Additional Mathematics Module

Exercise 2 [ x = -1.975 , y = 2.171, x = -10.02 y = 0.829 b) Solve the following simultaneous equations and give the
a) Solve the following simultaneous equations : answer correct to two decimal places.

x+y =3 x  2 y  7
x2 + y2 = 29 3

y2 – xy = 9

c). x + y = 7 d) x2 – y + 2y2 = 12

4  15  5 Ans [ x  14 , 2 y  21,5] 3x + 2y = 12 [ ans 19. x  36 , 3 y  12 , 3 ]
xy 5 5 11 11 2

Homework : Text Book Exercise 4.1.1 page 65

44

1.2 Solve simultaneous equations involving real-life situations

Real- life problems involving two unknown can be solved as described in the following steps
1. Identify the two unknowns described in the given problem. Then choose a suitable letter to
represent each unknown.
2. Form two equations using these two letters based on the information described in the problem.
3. Solve the simultaneous equations accordingly and obtain the final answer as required

Example 3 Example 4

a) Find the coordinate of the intersection points of the b) Given that the perimeter of a rectangle is 34 cm
and its area is 72 cm2 . Find the length and the
curve x  2 y  1 and the straight line 2x + y = 3 breadth of the rectangle.[ 9 and 8 ]
yx

Exercise 3 Exercise 4

The straight line x - y = 5 intersects the curve x2 + 2xy + y2 Given that the perimeter and the area of a rectangular
field are 80 m and 396 m2 respectively. Find the length
= 9 at point P and point Q. Find the coordinates of P and Q
and the breadth of the field. [22 , 18]
[ P(4,-1) , Q (1,-4) or P(1,-4) , Q (4, -1) ]

Homework : Text Book exercise 4.1.2 Page 66
47

1.So lve x + y = 3, xy = – 10 . SPM Questions
2. Solve x + y = 5, xy = 4 .
x + y = 3 ........ (1)
xy = – 10 ........ (2) (Ans : x = 1, y = 4 ; x = 4, y = 1)

Fromi (1), y = 3 – x ......... (3) 4. Solve 2x + y = 6, xy = – 20 .
Substitute (3) into (2),

x (3 – x) = – 10
3x – x2 = – 10
x2 – 3x – 10 = 0
(x + 2) (x – 5) = 0
x = – 2 atau x = 5
From (3), when x = – 2 , y = 3 – (-2) = 5

x = 5, y = – 2

Ans wers: x = – 2, y = 5 ; x = 5 , y = – 2 .

3.So lve x + y = – 2 , xy = – 8 .

(Ans : x = – 4 , y = 2 ; x = 2, y = – 4 ) (SPM 2000) (Ans : x = – 2 , y = 10 ; x = 5, y = – 4 ) (SPM 2002)

4. Solve the simultaneous equations 5. Solve the simultaneous equations :

3x – 5 = 2y x+y– 3 = 0
y(x + y) = x(x + y) – 5 x2 + y2 – xy = 21

(Ans : x = 3 , y = 2 ) (Ans : x = – 1, y = 4 ; x = 4, y = – 1 )

48

6. Solve the simultaneous equations : (SPM 2003) 7. So lve the simultaneous equations p – m = 2 and
p2 + 2m = 8. Give your answers correct to three
4x + y = – 8
x2 + x – y = 2 decimal places . (SPM 2004)

(Ans : x = – 2 , y = 0 ; x = – 3 , y = 4 ) (Ans : m = 0.606, p = 2.606 ; m = – 6.606 , p = – 4.606 )

8. Solve the simultaneous equations (SPM 2005) 9. So lve the simultaneous equations 2x + y = 1 and
2x2 + y2 + xy = 5. Give your answers correct to three
x  1 y  1 and y2 – 10 = 2x
2 decimal places . (SPM 2006)

(Ans : x = – 4 , y = 3 ; x = – ½ , y = 3 ) (Ans : x = 1.618, y = – 2.236 , x = 0.618, y = – 0.236)

Homework : Text Book Review Exercise page 67

49

SIMULTANE OUS EQUA TIONS – SPM QUES TIONS [5 marks]
1. Solve the equations x2  y + y2 = 2x + 2y = 10. 93P1.5

2. Solve the following simultaneous equations, give your answer correct to two decimal [5 marks]
places.. 94P1.9
2x + 3y + 1 = 0
x2 + 6xy + 6 = 0. [5 marks]
95P.1. 9
3. Solve the equations 4x + y + 8 = x2 + x - y = 2.
[5 marks]
4. Given that (-1, 2k ) is a solution for the simultaneous equation 96P.1.9
x2 + py  29 = 4 = px  xy where k and p are constants. Find the value of k and of p.
[5 marks]
5. Given that (3k , 2p) is a solution for the simultaneous equation x  2y = 4 and 97P.1.8

23 [4 marks]

+ = 1. Find the value of k and of p.

x 2y

6. Solve the following simultaneous equations:

x2

+ =4

3y

x + 6y = 3

7. Given the curve y2 = 8(1  x) and the straight line y 98P.1.8

= 4. Without drawing the graph, [4 marks]
x 99P.1.8

calculate the points of intersection of the two graphs . [6 marks]
99P.2.14(a)
8. Solve the simultaneous equations 2x + 3y = 9 and 6 y  x = 1.
xy 00P.1.8
00P.2.14(b)
9. Solve the following simultaneous equations:

3x  5 = 2y

y(x + y) = x(x + y)  5

xy 3 21
10. Solve the simultaneous equations  + 3 = 0 and +  = 0.
32 x y2

11. Given x + y  3 = 0 is a straight line which int ersects the curve x2 + y2  xy = 21 at [4 marks]
two different point. Find the coordinates of the points. 02P.1.8

12. y m

YAM

x m PADI

5m

15 m

Pak Amin own a piece of land which is rectangular in shape . On this land, he planted
padi and yam in the area as shown in the diagram. The area for planting yam is

rectangular in shape. The area for planting padi is 115 m2 and the perimeter of the

50

area for yam is 24 m. Find the area for planting yam. . [4 marks] 02P.2.14(b)

13. Solve the simultaneous equations 4x + y = 8 and x2 + x  y = 2. [5 marks]
03P.2.1
14. Solve the simultaneous equations p  m = 2 and p2 + 2m = 8.
Give your solutions correct to 3 decimal places.. [5 marks]
04P.2.1

15. Solve the simultaneous equations x + 1 y = 1 and y2  10 = 2x. [5 marks]
2 05P.2.1

16. Solve the simultaneous equations 2 x + y = 1 and 2x2 + y2 + xy = 5. Give your answer [5 marks]
correct to three decimal places. 06P.2.1

17. Solve the following simultaneous equation: [5 marks]

2x  y  3  0 , 2x2 10x  y  9  0

18. Solve the following simultaneous equations; 07P.2.1

x  3y  4  0 , x2  xy  40  0 [5 marks]
08P.2.
Answers : Simultaneous equations
4. k = 4, p = 4;
1. x = 2, y = 3; 2. x = 1.79, y = 1.08; 3. x = 1, y = 2; k = 2, p = 8

55 x = 1.12, y = 0.86 x = 4, y = 10 8. x = 3, y = 1;
x = 18, y = 9
x= ,y=
12. 35 m2
22 (x = 10, y = 8)

13 1 7.  1 ,2 , (1, 4) 16. x = 1.443,
2  y = –1.886;
5. k = , p = ; 6. x = 0, y = ; x = –0.693,
y = 2.386
34 2

8 p = 1 x = 15, y = 2

k= ,

3

9. x = 2, y = 2 10. x = 6, y = 2; 11. (- 1, 4) and (4, - 1)
x = 9, y = 12
13. x = 2, y = 0; 15. x = 3, y = –4;
x = 3, y = 4 14. m = 0.606,
p = 2.606; x= –1 , y= 3
m = 6.606, 2
p = 4.606
17. x = 3 , x = 1 ;
y= 3,y= -1 18. x = - 6 , x = 5 ;

y=  2,y 3
3

51

Chapter 5- Logarithms and Indices

Students will be able to:
1. Understand and use the concept of indices and laws of indices to solve problems.
1.1 Find the value of numbers given in the form of:

a) integer indices.
b) fractional indices.
1.2 Use laws of indices to find the value of numbers in index form that are multiplied, divided or raised
to a power.
1.3 Use laws of indices to simplify algebraic expressions.

