Cambridge IGCSE and O Level Additional Mathematics Workbook by Val Hanrahan, Jeanette Powell

Cambridge AssW orking for over
onal Education
25

YEARS
essmeWnItTIHnternati

Cambridge

IGCSE® and O Level

Additional
Mathematics

Val Hanrahan

Jeanette Powell

Welcome to the Cambridge IGCSE and O Level Additional Mathematics Workbook. The aim of this
Workbook is to provide you with further opportunity to practise the skills you have acquired while
using the Cambridge IGCSE and O Level Additional Mathematics Student’s Book. It is designed to
complement the Student’s Book and to provide additional exercises to support you throughout the
course and help you to prepare for your examination.
The chapters in this Workbook reflect the topics in the Student’s Book. There is no set way to
approach using this Workbook. You may wish to use it to supplement your understanding of the
different topics as you work through each chapter of the Student’s Book, or you may prefer to use
it to reinforce your skills in dealing with particular topics as you prepare for your examination. The
Workbook is intended to be sufficiently flexible to suit whatever you feel is the best approach for
your needs.

All exam-style questions and sample answers in this title were written by the author(s). In
examinations, the way marks are awarded may be different.
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overlooked, the Publishers will be pleased to make the necessary arrangements at the first
opportunity.
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© Roger Porkess, Val Hanrahan & Jeanette Powell 2018
First published 2018 by
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A catalogue record for this title is available from the British Library.
ISBN: 978 1 5104 2165 3

2 Photocopying prohibited Cambridge IGCSE™ and O Level Additional Mathematics Workbook

Contents 4
11
CHAPTER 1 Functions 16
CHAPTER 2 Quadratic functions 24
CHAPTER 3 Equations, inequalities and graphs 29
CHAPTER 4 Indices and surds 34
CHAPTER 5 Factors of polynomials 38
CHAPTER 6 Simultaneous equations 46
CHAPTER 7 Logarithmic and exponential functions 51
CHAPTER 8 Straight line graphs 54
CHAPTER 9 Circular measure 62
CHAPTER 10 Trigonometry 64
CHAPTER 11 Permutations and combinations 70
CHAPTER 12 Series 75
CHAPTER 13 Vectors in two dimensions 85
CHAPTER 14 Differentiation 92
CHAPTER 15 Integration
CHAPTER 16 Kinematics

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1 Functions

1 For the function f(x) = 2x − 5 find:

a) f(2) b) f(0) c) f(−3)

2 For the function g(x) = (2x − 2)2 find:

a) g(0) b) g(0.5) c) g(−2)
c) h(0)
3 For the function h:x → x2 − 2x find:

a) h(2) b) h(−2)

4 For the function f:x → 5 − 3x find:

a) f(3) b) f(−3) c) f(1)

5 For the function g(x) = 2x + 1 find: Remember that the
√ symbol means the
a) g(0) b) g(4) c) g(12) positive square root
of a number.

6 For the function f(x) = x2 − 2 b) the numbers in the set {−1, −2, −3}.
Draw a mapping diagram when the inputs are:

a) the numbers in the set {1, 2, 3}

4 Photocopying prohibited Cambridge IGCSE™ and O Level Additional Mathematics Workbook

Questions 1–8

7 Find the range of the following functions:

a) f(x) = 2x + 1 Domain {0, 1, 2}

b) g(x) = 3x2 − 1 Domain {2, 4, 6}

c) h(x) = 2x − 1 Domain {1, 3, 6}
5

d) f(x) = 4x Domain ℤ+ The symbol ℤ means the
set of all integers, and ℤ+
means the set of positive
integers.

e) f:x → x2 − 5 Domain ℤ

8 What values must be excluded from the domain of the following functions and
why must they be excluded?
a) f(x) = 2x − 1

b) f(x) = 3
x2

c) f(x) = 2x − 1
x+1

Photocopying prohibited Cambridge IGCSE™ and O Level Additional Mathematics Workbook 5

1 Functions

9 a) On the axes provided, plot the graph of y = f(x), y
where f(x) = x2 − 4, for 0 ≤ x ≤ 4.
5
b) Add the line y = x to your graph. 4
3
2 1 2 3 4 5x
1

–5 –4 –3 –2 –1–10
–2
–3
–4
–5

c) Given that f−1(x) = x + 4 , calculate the values of:

(i) f−1(−4) (ii) f−1(−3) (iii) f−1(0) (iv) f−1(5)

d) Use these to add the graph of y = f−1(x) to the graph you drew in part a).
e) What you notice?

