SPM Mathematics Shortnote

NOTES AND FORMULAE
SPM MATHEMATICS

FORM 1 – 3 NOTES Cone
1. SOLID GEOMETRY
V= 1  r2h
(a) Area and perimeter 3
Triangle

A= 1  base  height Sphere
2

= 1 bh 4
2 3
V= r3

Trapezium

Pyramid

A= 1 (sum of two
2
1  
parallel sides)  height V= 3 base area

= 1 (a + b)  h height
2

Circle Prism

Area = r2 V = Area of cross section
Circumference = 2r  length

2. CIRCLE THEOREM

Sector Angle at the centre
= 2 × angle at the
Area of sector =   circumference
360 x = 2y

r2

Length of arc =

  2r
360

Cylinder Angles in the same
segment are equal
Curve surface area x=y
= 2rh

Sphere Angle in a
Curve surface area = semicircle
4r2
(b) Solid and Volume ACB = 90o
Cube:
V = x  x  x = x3 Sum of opposite
angles of a cyclic
Cuboid: quadrilateral = 180o
V=lbh
a + b = 180o
= lbh
The exterior angle
Cylinder of a cyclic
V =  r2h quadrilateral is
equal to the interior
opposite angle.

b=a

Angle between a
tangent and a radius
= 90o

OPQ = 90o

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The angle between a 2x2 – 6x + x – 3 = 2x2 – 5x − 3
tangent and a chord
is equal to the angle (b) (x + 3)2 = x2 + 2 × 3 × x + 32
in the alternate = x2 + 6x + 9
segment.
(c) (x – y)(x + y) = x2 + xy – xy – y2 = x2 – y2
x=y
6. LAW OF INDICES
If PT and PS are (a) xm  x n = xm + n
tangents to a circle,
PT = PS (b) xm  xn = xm – n
TPO = SPO
TOP = SOP (c) (xm)n = x m  n

3. POLYGON 1
(a) The sum of the interior angles of a n sided polygon (d) x-n = x n

= (n – 2)  180o 1
(b) Sum of exterior angles of a polygon = 360o
(e) x n  n x
(c) Each exterior angle of a regular n sided polygon =
(f) m (n x)m
3600
n xn

(d) Regular pentagon (g) x0 = 1

Each exterior angle = 72o 7. ALGEBRAIC FRACTION
Each interior angle = 108o
Express 1  10  k as a fraction in its simplest
(e) Regular hexagon 2k 6k 2

Each exterior angle = 60o form.
Each interior angle = 120o
Solution:
(f) Regular octagon
1  10  k  1 3k  (10  k)
Each exterior angle = 45o 2k 6k 2 6k 2
Each interior angle = 135o
= 3k 10  k  4k 10  2(k  5)  k  5
4. FACTORISATION 6k 2 6k 2 6k 2 3k 2

(a) xy + xz = x(y + z) 8. LINEAR EQUATION
(b) x2 – y2 = (x – y)(x + y)
Given that 1 (3n + 2) = n – 2, calculate the value
(c) xy + xz + ay + az 5
= x (y + z) + a (y + z)
= (y + z)(x + a) of n.

(d) x2 + 4x + 3 Solution:
= (x + 3)(x + 1)
1 (3n + 2) = n – 2
5. EXPANSION OF ALGERBRAIC 5
EXPRESSIONS
(a) 5 × 1 (3n + 2) = 5(n – 2)
5

3n + 2 = 5n – 10

2 + 10 = 5n – 3n 2n = 12 n=6

9. SIMULTANEOUS LINEAR EQUATIONS

(a) Substitution Method:
y = 2x – 5 --------(1)

2x + y = 7 --------(2)

Substitute (1) into (2) 4x = 12 x = 3
2x + 2x – 5 = 7 y=6–5=1

Substitute x = 3 into (1),

(b) Elimination Method:

Solve:

3x + 2y = 5 ----------(1)
x – 2y = 7 ----------(2)

(1) + (2), 4x = 12, x = 3

Substitute into (1) 9 + 2y = 5
2y = 5 – 9 = −4

2

y = −2 3. A bar chart uses horizontal or vertical bars to
represent a set of data. The length or the height of
10. ALGEBRAIC FORMULAE each bar represents the frequency of each data.
Given that k – (m + 2) = 3m, express m in terms of

k.