1.1 Finding the value of numbers given in the form of:

a) integer indices. a n = a x a x a x a.......a

n times

1m

b) fractional indices ; a n = n a ; a n = n am

Example 1

Find the values each of the following

i) 0.23 = ii)  1 4  iii) 3 6 = iv)   2 4 
5  3

2 1 1 4

v)  8 3  vi)  9 2 = vii)  8  3 = vii) 27 3 =
125   64   27 

Exercises 1 b)   1 3  c) (4.5) 3 = d)  1 2  e) (5-1)4 =
 3 4
a) 103 = 1 3
 32  3 4
1 5 c) ( 0.16 ) 2 = e) 4 2
d) 27 3
f) ( 0.125 ) 3 =

b)  243  =

Homework : Text Book Exercise 5.1.1 page 71

52

1.2 Using laws of indices to find the value of numbers in index form that are multiplied, divided or raised to a
power

Laws of indices

a) am  an  am n b) a m  a n  a m n c) (a m )n  a mn

Example 2 (ii) (p6 q-2 ) 3 x6n
2 (iii) x2n
Simplify the following indices:
(i) a2 x a5 =

Exercises 2
Simplify the following indices: [ Ans a8 b) a3 b –9 c) b2n/3-m/2 d) b2n-3/2m ]

(a) a 6 x a 2 (b) (a4 b-12 ) 3/4 8n 8n 3

(c) ( b 3  b2m ) 1/4 (d) ( b 3  b2m ) 4

Homework : Text Book Exercise 5.1.2 page 73

Example 3: Simplify each of the following :

a 2n  a3n (ii) 23n1  4  8n1 (iii) 52n  523n
(i) a 2n5  a 4

1

Exercises 3 : Simplify each of the following [ Ans a) 2a 3b2 b) 264n c) 56n4 ]

2 (b) 4n  82n  23n (c) 53n5  52n  512n

(a) (2ab2 )3  (8a4b6 )3

Example 4 : Show that : (b) 11n2  11n3 is divisible (c) 3n  3n1  3n2 is divisible

(a) 5n  5n1  5n2 is divisible by 12 for all positive integers of n by 13 for all positive integers of n

by 31 for all positive integers of n

Homework : Text Book Exercise 5.1.3 page 74

53

Students will be able to:
2. Understand and use the concept of logarithms and laws of logarithms to solve problems

2.1 Express equation in index form to logarithm form and vice versa.
2.2 Find logarithm of a number.
2.3 Find logarithm of numbers by using laws of logarithms.
2.4 Simplify logarithmic expressions to the simplest form.

2.1 Expressing equation in index form to logarithm form and vice versa.

Definition of logarithm

If a is a positive number and a  1, then N = a x  log a N = x

(log a N is read as logarithm of N to the base of a )

Note : log a 1 = 0 ( The logarithms of 1 to any positive number base is zero )
a) a 0 = 1 log a a = 1 ( The logarithms of any positive number a to the base a is 1)
b) a1 = a

Example 5 : Convert the following to logarithm form

(a) b-2 = 0.01 1 (c) 81 = 33

(b) c = 643

Exercises 5 : Convert the following to logarithm form

(a) 64 = 4 3 (b) 3 b = 1 c) f = ( 2 ) -2
9 3

Example 6 : Convert the following to index form.

(a) log a 64 = 3 1 c) log a 5 = p

(b) =log x 9

2

Exercises 6: Convert the following to index form

1 (b) 4 =log 5 625 c) log 10 100 = 2

(a) log x = - 2

9

Example 7: Solve the following equations ;

1 (c) log x 4 = 2 d) log 5 x = 0

(a) log x 3 = 5 (b) log x 4 =

2

Exercises 7: Solve the following equations { Ans a) a b) 125 c) 4 d) 3

(a) log 3 x = 2 11 1 d) log x 27  3

(b) log x = - c)log x = -2

53 16

Homework : Text Book Exercise 5.2.1 page 76

54

2.2 Finding Logarithm of a Number.

Logarithms to the base of 10 are called common logarithms . The values of common logarithms can
be determined using a scientific calculator.5
Logarithm of negative numbers and zero are undefined Eg log 6 0 and log 10 (-8 )are undefined .

If log 10 N = x , then antilog N = 10x

Example 8 Use a calculator to evaluate each of the following

(a) log 10 16 (b) log 10 ( 2 )3 (c) antilog 0.1383 (d) antilog(-0.279)
3

Exercises 8 : Use a calculator to evaluate each of the following

(a) log 10 61 (b) log 10 ( 4 )3 (c) antilog 1.1383 (d) antilog(-0.979)
3

Example 9 Find the value of the following logarithms

(a) log4 16 81 (c) log5 125 (d) log6 216

(b) log 3 ( )

27

Exercises 9 Find the value of the following logarithms

(a) log2 64 (b) log 3 81 (c) log4 1024 1
(d) log 5 ( 625)

Homework : Text Book Exercise 5.2.2 page 77
2.3 Finding Logarithm of Numbers by Using Laws of logarithms

Laws of Logarithms = log a x – log a y
x

(i). Log a xy = log a x + log a y (ii) . Log a ( )
y

(iii) Log a x n = n log a x

Example 10: Evaluate each of the following without using a calculator Ans : a) 4 b) –2 c) 0 d) 1
3

(a) log 2 16 1 (c) log a 1 (d) log c 3 c

(b) log 5

25

Exercises 10: Evaluate each of the following without using a calculator Ans : a) 4 b) -2 c) 3 d) 1
(a) log 3 81 (b) log 27 3 3

1 (c) log 0..5 0.125
(b) log 2 4

55

Example 11: Ans; a) 2 b) i)1011 ii)025 iii)2.65 b) Given that log 5 2 = 0.43 and log 5 3 = 0.8
a) Find the value of log 2 7 + log 2 12 –log 2 21 evaluate the following i) log 5 6 ii) log 5 1.5
log 5 72
iii)

Exercises 11 : A) Evaluate each of the following Ans : a)3 b)2 c)2

(a)log 3 21 + log 3 18 –log 3 14 (c) 2log 4 2 – 1 log 4 9 + log 4 12 (a) log 8 45 – 1 log 8 81
2 2

+ 7log 8 2 – log 8 10

B) Given that log 7 4 = 0.712 and log 7 5 = 0.827. Evaluate each of the following
[ Ans : a)0.115 b)1.712 c)2.366 d) –0.712 ]