6 Photocopying prohibited Cambridge IGCSE™ and O Level Additional Mathematics Workbook

Questions 9–10

10 Given that f(x) = 2x − 3; g(x) = x2 and h(x) = 3x − 2 find the following:

a) fg(3) e) fgh(0)

b) gf(3) f) hgf(0)

c) fh(2) g) fgh(x)

d) hf(2) h) hgf(x)

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1 Functions c) fg(0)

11 Given f(x) = 2x + 1 and g(x) = 3x + 1 find:
a) fg(5)

b) gf(5) d) gf(0)

12 Given f(x) = x + 3, g(x) = x2 − 3 and h(x) = x 1 3 for x ≠ −3, find:
+

a) fg(x) d) hf(x)

b) gf(x) e) hfg(x)

c) fh(x) f) hgf(x)

8 Photocopying prohibited Cambridge IGCSE™ and O Level Additional Mathematics Workbook

13 a) Find the inverse for the following functions: Questions 11–14
(i) f(x) = 2x − 3
(ii) f(x) = x2 − 4 for x  0

b) On the axes below, plot the graphs of y = f(x) and y = f−1(x).

y

5
4
3
2
1

–5 –4 –3 –2 –1–10 1 2 3 4 5 x

–2
–3
–4
–5



14 The first graph shows the line y = 2 − x and the other graphs are related to this.
Write down their equations.

y y y a) y y y y b)y y

6 6 6 6 6 66 6 6
5 5 5 5 5 55 5 5
4 4 4 4 4 44 4 4
3 3 3 3 3 33 3 3
2 2 2 2 2 22 2 2
1 1 1 1 1 11 1 1

–3 –2 ––110 –3 1–2 2––1130 –341–252–x–1130–34–125 2–x1–103–341–252–1x–130 –341–252–x1–130–34–1252–x1–103–341–252–1x–310 –431–252–x1–310 4 1 5 2x 3 4 5 x

–2 –2 –2 –2 –2 –2 –2 –2 –2

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1 Functions

15 The graph below shows part of a quadratic curve, defined for values of x  0.

a) Determine the equation of the curve. b) Sketch the inverse of the curve on the
same axes and give its equation.

y

10
9 (2,9)
8
7
6
5
4
3 (1,3)
2
1

–1–10 1 2 3 4 5 6 7 8 9 10 x

16 Draw the following graphs. c) y = – |x – 2|
a) y = x – 2 d) y = 2 – |x – 2|
b) y = |x – 2|

a) y 1 2 3 4 5 6 7x c) y 1 2 3 4 5 6 7x

4 4
3 3
2 2
1 1

–3 –2 –1–10 –3 –2 –1–10
–2 –2
–3 –3
–4 –4

b) y 1 2 3 4 5 6 7x d) y 1 2 3 4 5 6 7x
4
4
3 3
2 2
1 1

–3 –2 –1–10 –3 –2 –1–10
–2 –2
–3 –3
–4 –4

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2 Quadratic functions

1 Solve the following equations by factorising: c) x2 - 11x - 12 = 0
a) x2 + 4x + 4 = 0

b) x2 - 3x + 2 = 0 d) x2 + 2x - 15 = 0

2 Solve the following equations by factorising: c) 4x2 - 12x + 9 = 0
a) 2x2 + 11x + 12 = 0

b) 3x2 - 17x - 6 = 0 d) 4x2 + 5x - 6 = 0

3 Solve the following equations: c) 49 - 16x2 = 0
a) x2 - 64 = 0 d) 64x2 - 100 = 0

b) 9x2 - 144 = 0

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2 Quadratic functions

4 For each of the following functions:

(i) factorise the function

(ii) work out the coordinates of the stationary point

(iii) state whether the stationary point is a maximum or a minimum.

a) y = x2 + 7x + 12 c) f(x) = 2x2 + x - 28

b) y = 12 + 2x - 2x2 d) f(x) = 6 + 2x - 8x2

5 For each of the following

(i) write the left hand side in the form c(x + a)2 + b

(ii) solve the equation.