Solution:

k – (m + 2) = 3m k – m – 2 = 3m

k – 2 = 3m + m = 4m

m= k2
4

11. LINEAR INEQUALITIES 4. A pie chart uses the sectors of a circle to represent
1. Solve the linear inequality 3x – 2 > 10. the frequency/quantitiy of data.

Solution: 3x > 10 + 2
3x – 2 > 10

3x > 12 x>4

2. List all integer values of x which satisfy the

linear inequality 1 ≤ x + 2 < 4

Solution: 1−2≤x+2–2<4–2
1≤x+2<4 −1 ≤ x < 2
Subtract 2,
A pie chart showing the favourite drinks of a group
 x = −1, 0, 1 of students.

3. Solve the simultaneous linear inequalities

4p – 3 ≤ p and p + 2  1 p FORM FOUR NOTES
2 1. SIGNIFICANT FIGURES AND STANDARD

Solution: 4p – p ≤ 3 3p ≤ 3 FORM
4p – 3 ≤ p Significant Figures
p≤1 1. Zero in between numbers are significant.

1 × 2, 2p + 4  p Example: 3045 (4 significant figures)
2. Zero between whole numbers are not
p+2 p
significant figures.
2 Example: 4560 (3 significant figures)
3. Zero in front of decimal numbers are not
2p – p  −4 p  −4 significant.
Example: 0.00324 ( 3 significant figures)
 The solution is −4 ≤ p ≤ 1. 4. Zero behind decimal numbers are significant.
Example: 2.140 (4 significant figures)
12. STATISTICS Standard Form
Standard form are numbers written in the form A ×
Mean = sum of data 10n, where 1 ≤ A < 10 and n are integers.
number of data Example: 340 000 = 3.4 × 105

Mean = sum of(frequency  data) , when the data 0.000 56 = 5.6 × 10-4
sum of frequency 2. QUADRATIC EXPRESSION AND

has frequency. QUADRATIC EQUATIONS
Mode is the data with the highest frequency 1. Solve quadratic equations by factorization.
Median is the middle data which is arranged in
ascending/descending order.
1. 3, 3, 4, 6, 8

Mean = 3  3  4  6  8  4.8 Example: Solve 5k 2  8  2k
5 3

Mode = 3 5k2 – 8 = 6k 5k2 – 6k – 8 = 0

Median = 4 (5k + 4)(k – 2) = 0

2. 4, 5, 6, 8, 9, 10, there is no middle number, k= 4,2
5
the median is the mean of the two middle
2. Solve qudratic equation by formula:
numbers. Example: Solve 3x2 – 2x – 2 = 0

Median = 68 =7

2 x = b  b2  4ac = 2  4  4(3)(2)
2a 6
2. A pictograph uses symbols to represent a set of
= 2  28 x = 1.215, −0.5486
data. Each symbol is used to represent certain 6
frequency of the data.  - union
3. SET  - universal set
January (a) Symbol - is a member of

February  - intersection
 - subset
March  - empty set

Represents 50 books

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n(A) –number of element in set A. Type III
A – Complement of set A. Premise 1: If A, then B
(b) Venn Diagram Premise 2: Not B is true.
Conclusion: Not A is true.
AB
5. THE STRAIGHT LINE
(a) Gradient

AB Gradient of AB =

A m = y2  y1
x2  x1
Example:
(b) Equation of a straight line
n(A) = 7 + 6 = 13
n(B) = 6 + 10 = 16 Gradient Form:
n(A  B) = 6 y = mx + c
n(A  B) = 7 + 6 + 10 = 23 m = gradient
n(A  B‟) = 7 c = y-intercept
n(A‟  B) = 10
n(A  B) = 7 + 10 + 2 = 19 Intercept Form:
n(A  B) = 2
x  y 1
4. MATHEMATICAL REASONING ab
(a) Statement
a = x−intercept
A mathematical sentence which is either true or b = y−intercept
false but not both.
Gradient of straight line m =  y-int ercept
(b) Implication x-intercept
If a, then b
a – antecedent = b
b – consequent a

„p if and only if q‟ can be written in two 6. STATISTICS
implications: (a) Class, Modal Class, Class Interval Size, Midpoint,
If p, then q
If q, then p Cumulative frequency, Ogive
Example :
(c) Argument The table below shows the time taken by 80
Three types of argument: students to type a document.
Type I
Premise 1: All A are B Time (min) Frequency
Premise 2 : C is A 10-14 1
Conclusion: C is B 15-19 7

Type II
Premise 1: If A, then B
Premise 2: A is true
Conclusion: B is true.