1 (b) log 7 28 (c) log 7 100 (b) log 7 0.25

(a) log 7 1

4

Homework : Text Book Exercise 5.2.3 page 78

2.3 Simplifying logarithmic expressions to the simplest form.

Example 12 : Express each of the following in term of log a x and log a y

[ Ans : a) 2log a x – 1 b) log a x + 3log a y c) 1 log a y – 3 –1]

log a y log a x
2 22

x2 (b) loga xy3 (c) loga y
(a) loga y a2x3

Exercises 12 : Express each of the following in term of log a x and log a y

[ Ans : a) 3log a x + 2log a y b) 3log a x – log a y c) 3 log a x – log a y – 2 ]
2

(a) log a x3 y 2 (b) log a x3 (c) loga x3
y y2a4

Example 13 : Express each of the following as a single logarithm.

xy x2 y c) log 3 xy 2 ]
[ Ans : a) log a a b) log a a 3

(a) log a x + log a y – 1 (b) 2 log a x – 1 + 1 (c) log 3 x + 2log 3 y – 1
2 log a y

56

Exercises 13 : Express each of the following as a single logarithm

[ Ans : a) log 2 x y2 (b) logb xyb x ]
(c) ) log 4 16 y 3

(a) log 2 x + log 2 y 2 (b) log b x + log b y + 1 (c) 1 log 4 x – 2 – 3log 4 y
2

Homework : Text Book Exercise 5.2.4 page 79 and skill practice 5.2 page 79

Students will be able to:
3 Understand and use the change of base of logarithms to solve problems

3.1 Find the logarithm of a number by changing the base of the logarithm to a suitable base.
3.2 Solve problems involving the change of base and laws of logarithms.

3.1 Finding the logarithm of a number by changing the base of the logarithm to a suitable base

If a, b, and c are positive number and a  1 then

(i) log a b = logc b 1
logc a (ii) . log a b =

logb a

Example 14 Find the values of the following

Ans : a) 0.4076 b) -1.152 c) 3.5522

(a) log 30 4 (b) log2 0.45 c) log 5 304

Exercises 14: Find the values of the following
Ans : a)1.2925 b)0.8149 c)0.4512

(a)log 5 28 (b)log 200 75 (c)log 0.06 0.32

Homework : Text Book Exercise 5.3.1 page 80
3.2 Solving problems involving the change of base and laws of logarithms.

Example 15 Given that log5 x = p , express each of the following in term of p [ Ans i)p/2 ii)1/p iii) 2/p + 3 ]
iii) log x 25x3
i) log 25 x ii)log x 5

57

Exercises 15 Given that log2 5 = t express each of the following in term of t [ Ans i)3/t ii) 1/t iii) 2/t –1iv) 1- 1/t

i) i) log 5 8 ii)log 5 2 iii)log 5 0.8 iv)log 5 2.5

Example 16 [ Ans 2r/s , 4/r+2s ] Exercises 16 [ Ans ( a) 2m + n)/2 b) (m+2n)/(2m+n)

If log 3x = r and log3y = s , express each of the following in If log 2 x = m and log2y = n , express each of the following
terms of r and s . in terms of m and n

a) log y x2 b) log xy2 81 a) log4 x2 y b) log x2 y xy 2

Homework : Text Book Exercise 5.3.2 page 81 skill practice 5.3 page 81

Students will be able to:
4. Solve equations involving indices and logarithms
4.1 Solve equations involving indices.
4.2 Solve equations involving logarithms.
.

4.1 Solving equations involving indices

Comparison of indices or bases
If a x = a y then x = y or a x = b x then a = b

Example 17

Solve the following equations : [ Ans a) x = ¾ b) 5/3 c) x = 1 , x = 3

a) 16 x = 8 b) 9 x  3 x-1 = 81 c) 2 x2 3 - 4 2x = 0

Exercises 17: Solve the following equations: J: a)2/3 b) – 1 c) 7/5

a)9 x-2 = 27 x-1 b) 8 x = 4 c) 2 3x - 4 x-1 = 32

58

Example 18

Solve the following equations : [ Ans (a) 0.8614 b) 0.8174 c) 0.1123 ]
b) 0.8 2x –1 = 25 x + 1 c) 2 2x  5 x+1 = 7
a) 5x=4

Exercises 18: Solve the following equations: Ans : (a) x = 0.7713 b) x = 21.67 (c) 8.837

a) 3 x+1 = 7 b) 2x 3 x = 9 x-4 c) 2x 3 x = 5 x+1

Example 19 Solve the following simultaneous equations : [ Ans x = -1 , y = 1 b) x = 2 y = 1 ]

a) 2 x  4 2y – 8 and 3x  1 (b) 2 ( 4 x) = 32 y and 27x  81
9y 27 9y

Exercises 19: Solve the following simultaneous equations : Ans : (a) x = 3 , y = 2 b) x = 3, y = 2

a)2x  4 y-x = 32 and 32x  1 b) 3x . 9 y –1 = 243 and 23x  32
35y 81 4y

Homework : Text Book Exercise 5.4.1 page 83

59

4.2 Solve equations involving logarithms

For two logarithms of the same base,
if loq am = log a n then m = n

Example 20 : Solve the following equations: Ans a) x = 14/3 b) x = 2 c) x = 3
c) log 4 (x + 6) = log 23
a) log 5 (3x –5) = 2 log5 6 – log 5 4 b) log x 8 = 5 – log x 4

Exercises 20 : Solve the following equations: Ans a) :a) x = 11 b) x = 2 c) x = 3 / 2

a) log 3 2 + log 3 (x + 5) = log 3 (3x –1) b) b) log 4 ( x + 6) = log 2 3 c) 3 + log x 4 = 5 log x 2

Homework : Text Book Exercise 5.4.2 page 83 Skill Practice 5.4 Pg 84

Solve the equation Given x = log3 k  m and Solve the equation 52x1  9x
log 2 k  n . Find the value of
log5 (8x  4)  2log5 3  log5 4 log k 24 in term of m and n

Ans: x  5 Ans: 1  3 Ans: x  1.5753
mn

60

SPM QUESTIONS 2003

1. Given log 2 T – log4 V = 3, express T in term of V. [4 marks]

2. Solve the equation 4 2x -1 = 7x.

2003

[4 marks]

3. Solve the equation 32 4x = 4 8x + 6.

5. Given that log5 2 = m and log5 7 = p, express log5 4.9 in terms of m and p 2004

[3 marks]

2004

6. Solve the equation 2  4  2  3  1 [4 marks]
[3 marks]
61

7. Solve the equation log3 4x  log3 (2x  1)  1

2005

[3 marks]

8. Given that logm 2  p and logm 3  r , express logm  27m  in terms of p and r.
4

9. Solve the equation 82x  3  1 [42m00a5rks]
4x  2
[3 marks]
10. Given that log2 xy  2  3 log2 x  log2 y , express y in terms of x
2006

[3 marks]

62

11. Solve the equation 2  log3 (x  1)  log3 x 2006

[3 marks]

12.  8b 
 c 
Given that log2 b = x and log2 c = y, express log4 in terms of x and y.