a) 2x2 - 10x + 15 = 0 c) 2x2 + 8x - 8 = 0

b) 3x2 - 6x + 10 = 0 d) 5x2 + 15x + 9 = 0

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Questions 4–7

6 For each of the following functions:

(i) use the method of completing the square to find the coordinates of the stationary point

(ii) state whether the stationary point is a maximum or a minimum.

a) y = x2 + 2x - 12 c) f(x) = 15 + 2x - x2

b) f(x) = x2 + 3x + 9 d) y = 3 + 4x - 2x2

7 Draw the graph and find the corresponding range for each function and domain.

a) y = x2 - 2x - 8 for the b) f(x) = 4x2 - 2x - 12 for the
domain -3  x  5 domain -3  x  3

y 1 2 3 4 5x y 1 2 3x

8 30
7 25
6 20
5 15
4 10
3
2 5
1
–3 –2 –1 0
–3 –2 –1–10 –5
–2
–3 –10
–4 –15
–5
–6
–7
–8
–9

–10

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2 Quadratic functions

8 For each of the following equations, use the discriminant to decide

if there are two real and different roots, two equal roots or no real roots.

Solve the equations with real roots.

a) 3x2 - 6x = 0 c) r2 + 5r - 14 = 0

b) m2 + 6m + 9 = 0 d) 2x2 - 7x + 6 = 0

9 Solve the following equations by (ii)  using the quadratic formula.
(i) completing the square c) 2r2 + 2r - 1 = 0
a) x2 - 4x - 9 = 0
(i) (i)

(ii) (ii)

b) y2 + 3y = 5 d) 3m2 - 12m + 7 = 0
(i) (i)

(ii) (ii)

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Questions 8–11

10 For each pair of equations:

(i) determine if the line intersects the curve, is a tangent to the curve
or does not meet the curve

(ii) give the coordinates of any points where the curve and line touch
or intersect.

a) y = x2 + 2x - 3; y = x - 1 c) y = 2 - x - x2; y = 2 - x

b) y = x2 - 3x - 3; y = x - 8 d) y = x2 + 2x - 5; y = 4 - 2x

11 Solve the following inequalities and illustrate each solution on a number line:
a) x2 - 5x + 6 > 0

b) p2 + 3p - 10 < 0

c) 4  m2 + 3m

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3 Equations, inequalities and graphs

1 a) Plot the graphs of y = 3x and c) Plot the graphs of y = 2x - 4 and
y = |3x| on the same axes. y = |2x - 4| on the same axes.

y y

6 6
5 5
4 4
3 3
2 2
1 1

−3 −2 −1−10 1 2 3x –3 –2 –1–10 1 2 3 4 5 6 7x
−2 –2
−3 –3
−4 –4

b) Plot the graphs of y = x - 4 and d) Plot the graphs of y = 5 - x and
y = |x - 4| on the same axes. y = |5 - x| on the same axes.

y y

6 6
5
5 4
3
4 2
1
3

2
1

−2 −1−10 1 2 3 4 5 6 7 8 9x −2 −1−10 1 2 3 4 5 6 7 8 9x
−2 −2
−3 −3
–4 –4

16 Photocopying prohibited Cambridge IGCSE™ and O Level Additional Mathematics Workbook

2 a) On the axes below, draw the graph of Questions 1–4
y = |x + 2|.
b) Use the graph to solve |x + 2| = 2.
y
c) Use algebra to verify your answer to b).
6
5 1 2 3 4 5 6 7x
4
3
2
1

–5 –4 –3 –2 –1–10
–2
–3
–4

3 a) On the axes below, draw the graph of b) Use the graph to solve |2x - 3| = 1.
y = |2x - 3|.

y c) Use algebra to verify your answer to b).

6
5
4
3
2
1

–5 –4 –3 –2 –1–10 1 2 3 4 5 6 7x
–2
–3
–4

4 Solve the equation |x - 2| = |x + 2|

a) graphically b) algebraically.

y 1 2 3 4 5x

5
4
3
2
1

–5 –4 –3 –2 –1–10
–2
–3
–4
–5

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3 Equations, inequalities and graphs

5 Solve the equation |2x + 3| = |2x - 3|

a) graphically b) algebraically.

y 1 2 3 4 5x

5
4
3
2
1

–5 –4 –3 –2 –1–10
–2
–3
–4
–5

6 Write each of the following inequalities in the form |x - a|  b:

a) -2  x  12 b) -5  x  25 c) -16  x  8.

