × 105
0
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6 × 10-4
2
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Solve quadratic equations by factorization
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Solve qudratic equation by formula:
Example: Solve 3x2 – 2x – 2 = 0
2
x = b  b  4ac = 2  4  4(3)( 2)
2a
6
2

28
=
x = 1
...
5486
6

3
...

A – Complement of set A
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Conclusion: Not A is true
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THE STRAIGHT LINE
(a) Gradient

AB

AB

Gradient of AB =
m =

(b)

A

y 2  y1
x 2  x1

Equation of a straight line

Example:

Gradient Form:
y = mx + c

n(A) = 7 + 6 = 13
n(B) = 6 + 10 = 16
n(A  B) = 6
n(A  B) = 7 + 6 + 10 = 23
n(A  B‟) = 7
n(A‟  B) = 10
n(A  B) = 7 + 10 + 2 = 19
n(A  B) = 2

4
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Intercept Form:

x y
 1
a b

Implication
If a, then b
a – antecedent
b – consequent

a = x−intercept
b = y−intercept

„p if and only if q‟ can be written in two
implications:
If p, then q
If q, then p
(c)

Argument
Three types of argument:
Type I
Premise 1: All A are B
Premise 2 : C is A
Conclusion: C is B
Type II
Premise 1: If A, then B
Premise 2: A is true
Conclusion: B is true
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(a)

STATISTICS
Class, Modal Class, Class Interval Size, Midpoint,
Cumulative frequency, Ogive
Example :
The table below shows the time taken by 80
students to type a document
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5 min
Upper boundary = 14
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5 – 9
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TRIGONOMETRY
sin o = Opposite  AB
hypotenuse

Modal class = 25 – 29 min

Midpoint of modal class = 25  29 = 27

AC

cos o = adjacent  BC

2

hypotenuse

To draw an ogive, a table of upper boundary and
cumulative frequency has to be constructed
...
5
14
...
5
24
...
5
34
...
5
44
...
5

0
1
8
20
42
60
72
78
80

AC

tan o = opposite  AB
adjacent

Acronym:

BC

“Add Sugar To Coffee”

Trigonometric Graphs
1
...
5 min
First quartile = 24
...
5 = 9
...

(b)

Histogram, Frequency Polygon
Example:
The table shows the marks obtained by a group of
students in a test
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y = tan x

Frequency
2
8
16
20
4

8
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ANGLE OF ELEVATION AND DEPRESSION
(a) Angle of Elevation

5

(a)

2 34
2 17
2 8
2 4
2 2
2 1
0

The angle of elevation is the angle betweeen the
horizontal line drawn from the eye of an observer
and the line joining the eye of the observer to an
object which is higher than the observer
...

Method: Repeated division
...

The angle of depression of B from A is BAC
...


(b)

Convert number in base 2, 5, 8 to number in base
10
...

Method: Number in base m  Number in base
10  Number in base n
...


LINES AND PLANES
(a) Angle Between a Line and a Plane

In the diagram,
(a) BC is the normal line to the plane PQRS
...

(c) The angle between the line AC and the plane
PQRS is BAC
(b)

25 24 23 22 21 1
1 1 0 0 1 12
= 25 + 24 + 2 + 1
= 5110
5 51
5 10 1
5 2 0
0 2
Therefore, 1100112= 2015

Angle Between Two Planes

(d)

In the diagram,
(a) The plane PQRS and the plane TURS
intersects at the line RS
...

The angle between the plane PQRS and the plane
TURS is MNK
...

Using a conversion table
Base 2
000
001
010
011
100
101
110
111

Base 8
0
1
2
3
4
5
6
7

FORM 5 NOTES
10
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(a)

GRAPHS OF FUNCTIONS
Linear Graph
y = mx + c

(b)

Quadratic Graph
y = ax2 + bx + c

Reflection
Description: Reflection in the line __________
Example: Reflection in the line y = x
...

Example: A clockwise rotation of 90o about the
centre (5, 4)
...


a
x

12
...


Area of image
k2
Area of object

h

Description: Translastion  
k 

 
 4 
Example : Translastion 
  3 
 

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k = scale factor
(e)

Combined Transformtions
Transformation V followed by transformation W is
written as WV
...


MATRICES

(a)

a  c   a  c 
      

 b   d  b  d 

(b)

 a   ka 
k     
 b   kb 

7

(c)

 a b  e


 c d  g

(d)

If M = 


(e)

f   ae  bg af  bh 


h   ce  dg cf  dh 

 a b  , then

c d 
1  d  b
M-1 =


ad  bc   c a 
If

Gradient = Rate of change of speed
=

ax + by = h
cx + dy = k

= acceleration

 a b  x   h 

    
 c d  y   k 

(f)

14
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Inverse Variation
If y varies inversely as x,
Written in mathematical form:
Written in equation form:

(c)

y

y

1
x

k
, k is a constant
...

(a)

Written in mathematical form:

y

y

(b)

Complementary Event
P(A) = 1 – P(A)

(c)
(i)

Probability of Combined Events

(ii)

P(A and B) = P(A  B)

17
...
Bearing is written in 3 digits
...

15
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(a)

THE EARTH AS A SPHERE
Nautical Miles
1 nautical mile is the length of the arc on a great
circle which subtends an angle of 1 at the centre
of the earth
...

Distance =   60 nautical miles
 = angle between the parallels of latitude
measured along a meridian of longitude
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 = angle between the meridians of longitude
measured along the equator
...

Distance =   60  cos o
 = angle of the parallel of latitude
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(e)

Knot
1 knot = 1 nautical mile per hour
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PLAN AND ELEVATION
(a) The diagram shows a solid right prism with
rectangular base FGPN on a horizontal table
...
The
rectangular surface EKLM is a slanting plane
...
The edges
EF, KJ and HG are vertical
...


(ii)

D-elevation

(b) A solid in the form of a cuboid is joined to the solid
in (a) at the plane PQRLMN to form a combined
solid as shown in the diagram
...

Draw to full scale
(i) the elevation of the combined solid on the
vertical plane parallel to FG as viewed from C,

zefry@sas
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