[420m0a7rks]

13. Given that 9(3 n  1) = 27 n, find the value of n.
.

14. Solve the equation 16 2x  3 = 8 4x 2007

63 [3 marks]
2008

[3 marks]

15. Given that log4 x = log2 3 , find the value of x.

2008
[3 marks]

Answer

PAPER 1

1. T  8 V
2. x  1.677
3. x  3
4. 2 p  m 1
5. x  3

6. x  3
2

7. 3r  2 p 1
8. x  1
9. y  4x
10. x  11

8

64

Chapter 6 – Coordinate Geometry

Students will be able to:

1. Find distance between two points
1.1 Find the distance between two points using formula.

1. Finding Distance Between Two Points

y B(x2, y2) Recall : !!!
y2 – y1 Theorem
A(x1, y1) Pythagoras
0
x2 - x1 x
Diagram 1

Distance between A and B given by , AB  (x2  x1)2  ( y2  y1)2

Example 1 [ Ans a) 15 unit b) p = 0 or p = 10 ] b) Distance between A (p,-6 ) and B (-5,6 ) is 13 unit.
a) Find the distance between A (4,15 ) B (-5,3) Find the possible value of p

Exercise 1 [ Ans 13 unit b) b = -3 or 9 ] b) Given points P ( 2,6 ) , Q (7,3 ) and R (-3,b ). Find the
a) Find the distance between A ( 7, -4) and B (2 , 8)
value of b if PQ = 1 QR
2

c) Find the distance between A ( 19, a) and A ( 4, 3) c) If the point A( p, q) is equidistant form the points

is 17 units. Find the possible values of a B(2,-1) and C (3,6) ,show that p = 20 – 7q

Homework Text Book Skill Practice 6.1 page 91
65

Students will be able to: m : n.

2. Understand the concept of division of a line segments .
2.1 Find the midpoint of two given points.
2.2 Find the coordinates of a point that divides a line according to a given ratio

y n B(x2, y2) Given A (x1, y1 ) and B (x2 , y2) , the mid point of AB
 x
m P(x, y) = ( x1  x2 , y1  y2 )
A(x1, y1) 22
0
T he coordinates of a point , P that internally divides
a line segment in the ratio m : n =

(x. y) = ( nx1  mx2 , ny1  my2 )

mn mn

Example 2 [ Ans a) (-1, 1) b (0,9) ]

a) Find the coordinates of midpoint of the pair of a b) ABCD is a parallelogram . Given that the diagonal
intersection is at (I,6) and point D is (2,3) . Find
points A (6,-5) and B (-8 , 7) the coordinates of B

Exercise 2 [Ans a) x = 8 , y = 14 b) a= 5 , b= - 1/12, c = -23/12 ]

a) Given that the midpoint AB is (3,4), A ( -2,-6) and B b) Given that vertices of a rhombus are A (-1,2 ) , B(

( x, y) . Find the value of x and y a,b), C (0,-4) and D (-6,c) , Find the value of a, b,

and c

Homework Text Book exercise 6.2.1 page 93 and exercise 6.2.1 page 94
66

Example 3 [ Answer ( 3 , 3/2) b) m : n = 2 : 3 ] b) The point P ( 6/5 , -1) internally divides the line
a) The point G (x, y) internally divides the line segment joining points S (4,3 ) and T (-3,-7) in the ratio
m : n. Find the ratio m : n .
segment joining points A(6,3) and B(2,1) in the ratio
3 : 1. Find the coordinates of point G

Exercise 3 [ Answer (-1 , - 6 1 ) b) 1 : 5 , a = -1 c) (-9, 22 1 ) d) ( 8 , 2 2 ) ]
2 23

a) Point C internally divides the line AB in the ratio b) The point P (-3, a ) internally divides the line

5 : 3. Given that point A and B are ( -6, -9) and segment joining points A (-6, -2) and B(12,4 ) in the
(2, -5). Find the coordinates of C
ratio m : n. Find the ratio m : n and the value of a

c) Point P internally divides the line AB so that d) The coordinates of points A and B are (11, 1) and

PA = 2 PB. If the coordinates of P is (-3, 12) and (2,6) respectively. Point Q lies on the straight line AB
3 such that 2AQ = QB . Find the coordinates of point Q.

point A is (1,5), Find the coordinates of B.

Homework Text Book Skill Practice 6.2 page 95
67

Students will be able to:

3.0 Find areas of polygons.

3.1 Find the area of a triangle based on the area of specific geometrical shapes.

3.2 Find the area of a triangle by using formula.
3.3 Find the area of a quadrilateral using formula.

Notes

The area of a polygon formed by the points A (x 1, y 1 ) , B ( x2, y2), C (x3, y3) ….. G ( xn, yn ) as
vertices is given by the positive values of the formula

All points must arranged in order i.e

Area  1 x1 x2 x3 ....... xn x1 point ABCD or ADCB not ACDB .
2 y1 y2 y3 yn y1

Example 4 [ Answer 11 unit 2 b) 34 unit2 ] b) The vertices of a quadrilateral are A (1 ,-2 ) B(6,2), C
a) Find the areas of a triangle with vertices are (5,6) and D (-2,3). Find the area of a quadrilateral ABCD.

A (5,2), B (1,3) and C(-5 - 1)

Exercise 4 [ Answer 10 unit 2 b) 35 unit 2 ] d) The vertices of a quadrilateral are A (5,10 ) B(10,11)
c) Find the areas of a triangle with vertices are C (12,6) and D (3,5). Find the area of a quadrilateral ABCD.

A (2,6) , B (-5,5) and C (1,3)

Homework Text Book Exercise 6.3.2 pg 99 and Exercise 6.3.3 pg 100
68

Example 5 [ Answer a = 15 or -6 1 ] b)Show that A(-4,1 ) B(1,-2) and C(6,-5) lie on a straight
2 line

a) The vertices of a triangle are (2a,a ) (5,6) and

(9,4) . Find the value of a. if the area of the
triangle is 43 unit 2.

Exercise 5 [ Answer a = 3 b) q = 3 or  11 ] b) Find the value of q if the points A ( 2, 1) , B( 6,q ) and
3
9
a) The vertices of a quadrilateral are (- a , 4a ) (9,11)
C ( 3q , ) are collinear .
and (1,2 ) and (-11, 3) . Find the value of a. if the area of
quadrilateral is 116 unit 2. 2

Homework Text Book Skill Practice 6.3 pg 100

Students will be able to:

4. Understand and use the concept of equation of a straight line.

4.1 Determine the x-intercept and the y-intercept of a line.
4.2 Find the gradient of a straight line that passes through two points.
4.3 Find the gradient of a straight line using the x-intercept and y-intercept.
4.4 Find the equation of a straight line given:

a) gradient and one point; b) two points; c) x-intercept and y-intercept.
4.5 Find the gradient and the intercepts of a straight line given the equation.
4.6 Change the equation of a straight line to the general form.
4.7 Find the point of intersection of two lines.

69

4.1 Determining the x-intercept and the y-intercept of a line

Example 6

a) State the x – intercept and the y – intercept of b) Find the intercept of the following graphs

the straight line passing through each of the y
following pairs of points (0, -9 ) and ( 8, 0).