7 Write each of the following expressions in the form a  x  b.

a) |x + 1|  3 b) |x + 2|  4 c) |x + 3|  5

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8 (i) Solve the following inequalities and Questions 5–8
a) |x + 1| < 5
(ii) illustrate the solution on a number line:

b) |x + 1| > 5

c) |3x + 2|  7

d) |3x + 2|  7

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3 Equations, inequalities and graphs

9 Illustrate the following inequalities by shading out the unwanted region:

a) y + 2x < 0 c) 2y - 3x < 0

y y

6 6
5 5
4 4
3 3
2 2
1 1

–5 –4 –3 –2 –1–10 1 2 3 4x –5 –4 –3 –2 –1–10 1 2 3 4x
–2 –2
–3 –3
–4 –4

b) y - 2x > 0 d) 2y + 3x > 0

y y

6 6
5 5
4 4
3 3
2 2
1 1

–5 –4 –3 –2 –1–10 1 2 3 4x –5 –4 –3 –2 –1–10 1 2 3 4x
–2 –2
–3 –3
–4 –4

20 Photocopying prohibited Cambridge IGCSE™ and O Level Additional Mathematics Workbook

10 a) Draw the lines y = x - 3 and y = x + 3 on the grid Questions 9–11

y b) Hence solve these inequalities
(i) |x - 3| < |x + 3|
6 (ii) |x - 3| > |x + 3|
5
4
3
2
1

–5 –4 –3 –2 –1–10 1 2 3 4 5 6 7x
–2
–3
–4



11 Solve the following inequalities algebraically: b) |2x - 3| > |x + 3|
a) |2x - 3| < |x + 3|

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3 Equations, inequalities and graphs

12 The unshaded region of each graph illustrates an inequality of a modulus function.
In each case write the inequality.

a) y = |2x - 1| b) y = |3 - 2x|

y y

6 6
5 5
4 4
3 3
2 2
1 1

−4 −3 −2 −1−10 1 2 3 4 5 6 7x –3 –2 –1–10 1 2 3 4 5 6 7x
−2 –2
−3 –3
−4 –4

13 Sketch the following graphs on the axes provided, indicating the points where they
cross the co-ordinate axes:

a) y = x(x + 1)(x + 2) b) y = |x(x + 1)(x + 2)|

y y

6 6
5 5
4 4
3 3
2 2
1 1

–5 –4 –3 –2 –1–10 1 2 3 4 5 6 7x –5 –4 –3 –2 –1–10 1 2 3 4 5 6 7x
–2
–3 –2
–4
–3

–4

22 Photocopying prohibited Cambridge IGCSE™ and O Level Additional Mathematics Workbook

Questions 12–15

14 Identify the following cubic equations from their graphs:

a) b) c)

y y y

6 6 6
5 5 5
4 4 4
3 3 3
2 2 2
1 1 1

−3 −2 −1−10 1 2 3 4 5x −3 −2 −1−10 1 2 3 4 5x −3 −2 −1−10 1 2 3 4 5x
−2 −2 −2
−3 −3 −3
−4 −4 −4

15 Find an equation for each of the following modulus graphs.
All represent the moduli of cubic graphs.

a) b) c)

y 1 2 3 4 5x y 1 2 3 4 5x y

6 6 20 1 2 3 4x
5 5 18
4 4 16
3 3 14
2 2 12
1 1 10

–4 –3 –2 –1–10 –2 –1–10 8
–2 –2 6
–3 –3 4
–4 –4 2

–3 –2 –1–20

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4 Indices and surds

1 Simplify the following; give your answers in the form xn.

a) 34 × 38 c) 54 ÷ 53 e) (24)2

b) 4-2 × 47 d) 65 ÷ 6-2 f) (72)-3

2 Simplify the following; leave your answers in standard form.

a) (5 × 104) × (3 × 103) c) (8 × 104) ÷ (2 × 102)

b) (2 × 106) × (4 × 10-3) d) (6 × 109) ÷ (2 × 10-3)