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20-24 12
25-29 21
30-34 19
35-39 12
40-44 6
45-49 2
For the class 10 – 14 :
Lower limit = 10 min
Upper limit = 14 min

Lower boundary = 9.5 min
Upper boundary = 14.5 min

Class interval size = Upper boundary – lower 7. TRIGONOMETRY
boundary = 14.5 – 9.5 = 5 min sin o = Opposite  AB
hypotenuse AC
Modal class = 25 – 29 min
cos o = adjacent  BC
Midpoint of modal class = 25  29 = 27 hypotenuse AC

2 tan o = opposite  AB
adjacent BC
To draw an ogive, a table of upper boundary and

cumulative frequency has to be constructed.

Time Frequency Upper Cumulative
(min) boundary frequency

5-9 0 9.5 0

10-14 1 14.5 1

15-19 7 19.5 8

20-24 12 24.5 20

25-29 21 29.5 42

30-34 19 34.5 60

35-39 12 39.5 72

40-44 6 44.5 78

45-49 2 49.5 80

Acronym: “Add Sugar To Coffee”

Trigonometric Graphs
1. y = sin x

From the ogive : 2. y = cos x

Median = 29.5 min 3. y = tan x

First quartile = 24. 5 min 8. ANGLE OF ELEVATION AND DEPRESSION
(a) Angle of Elevation
Third quartile = 34 min
Interquartile range = 34 – 24. 5 = 9.5 min.

(b) Histogram, Frequency Polygon
Example:
The table shows the marks obtained by a group of
students in a test.

Marks Frequency
2
1 – 10 8
11 – 20 16
21 – 30 20
31 – 40 4
41 – 50

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(a) Convert number in base 10 to a number in base 2, 5

or 8.

Method: Repeated division.

Example:

2 34

2 17 0

The angle of elevation is the angle betweeen the 28 1
horizontal line drawn from the eye of an observer
and the line joining the eye of the observer to an 24 0
object which is higher than the observer.
The angle of elevation of B from A is BAC 22 0
(b) Angle of Depression 0
21
The angle of depression is the angle between the 0 1
horizontal line from the eye of the observer an the
line joining the eye of the observer to an object 3410 = 1000102
which is lower than the observer.
The angle of depression of B from A is BAC. 8 34 2
9. LINES AND PLANES 84 4
(a) Angle Between a Line and a Plane
0
In the diagram,
(a) BC is the normal line to the plane PQRS. 3410 = 428
(b) AB is the orthogonal projection of the line
(b) Convert number in base 2, 5, 8 to number in base
AC to the plane PQRS. 10.
(c) The angle between the line AC and the plane Method: By using place value
Example: (a) 110112 =
PQRS is BAC 24 23 22 211
(b) Angle Between Two Planes 1 1 0 1 12
= 24 + 23 + 21 + 1
In the diagram, = 2710
(a) The plane PQRS and the plane TURS (b) 2145 =
52 51 1
intersects at the line RS. 2 1 45
(b) MN and KN are any two lines drawn on each = 2  52 + 1  51 + 4  1
= 5910
plane which are perpendicular to RS and
intersect at the point N. (c) Convert between numbers in base 2, 5 and 8.
The angle between the plane PQRS and the plane Method: Number in base m  Number in base
TURS is MNK. 10  Number in base n.
FORM 5 NOTES Example: Convert 1100112 to number in base 5.
10. NUMBER BASES
25 24 23 22 21 1
1 1 0 0 1 12
= 25 + 24 + 2 + 1
= 5110
5 51
5 10 1
520

02
Therefore, 1100112= 2015

(d) Convert number in base two to number in base
eight and vice versa.
Using a conversion table