3

c) State the x – intercept and the y – intercept of x
the straight line passing through each of the 4
following pairs of points (- 4, 0 ) and ( 0 , -6 ).
d) Find the intercept of the following graphs

y

2

-3 x

Home work Text Book exercise 6.4.1 pg 101

4.2 Finding the gradient of a straight line that passes through two points.

The gradient of a straight line that passes through two points is given m = y1  y2
x1  x2

Example 7 b) Given that the gradient of the straight line passing
through P ( 1, a) and Q ( 4p , 9 ) is 3, Find the
a) Find the gradient of the straight line that passing value of a
the points (- 2 , -9 ) and ( 8, 5 ).

c) Find the gradient of the straight line that passing d) Given that the gradient of the straight line passing

the points (- 4 , -7 ) and ( 3, 5 ). through A ( a , 3 ) and Q ( 4 , 9 ) is 2 , Find the

value of a

Homework Text Book exercise 6.4.2 pg 103

4.3 Finding the gradient of a straight line using the x-intercept and y-intercept.

Gradient, m = - y  int ercept
x  int ercept

Find the gradient of each line in 4.1 c) d)

a) b)

Homework Text Book exercise 6.4.3 pg 105

70

4. 4 Finding the equation of a Straight Line

1. If the gradient m and a point (x1 , y1) lie on a straight line 3. Given two point (a,0) and (0,b) where
The equation of a straight line is given by y - y1 = m (x - x1 ) a is x – intercept and b is y- intercept

2 If two point (x1 , y1) and ( x2, y2 ) lies on a straight line is given The equation of a straight line is given

The equations of a straight line is y  y1  y2  y1 by x  y  1
x  x1 x2  x1 ab

Example 6 [Answer y = 2/3x + 5 , b) y = 3 - x c) 2x + y = 10 ]

a) Find the equation of a straight b) Find the equation of the straight c) Find the equation of the straight line
that passes through the points (5,0)
2 line that passes through the points and (0,10)
line where the gradient is (-2,5) and (4, -1)

3

and passing through the point

(-6,1)

Exercise 6 [ Answer y = 3x + 11 b) x + 5y = 16 c) 5x + 3y = 15 ] b) Find the equation of the straight
a) Find the equation of a straight b) Find the equation of the straight line line that passes through the points
line where the gradient is 3 and that passes through the points (1,3) (3,0) and (0,5)
passing through the point and (6,2 )
( -2,5)

Home work Text Book exercise 6.4.4 pg 107
4.5 Finding the gradient and the intercepts of a straight line given the equation.

The equations of a straight line can be expressed in gradient form or intercept form and subsequently
determine the gradient and the intercept of the straight line

a) Gradient form, y = mx + c, where m is the gradient and c is the y – intercept

b) Intercept form x  y  1 where a is the x – intercept and b is the y – intercept
ab

71

Example 7 [Answer y = - 3/2x + 7 , b) y = 5 x – 7 c) x  y  1 ]
3  10 2

a) Express the equation of the 4 c) Express the equation of the straight
line 5y + 4x + 10 = 0 in intercept form.
straight line 2y +3x = 14 in b) Write the equation of the straight Hence, state the x- intercept and y-
intercept
gradient form. Hence state the 5
gradient and the y –intercept of the
line with a gradient of and
line
3

y – intercept of - 7 in gradient form .

Exercise 7 y =  5 x - 3 , b) y = – 7x + 11 c) x  y  1 ]
3 4 2

a) Express the equation of the b) Write the equation of the straight c) Express the equation of the straight
line 2 x + 8 = 4y in intercept form.
straight line 3y +5x + 9 = 0 in line with a gradient of  7 and Hence, state the x- intercept and y-
intercept
gradient form. Hence state the y – intercept of 11 in gradient form .

gradient and the y –intercept of the

line

Home work Text Book exercise 6.4.5 pg 109

4.6 Changing the equation of a straight line to the general form.
The equation of a straight line in general form is written as ax + by + c = 0

Example 8 b) Express the equation of the straight c) Express the equation of the straight
a) Express the equation of the
straight line 2y +3x = 14 in line y =  3 x - 14 in general form line x  y  1 in general form
general form 2 54

Exercise 8 b) Express the equation of the straight c) Express the equation of the straight

a) Express the equation of the line y = 7 x - 12 in general form line 3x  2 y  1 in general form
straight line 2y + 8 = 7x in 5 53
general form

Home work Text Book exercise 6.4.6 pg 109

72

4.7 Find the point of intersection of two lines.

When two line intersect, the point of intersection is the point that lies on both lines. Hence, we can
find the point of intersections by solving the equations of both lines simultaneously

Example 8 [ Answer a) (0,9) b) y = 3x ] b) Find the equation of the straight line that pass through
a) Find the point of intersection of the straight origin and the intersections point of 3x - 2y + 3 = 0 and 3x
+ y - 6 = 0.
lines y = 4x - 9 and x  y  1
18 9

Exercise 8 [ a) Answer (11,-3) b) x - 4y + 3 = 0 Find the equation of a straight line that has a gradient of
a) Find the point of intersection of the straight b) 1/4 and passes through the point of intersection of the
straight lines y = 3x - 2 and 2x + 3y - 5 = 0
lines x + 2y = 5 and 2x + y = 19

Home work Text Book exercise 6.4.7 pg 111 Skill practice 6.4 pg 111

Students will be able to:

5. Understand and use the concept of parallel and perpendicular lines
5.1 Determine whether two straight lines are parallel when the gradients of both lines are known and vice

versa.
5.2 Find the equation of a straight line that passes through a fixed point and parallel to a given line.
5.3 Determine whether two straight lines are perpendicular when the gradients of both lines are known

and vice versa.
5.4 Determine the equation of a straight line that passes through a fixed point and perpendicular to a

given line.
5.5 Solve problems involving equations of straight lines.

5.1 Determining whether two straight lines are parallel when the gradients of both lines are known
and
vice versa

If two lines have an equal gradient, they must be parallel. Conversely if two lines are parallel, they have an
equal gradient. y = m1 x + c1 and y = m2x + c2 are parallel if and only if m1 = m2

73

Example 9 [ b) Answer k = -10/3 ] b) Given that the straight line 5x + ky = 3 and
a) Show that A( -1 , 2 ) , B ( 2,3 ) and c (5,4) are 2y - 3x -8 = 0 are parallel .Find the value of k

collinear

Exercise 9 [ Answer k = 4 b) If A ( -2,4) , B ( 1, k ) and C (2, - 8) are collinear find the
value of k
a) Given that the straight line kx  2y 1  0 and
8x  4y  3  0 are parallel Find the value of k

Homework Text Book exercise 6.5.1 pg 114

5.2 Finding the equation of a straight line that passes through a fixed point and parallel to a given line.

Example 10 Answer 2y = 5x - 17 b) 6y  5x  64 ]

a) Find the equation of the straight line that passes b) Find the equation of the straight line that passes through
P(8,4) and parallel to the line which joins A(-1,2) and
through A ( 5 , 4 ) and parallel to the line B(5,-3)
5x – 2y – 1 = 0

3 c) Find the equation of the straight line that passes through
A(2-3) and parallel to the line which joins B (2,0) and C
Exercise 10 Answer a) y = x +7 b) y = -1/2x - 2
(-6,4)
2

a) Find the equation of the straight line that passes

through A ( -2 , 4 ) and parallel to the line
3x – 2y – 1 = 0

Homework Text Book exercise 6.5.2 pg 115

74

5.4 Determining whether two straight lines are perpendicular when the gradients of both lines are
known and vice versa.

Two straight lines with gradient m1 and m2 are perpendicular if and only if m1 m2 =  1

Example 11 [ Answer p = - 6 ] b) Given the point P ( -3,3) , Q ( 3,1) and R ( -2,4 ) and
S ( (1,5) , show that PQ is perpendicular to RS
a) Given that the straight line px 10y  7  0
and 5x  3y  4  0 are perpendicular to each
other Find the value of p

Exercise 11 [ Answer p =2/3 ] b) Given the point A (4,3 ) , B ( 8,4) and R ( 7,1 ) and S (
a) Given that the straight line y + mx = 5 and (6,5 ) , show that AB is perpendicular to RS

2y = 3x + 4. are perpendicular to each other Find
the value of m.