3 Rewrite each of the following as a number raised to a positive integer power.

a) 2-4 ( )c) 1 -2
2

b) 4-3 ( )d) 2 -3
5

24 Photocopying prohibited Cambridge IGCSE™ and O Level Additional Mathematics Workbook

Questions 1–7

4 Find the value of each of the following. Answer as a whole number or fraction.

a) 53 × 5-2 d) (32)3 g) 16 − 1
2

b) 7-5 × 73 ( )e) 2 −2 3
c) 28 ÷ 23 3
h) 25 2
( )f) 43 i) 197 × 19-7
5

5 Rank each set of numbers in order of increasing size.

a) 33, 42, 25 b) 4-3, 7-2, 3-4

6 Find the value of x.

a) 2x ÷ 22 = 24 b) (75 × 7)x = 77
24 × 23
73 × 72

7 Simplify: d) 9b5 ÷ 3b2
a) 3n3 × 2n2

b) 5p2q4 × 3q-2 e) 6x2y-5 × 4x6y-4
c) (2a3)3
f) 20a 2b 3
4a4b5

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4 Indices and surds b) 10a ÷ 5b = 26 × 52.

8 Find integers a and b such that
a) 5a × 2b = 103

a= b = a= b=

9 Write each of the following in its simplest form.

a) 72 c) 32 + 128

b) 2 2 + 5 2 d) 3 112 − 2 28

10 Express each of the following as the square root of a single number:

a) 2 5 c) 9 2

b) 3 7 d) 6 3

11 Write each of the following in the form a b where a and b are integers
and b is as small as possible.

a) 72 c) 18 A rational number is
64 8 an integer or fraction.

b) 75 d) 32
16 144

12 Simplify the following by collecting like terms: b) 5( 2 + 1) - 3(1 - 2 )
a) (2 + 3 ) + (6 + 3 3 )

26 Photocopying prohibited Cambridge IGCSE™ and O Level Additional Mathematics Workbook

Questions 8–15

13 Expand and simplify: c) (6 + 3 )2
a) (5 + 5)(5 - 5)

b) 2 (6 + 2 ) d) (5 - 2 2 )(5 + 2 2 )

14 Rationalise the denominators. Give each answer in its simplest form.

a) 1 c) 1
5 6–2

b) 3 d) 5 + 2
3 3 − 2

15 Write the following in the form a + b c where c is an integer and a and b are rational numbers.

a) 7 + 3 b) 33
2− 3 3−1

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4 Indices and surds

16 Work out the length of AB. Answer in the simplest surd form.

A

6 + √2

B 6 – √2 C AB =

 
17 A rectangle has sides of length x cm and 2x cm and a diagonal of length 15 cm.

a) Use Pythagoras’ Theorem to find the exact value of x in its simplest surd form.

b) Work out the area of the rectangle.

18 An isosceles triangle has sides of length 5 cm, 5 cm and 2  cm. Work out:
a) the height of the triangle in its simplest surd form

b) the area of the triangle.

28 Photocopying prohibited Cambridge IGCSE™ and O Level Additional Mathematics Workbook

5[Page 31] Factors of polynomials


1 Multiply (x3 + 2x2 + x - 4) by (x + 2).

2 Multiply (2x3 - 3x2 + 2x + 2) by (x - 1).

3 Multiply (2x3 - 5x2 + 4) by (2x - 1).

4 Multiply (x3 + 3x2 - 5x + 5) by (x + 3).

5 Multiply (2x2 - 5x + 6) by (x2 + x - 2).

6 Divide (x3 + x2 - 4x - 4) by (x + 1).

Photocopying prohibited Cambridge IGCSE™ and O Level Additional Mathematics Workbook 29

5 Factors of polynomials

7 Divide (x3 + 7x2 + 16x + 12) by (x + 2).

8 Divide (x3 - 13x + 12) by (x - 1).

9 Simplify (2x2 - 7x + 1)2.

10 Determine whether the following linear functions are factors of the given polynomials or not.
a) (x3 + 9x2 - 2x + 4); (x - 1)

b) (x3 - x2 - x + 1); (x + 1)

c) (2x3 - 2x2 + 5x - 5); (x - 1)

30 Photocopying prohibited Cambridge IGCSE™ and O Level Additional Mathematics Workbook

Questions 7–11

11 For each equation:

(i) use the factor theorem to find a factor of each function

(ii) factorise each function as a product of three linear factors

(iii) draw its graph on the axes provided.