Base 2 Base 8
000 0
001 1
010 2
011 3
100 4
101 5
110 6
111 7

Example : = 238
10 0112

6

458 = 100 1012 (b) Reflection
11. GRAPHS OF FUNCTIONS Description: Reflection in the line __________
(a) Linear Graph Example: Reflection in the line y = x.

y = mx + c (c) Rotation
Description: Direction ______rotation of
(b) Quadratic Graph angle______about the centre _______.
y = ax2 + bx + c Example: A clockwise rotation of 90o about the
centre (5, 4).
(c) Cubic Graph
y = ax3+ c

(d) Reciprocal Graph (d) Enlargement
Description: Enlargement of scale factor ______,
ya with the centre ______.
x

12. TRANSFORMATION Example : Enlargement of scale factor 2 with the
centre at the origin.
(a) Translastion
Area of image  k 2
Description: Translastion  h  Area of object
k
k = scale factor
Example : Translastion  43
(e) Combined Transformtions
Transformation V followed by transformation W is
written as WV.

13. MATRICES

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

(b) k a    kkab
b

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(c)  a b  e f    ae  bg af  bh 
c d g h ce  dg cf  dh

(d) If M =  a b  , then
c d

M-1 = 1  d b  Gradient = Rate of change of speed
 c a
ad bc vu

(e) If ax + by = h =
cx + dy = k
t
 a b  x    h 
c d y k = acceleration

 x   ad 1 bc  d b  kh  Distance = Area below speed-time graph
y  c a
16. PROBABILITY
(f) Matrix a c  has no inverse if ad – bc = 0 (a) Definition of Probability
 d 
 b  Probability that event A happen,
P( A)  n( A)
14. VARIATIONS
(a) Direct Variation n(S )
S = sample space
If y varies directly as x,
Writtn in mathematical form: y  x, (b) Complementary Event
Written in equation form: y = kx , k is a constant. P(A) = 1 – P(A)

(b) Inverse Variation (c) Probability of Combined Events
If y varies inversely as x, (i) P(A or B) = P(A  B)

Written in mathematical form: y  1 (ii) P(A and B) = P(A  B)
x
17. BEARING
Written in equation form: y  k , k is a constant. Bearing
x Bearing of point B from A is the angle measured
clockwise from the north direction at A to the line
(c) Joint Variation joining B to A. Bearing is written in 3 digits.
If y varies directly as x and inversely as z,
Example : Bearing B from A is 060o
Written in mathematical form: y  x ,
z 18. THE EARTH AS A SPHERE
(a) Nautical Miles
Written in equation form: y  kx , k is a
z 1 nautical mile is the length of the arc on a great
circle which subtends an angle of 1 at the centre
constant. of the earth.

15. GRADIENT AND AREA UNDER A GRAPH (b) Distance Between Two Points on a Great Circle.
(a) Distance-Time Graph
Distance =   60 nautical miles
Gradient = distance = speed  = angle between the parallels of latitude

measured along a meridian of longitude.

time

Average speed = Total distance
Total time

(b) Speed-Time Graph

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(ii) the elevation of the combined solid on the
vertical plane parallel to GPS as viewed from
D.

 = angle between the meridians of longitude
measured along the equator.

(c) Distance Between Two Points on The Parallel of Solution:
(a)
Latitude.
Distance =   60  cos o (b) (i)

 = angle of the parallel of latitude.

Plan

(d) Shortest Distance C-elevation
The shortest distance between two points on the (ii)
surface of the earth is the distance between the two
points measured along a great circle. D-elevation

(e) Knot
1 knot = 1 nautical mile per hour.

19. PLAN AND ELEVATION
(a) The diagram shows a solid right prism with

rectangular base FGPN on a horizontal table. The
surface EFGHJK is the uniform cross section. The
rectangular surface EKLM is a slanting plane.
Rectangle JHQR is a horizontal plane. The edges
EF, KJ and HG are vertical.
Draw to full scale, the plan of the solid.

(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. The square base
FGSW is a horizontal plane.
Draw to full scale
(i) the elevation of the combined solid on the
vertical plane parallel to FG as viewed from C,

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