Homework Text Book exercise 6.5.3 pg 118

5.5 Determining the equation of a straight line that passes through a fixed point and perpendicular to a

given line

Example 12 [a) 3x - 4y + 23 = 0 b) 2x + y + 1 = 0 ]

a) Given that P(2,1) and Q (-4,9). Find an equation b) Find an equation of the straight line passing through

of the perpendicular bisector of PQ the point (1 , -3 ) and perpendicular to the line x - 2y +

6=0

Exercise 12 [a) x - 6y + 4 = 0 b) 3x +2y + 3 = 0 ]

a) Given that R(3,-5) and S (1,7). Find an equation b) Find an equation of the straight line passing through

of the perpendicular bisector of RS the point (3 , -3 ) and perpendicular to the line 2x - 3y +

6=0

Homework Text Book exercise 6.5.4 pg 119

75

5.6 Solve problems involving equations of straight lines. y P
Q
Example 12 [a) n = 6 b) 4y = x + 32 ]

a) b)
y

B

A

(n, 0) x 0R x
0
In the above diagram , PQ dan QR are a straight line that
D (3, -2)
perpendicular to each other at point Q. Given that the
The diagram shows a trapezium ABCD.Given that equation of QR ialah y = 8 – 4x, Find the equation PQ.
the equation of straight line of AB is
3y – 2x -1 = 0 . Find the value of n.

Homework Text Book Skill Practice 6.5 pg 121

Students will be able to:

6.0 Understand and use the concept of equation of locus involving distance between two points.

6.1 Find the equation of locus that satisfies the condition if:

a) the distance of a moving point from a fixed point is constant;
b) the ratio of the distances of a moving point from two fixed points is constant.
6.2 Solve problems involving loci.

Example 12 [ Answer x2 + y2 + 4x - 6y - 12 = 0 , b ) 3x - 5y - 5 = 0 ]

a) A point P moves in a Cartesian plane such that b) b) Find the equation of the locus of a moving point R
its distance from A(-2,3) is 5 unit Find the such that its distance from A(4,-2) is equal to its
equation of the locus of P distance from B (1,3)

Exercise 12 [ Answer a) x2 + y2 - 4x + 10y - 35 = 0 b) 5x2 + 5y2 - 64x - 2y + 189 = 0
76

a) Find the equation of locus of a moving P such b) A moving point R moves such that its distance from A(0,-3)
that its distance from point A( 2,-5 ) is 8 unit and B (6,0). are in ratio RA : RB = 4 : 1 . Find the equation of
the locus of R

c) A moving point A moves such that its distance d) P(2,6) and R(-4,-2) is a diameter of a circle. Point Q(x,y)
from P ( 2 , 1) and Q ( - 1, 3) are in the ratio moves along the arc of a circle. Find the equation of the locus
of the point Q [ Answer x2 + y2 + 2x - 4y - 20 = 0 ]
d) 1 : 2 . Find the equation of the locus of P
[ Answer 3x2 + 3y2 – 18x -2y + 10 = 0 ] 3
x

2

+

3

y

2

+

2
x

-

5

=

0
]

2

Homework Text Book Skill Practice 6.6 pg 126

77

SPM Questions

TOPIC: COORDINATE GEOMETRY
PAPER 1

YEAR 2003
1. The points A(2h, h), B(p, t) and C(2p, 3t)are on a straight line. B divides AC internally in the ratio 2

: 3. Express p in terms of t.
[3 marks]

2. The equation of two straight lines are y  x  1 and 5y  3x  24 . Determine whether the lines are
53

perpendicular to each other.
[3 marks]

YEAR 2004
3. Diagram 3 shows a straight line PQ with the equation x  y  1.

23
The point P lies on the x-axis and the point Q lies on the y-axis.

y
Q•

•
O Px

Diagram 3
Find the equation of the straight line perpendicular to PQ and passing through the point Q.

[3 marks]

78

4. The point A is (-1, 3) and the point B is (4, 6). The point P moves such that PA : PB = 2 : 3. Find
the equation of the locus of P.
[3 marks]

YEAR 2005
5. The following information refers to the equations of two straight lines, JK and RT, which are

perpendicular to each other.

JK : y  px  k
RT : y  (k  2)x  p

where p and q are constants.

Express p in terms of k.

[2 marks]

YEAR 2006
6. Diagram 6 shows the straight line AB which is perpendicular to the straight line CB at the point B.

y

• A(0, 4)

•B

O x
•C
[3 marks]
Diagram 6
The equation of the straight line CB is y = 2x – 1.
Find the coordinates of B.

79

Paper 2
YEAR 2003
1. Solutions to this question by scale drawing will not be accepted.

A point P moves along the arc of a circle with centre A(2, 3). The arc passes through Q(-2, 0) and
R(5, k).
(a) Find

(i) the equation of the locus of the point P,
(ii) the value of k.

[6 marks]

(b) The tangent to the circle at point Q intersects the y-axis at point T.
Find the area of triangle OQT.

[4 marks]

YEAR 2004
2. Diagram 7 shows a straight line CD which meet straight line AB at the point D. The point D lies

on the y-axis.
y

C

O B(9, 0) x

D
A(0, –6)

Diagram 7
80

(a) Write down the equation of AB in the form of intercepts. [1 marks]
(b) Given that 2AD = DB, find the coordinates of D. [2 marks]
(c) Given that CD is perpendicular to AB, find the y-intercept of CD. [3 marks]

YEAR 2005

3. Solutions to this question by scale drawing will not be accepted.

y
A(–4, 9 )

B

O x

2y + x + 6 = 0

C

Diagram 8

(a) Find
(i) the equation of the straight line AB.
(ii) the coordinates of B.

[5 marks]

81

(b) The straight line AB is extended to a point D such that AB : BD = 2 : 3. [2 marks]
Find the coordinates of D. [3 marks]

(c) A point P moves such that its distance from point A is always 5 units.
Find the equation of the locus of P.

YEAR 2006
4. Solutions to this question by scale drawing will not be accepted.

Diagram 9 shows the triangle AOB where O is the origin.
Point P lies on the straight line AB.

y

A(–3, 4 )

•C

Ox

B(6, –2)

Diagram 3

(a) Calculate the area, in unit2, of triangle AOB.
[2 marks]

(b) Given that AC : CB = 3 : 2, find the coordinates of C.
[2 marks]

(c) A point P moves such that its distance from point A is always twice its distance from point B.
(i) Find the equation of the locus of P.
(ii) Hence, determine whether or not this locus intercepts the y-axis.
[6 marks]

82

ANSWERS (COORDINATE GEOMETRY)

PAPER 1 (a) x  y  1
96
1. p = -2 t , 5y  3x  24
y  x 1 (b)
53 D  (3,  4)

m1   5 , y  3 x  24 (c) y- intercept  1 .
3 55 2
3
2. m2  5 3.
(a)(i) Equation of line AB, y  2x 17
m1  m2   5  3 (ii) B (8, 1)
3 5
 1 (b) D(14, 11)

 the lines are perpendicular to each other (c) x2  y2  8x 18y  72  0

. 4. (a) area  9 unit2.

3.
y  2x3
3

4.
5x2  5y2  50x  6y 118  0

5.
p  1 or p  1
k2 2k

(b)

6. B ( 2, 3) Coordinates of C   12 , 2
 5 5 

PAPER 2 (c)
(i) locus of P : 3x2  54x  3y2  24 y 135  0
1.
(i) x2  y2  4x  6 y 12  0 (ii) when x  0, 3y2  24 y 135  0
y2  8y  45  0
(a)
(ii) k  1 or 7 a  1, b  8, c  45
b2  4ac  64 180
(b) Area of OQT   8 unit2
3  244
b2  4ac  0,
2. locus of P intercepts the y-axis

83

Ungroup Data CHAPTER 7 – STATISTIK Group Data
IMPORTANT POINTS
Ungroup Data

(in a Frequency Table)

Data sets which are not grouped Data sets which are not grouped Data sets which are grouped into
into classes. into classes but are presented in classes and presented in Frequency
Frequency Table. Table.
Example: Example: Example:

The masses of six pupils in Number of Number of Number of Books Number of
kilogram: Books Read Students Read Students
50, 52, 55, 60, 55, 59. 0–1
0 5 2-3 11
1 6 4-5 12
2 8 6-7 15
3 4 8-9 8
4 2 7

Mode = The value which is Mode = The value of data which Modal Class = The class with
repeated the most number of has the highest frequency. highest frequency.
times in a set of data. Mode is obtained from the highest
bar of a histogram with the
procedure as shown below.

Modal class

frequency

mode

Mean, _ = x Mean, _ =  fx Mean, _ =  fx

x N x f x f

 x = sum of all the x = value of data x = class mid-point
f = frequency f = frequency
values of data.
N = number of values

of data.

Median, m = the value in the Median, m = the value in the Median, m
middle position of a set of data middle position of a set of data
after the data are arranged in after the data are arranged in  N F 
ascending order. ascending order.  2 
= Lm +   c
 fm 

Lm = lower boundary of
the median class.

N = sum of frequency.
F = cumulative

frequency of the class before
the

median class.
fm = frequency of the median class.
c = size of the median class.

84

1. Determining the mean, mode, median of ungrouped data
Ungroup data is a data sets which are not grouped into classes

The mean or arithmetic mean of a set of ungroup data (x) consisting of N values is given by

mean  sum of all data . In symbol x   x
number of values N

Example 1 [ Ans a) 40.56 b) 4.059 ] b) Find the mean of the data in the following table
a) Find the mean of the following set of data x234567
f 5 13 15 12 4 2
23, 43, 56, 27, 34, 37, 45, 43 , 57

Exercise 1 [ Ans 102857 b) 3.2174 ] d) Find the mean of the data in the following table
c) Find the mean of the following set of data x123456
f 3 10 15 12 4 2
3,7,9,11,11,15,16.

Homework Text Book : Exercise 7.1.1 page 137

The mode is the value with the highest number of occurrence in a set of data.

Example2 Determine the mode for each of the following set of data

a) 2,4,5,6,4,7,3 e) Find the mode of the data in the following table c) 3.2,3.3, 3.9, 3.3.,3.4,3.5, .3.6, 3.7
x12345
f 3 10 15 12 4

Homework Text Book : Exercise 7.1.2 page 137

The median is a single value at the centre of a set of data which has been arranged in ascending or

descending order such that the data is divided into two parts, each part consisting of the same number of data.

In general if a set of data has N values which are arrange in order then ,

a) the median is the ( N  1)th value if N is odd number.

2

b) the median is the mean of ( N )th value and ( N  1)th value if N an even number
22

85

Example 3 Determine the median for each of the following set of data

a) 2,4,5,6,4,7,3 f) Find the mode of the data in the following table c) 3.2,3.3, 3.9, 3.3.,3.4,3.5, .3.6, 3.7
x12345
f 3 10 15 12 4

Homework Text Book : Exercise 7.1.3 page 138
Exercise 3
Find the mean , mode and the median for the following set of data:

(a) 9, 5, 3, 3, 7, 13, 9 (b) 3, 4, 11, 3, 10, 11, 2, 3, 7 (c) 2, 8, 11, 9, 6, 5, 12, 11

d) No. of e) No. of f) No. of pupils
No. of classes No. of players Score 4
absentees goals 8 8
0 3 3 12 9 11
12 10
1 8 4 10 15 5
6 20 2
2 4 5 9 21
3
3 1 6 7

4 7 5

5

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Students will be able to:
1.4 Determine the modal class of grouped data from frequency distribution tables.
1.5 Find the mode from histograms.

.

1. The modal class of group data is the class having the highest frequency
2. The mode of a grouped data can be obtained from a histogram

Example 4 For each of grouped data below, determine the modal class and the mode

a

Marks 10-19 20- 30-39 40 - 50 - Mass(kg) 30-39 40- 50-59 60 - 70 -
49 69 79
29 49 59
Frequenc 5 8 10 6 3
Frequenc 5 7 9 83 y

y

Modal class = …………….. Modal class = ……………..

From the histogram, mode =

From the histogram, mode =
Homework Text Book Exercise 7.1.5 Page 141

Students will be able to:
1.6 Calculate the mean of grouped data.

The mean of grouped data given by x  fx

where x is mid-point and f is the frequency of the class

f

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Exemple 5 : Find the mean of each grouped data of the following.

Height / No. of
(a) cm pupils

141 – 145 7

146 – 150 9

151 – 155 16

156 – 160 6

161 – 165 2

Exercise 5 : Find the mean of each grouped data of the following

a) 1-20 21-40 41-60 61-80 81-100 b) 10-19 20-29 30-39 40 - 49 50 -59
3 5 16 42 57 9 83
Marks Marks
Frequency Frequency fx

Solutions Mid point x ,f fx Marks Mid point x ,f  fx 

Marks

1-20 105 3 315

21-40 5

41-60 16

61-80 4

81-100 2

f   fx  f 

Mean, x  Mean =

Homework Text Book Exercise 7.1.6 Page 143

Students will be able to:
1.7 Calculate the median of grouped data from cumulative frequency distribution tables.
1.9 Estimate the median of grouped data from an ogive.

The median can be obtained by

a) estimating from an ogive L : lower boundary of the median class
N : Frequency
a) using a cumulative frequency table and the formula C : size of the class interval
F : cumulative frequency before the median class
b)
median.
1 N F 
 fm  fm : frequency of the median class
M  L  2 C , where

 
 

c) estimating from an ogive

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Example 5. Find the median for the data in the following group data

a) Ans ( median = 4925) b)

Marks 1-20 21-40 41-60 61-80 81-100 Marks 10-19 20-29 30-39 40 - 49 50 -59
Frequency 35 16 42 Frequency 57 9 83

Cumulative, F

Exercise 5 : For each of the following sets of data, without drawing an ogive, calculate the median of the set of data

a) Number b) No. of
Height / of pupils Marks pupils
cm 20 – 29
141 – 145 7 30 – 39 2
40 – 49
146 – 150 9 50 – 59 4
60 – 69
151 – 155 16 5

156 – 160 6 10

161 – 165 2 6

Homework Text Book exercise 7.1.7 page 144 and exercise 7.1.8 page 148 ( you have learnt in mathematics)

Students will be able to:
1.9 Determine the effects on mode, median and mean for a set of data when:

a) each data is changed uniformly;
b) extreme values exist;
c) certain data is added or removed.
1.10 Determine the most suitable measure of central tendency for given data.

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1.9 Determining the effect on mode, median, and mean for a set of data when each data is changed uniformly
.
Example 6 Find Mode , mean and median for each of following : 2, 3, 3, 6, 9, 13.
Find the Mode , mean and median if

a) each of the data was added by 3, subtracted by 3, multiplied by 4 and divided by 2
Complete the table below

Mean Mode Median Conclusions :

Original data Original data mean Mod Median
Data + 3 S U V
Data -3  Original

Data  4 data
Data  2
 / Original

data

b) Extreme values exist

Example 7
Find mean ,mod and median for data A and data B and complete the following table

A : 16,17,18,19,18
B : 15,16,19,42 16

Solution Mean Mode Median
176 18 18
Data A
Data B

Compare your answer .
The extreme value of 42 in Data B causes the mean to increase and the value of mean does not accurately
represents the measure of central tendency .

C) Certain data is added or removed

Example 8 : Data A : 16,17,18,19,18

Find mean ,mod and median for data A

If certain data is added or removed , Find mean ,mod and median for new data .

mean Mode Median
18
Original data 176 18

Add 16 to the original

data

Ans: 17167 Ans: 18 Ans:175

Add 20 to the original
data

Ans:18 Ans:18 Ans:18

Removed 16 from the
original data

Removed 19 from the Ans:18 Ans:18 Ans:18
original data Ans:18 Ans:17.5
Ans:1725

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Conclusions :

1. If the added data is less than the mean , then the new mean is decreased
2. If the added data is more than the mean, then the new mean is increased .
3. If the removed data is less than the mean, then the new mean is increased .
4. If the removed data is more than the mean, then the new mean is decreased

Effects of uniform changes in a set of data on the mode, mean and median:

1. When a constant number k is added or subtracted to each data in a set, then
* the new mode = original mode  k
* the new mean = original mean  k
* the new median = original median  k

2. When a constant number k is multiplied to each data in a set, then
* the new mode = k x original mode.
* the new mean = k x original mean.
* the new median = k x original median

Example 9 [a Answer x = 22 , b ) i) b =15, q = 20 ii) mode =17, median = 19 ]

a)A set of number has a mean of 14. When two numbers b) The set of data 21, b, 24, 17, b+2, 26 has a mean , q

x and y are added to the set of data the mean of new set if the number b is removed from the set, the mean of the

of the data is 17 . Find the value of x remaining set of data is q +1 . find

i) the values of b and q

ii) the mode and the median of the original data

Homework Text Book exercise 7.1.9 page 149
Determining the most suitable measure of central tendency for given data.

1. Under normal condition, the mean is chosen over the mode and the median to represent a set of data
because ever value in the set is taken into account in the calculation of the mean . Hence the mean gives a
good overall picture of the data.

2. If there exits one or two values that are too large or two small, then the median is a better choice as the
measure of central tendency compared to mean.

3. If a certain value occurs frequently in a set of data, then mode is a better measure of central tendency

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Example 10 : b) i) Determining the mean, mode and median of the set
a) Calculate the mean and median of the set of data of data 4.1, 3.9, 4.2, 3.9, 7.9.
1 , 2,3,3,5,7,8,10,96. Explain why median is more suitable
average measure to represent the above set of data ii) State and explain which of the values in (i)
Solution represents a suitable measure to describe the efficiency
level of the typists
Mean =

Median =

Median is more suitable measure to represent the above
set of data compared to mean because
i) eight out of nine values of data are smaller than the
mean and it does not give an accurate information of the
average

ii) the median is not affected by extreme value , 96 .

Homework Text Book exercise 7.1.10 page 152 . Skill Practice 7.1 page 153

Students will be able to:

7.2 Understand and use the concept of measures of dispersion to solve problems.

7.2.1 Find the range of ungrouped data.
7.2.2 Find the interquartile range of ungrouped data.

7.2 MEASURE OF DISPERSION

Ungroup Data Ungroup Data Group Data
(in a Frequency Table)
Range = largest value of Range = midpoint of the
data – smallest Range = largest value of highest class –
value of data. data – smallest midpoint of the
value of data. lowest class.

Inter quartile range Inter quartile range Inter quartile range
= Q3 - Q1 = Q3 - Q1 = Q3 - Q1

Variance, Variance,

2 =  x2 - _ Variance,  fx 2 _

N x2 2  fx 2 _ 2 =  x2

= f  x2 f

where; where; where;
f = frequency. f = frequency.
 x2 = sum of square of
x = class midpoint.
the values of
data.

N = number of x = value of data.
values of data.
x = mean x = mean

x = mean

Standard deviation, Standard deviation, Standard deviation,

= x2 _ =  fx 2 _ =  fx 2 _

N x2 f  x2 f  x2

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Effects of uniform changes in a set of data on the range, inter quartile range, variance and standard
deviation.
1. When a constant number k is added or subtracted to each data in a set, then

* the new range, interquartile range, variance and standard deviation = original range range, interquartile
range, variance and standard deviation
respectively.

2. When a constant number k is multiplied to each data in a set, then

* the new range = k x original range.

* the new interquartile range = k x original interquartile range..
* the new variance = k2 x original varaince.

* the new standard deviation = k x original standard deviation.

When two sets of data, i.e. set X and set Y are combined, then;

 x2   y2  x  y  2
Ny   
Combined standard deviation = Nx  Nx Ny

2.1 Finding the range and interquartile range of ungrouped data.

For ungroup data
Range = maximum value – minimum value
Interquartile range = upper quartile – lower quartile

Example 11 Find the range and the interquartile range of each set of data below

a) 45,48,50,60,64 b) 11,18,13,15,21,17,14,20

Exercise 11 Find the range and the interquartile range of each set of data below

a) 11,13,14,15,17 b) 17,4,6,10,12,12 13,17

Example 11 Find the range and the interquartile range of each set of data below

a) b) 012345
x12345 x 10 14 20 26 20 10
f 3 10 15 12 4 f
c.f 3 13 28 40 44

Homework Text Book exercise 7.2.1 and 7.2.2 page 154 and 156

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2.2 Finding the range and interquartile range of grouped data.

For grouped data
Range = mid - point of highest class - mid point of the lowest class
Interquartile range can be obtained by using a formula or an ogive (refer to formula for a median )

Example 11 Find the range and the interquartile range of each set of data below {Ans 40 , 17.71] For each of data,
draw an ogive to determine the interquartile range . Compare your answer between using a formula and an ogive )

a) 20-29 30-39 40-49 50-59 60-69 b)
x 1 8 10 6 5 x 26- 31- 36- 41- 46- 26-
f 30 35 40 45 50 30
c.f f 14 18 26 30 12 14
c.f

Homework Text Book exercise 7.2.3 and 7.2.4 page 157 and 160
2.3 Determining the variance and the standard deviation of ungrouped data and grouped data

Variance and standard deviation
a) Ungroup data

 variance, 2  (x  x)2 or  2  x2  x 2 , where x = mean of the set of data , N = total frequency
NN

 Standard deviation, 
(x  x)2 x2  x2
or  
NN

b) Grouped data

variance, 2 fx 2  x 2 , where x = mid-point of the class
f

Standard deviation,   fx 2  x2
f

Example /Exercise 11 Find the variance and standard deviation of each set of data below

a) 1,4,6,10,14 [ Ans 20.8 , 4.56 ] b) 4,3,5,6,8,10 [Ans 5667 ; 2238 ]

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Example/ Exercise 12 Find the variance and standard deviation of each set of data below

a) [ answer 4.193 ] b)

x123 4 5
12 4
x 26 28 30 32 f 3 10 15

f 10 12 15 11

Example/ Exercise 12 Find the variance and standard deviation of each set of data below

a) b) [ ans 122.67 ; 11.08 ]
x
f 26-30 31-35 36-40 41-45 mass kg 20-29 30-39 40-49 50-59 60-69
14 18 26 30 10 6 5
No of 18

student

Homework Text Book exercise 7.2.6 page 165 and 7.2.7 page 168

Students will be able to:
2.8 Determine the effects on range, interquartile range, variance and standard deviation for a set of data

when:
a) each data is changed uniformly;
b) extreme values exist;
c) certain data is added or removed.
2.9 Compare measures of central tendency and dispersion between two sets of data.

2.4 Determining the effects on range, interquartile range, variance and standard deviation for a set of data when:
a) each data is changed uniformly;
b) extreme values exist;
c) certain data is added or removed.

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