a) x3 - 3x2 - x + 3

(i) (iii) y

5
4
3
2
1

(ii) –5 –4 –3 –2 –1–10 1 2 3 4 5x
–2
–3
–4
–5

b) x3 + 2x2 - 5x - 6 (iii) y
(i)
5 1 2 3 4 5x
(ii) 4
3
2
1

–5 –4 –3 –2 –1–10
–2
–3
–4
–5

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5 Factors of polynomials

12 Factorise as far as possible: x3 + 6x2 + 12x + 7.

13 For what value of a is (x - 3) a factor of x3 - ax2 + 18?

14 For what value of b is (2x + 1) a factor of 2x3 - 7x2 - bx - 6?

15 The expression x3 + px2 - 6x + q is exactly divisible by (x + 1) and (x + 4).
Find and solve two simultaneous equations to find p and q.

16 Find the remainder when each function is divided by the linear factor (shown in brackets).

a) x3 + 3x2 - 2x + 1; (x - 2) b) 3x3 + 3x2 - 4x - 14; (x - 1)

32 Photocopying prohibited Cambridge IGCSE™ and O Level Additional Mathematics Workbook

Questions 12–18

17 The equation f(x) = x3 + 3x2 - 10x - 24 has three integer roots. Solve f(x) = 0.

18 When x3 + ax2 + bx + 2 is divided by (x - 2) the remainder is 36.
When it is divided by (x + 3) the remainder is -4.

a) Find the values of a and b. b) Solve the equation x3 + ax2 + bx + 2 = 0.

Photocopying prohibited Cambridge IGCSE™ and O Level Additional Mathematics Workbook 33

6 Simultaneous equations

1 Solve the following pairs of simultaneous equations graphically.

a) y = x + 4 b) x + 3y = 9

y = 2x + 2 2x - y = 4

y y 1 2 3 4 5 6 7 8 9 10 x

8 1 2 3 4 5 6 7 8x 6
7 5
6 4
5 3
4 2
3 1
2
1 −4 −3 −2 −1−10
−2
−4 −3 −2 −1–10 −3
–2 −4
–3
–4

Use the substitution method to solve the simultaneous equations in questions 2 and 3.

2 2x + y = 13 3 3x + 2y = 6

y = 2x + 1 y=x-2

Use the elimination method to solve the simultaneous equations in questions 4 and 5.

4 x + y = 6 5 2x + y = 10

x-y=2 3x - y = 5

34 Photocopying prohibited Cambridge IGCSE™ and O Level Additional Mathematics Workbook

Questions 1–7

6 Four tins of soup and 3 packs of bread rolls cost $10:80.
Two tins of soup and 5 packs of bread rolls cost $9:60.
Find the cost of 3 tins of soup and 7 packs of bread rolls.

7 In a sale, 3 DVDs and 4 CDs cost $51 and 4 DVDs and 3 CDs cost $54.
a) Find the cost of a DVD and the cost of a CD.

b) I have $90 dollars to spend and would like to buy the same number of DVDs and CDs.
How many of each can I buy?

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6 Simultaneous equations

8 a) Solve this pair of simultaneous equations b) Illustrate the solution graphically.
algebraically:
y
x2 - 6x - y = -8
y-x+4=0 5
4
3
2
1

–5 –4 –3 –2 –1–10 1 2 3 4 5x
–2
–3
–4
–5

9 Solve this pair of simultaneous equations algebraically:
x2 - y2 + xy = 20
x = 2y

10 Two numbers, x and y, have a difference of 2 and a product of 15.
a) Write down two equations that are satisfied by x and y.

b) Find two possible values for the pair of numbers by solving the equations simultaneously.

36 Photocopying prohibited Cambridge IGCSE™ and O Level Additional Mathematics Workbook

Questions 8–11

11 a) Solve this pair of simultaneous equations b) Given that (x - 2)2 + (y - 3)2 = 9 is the
algebraically: equation of a circle, with centre (2,3)
and radius 9, illustrate the solution
(x - 2)2 + (y - 3)2 = 9 graphically.
x-y+4=0
y
1 2 3 4 5x
6
5
4
3
2
1

–5 –4 –3 –2 –1–10
–2
–3
–4
–5
–6

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7 L ogarithmic and exponential
functions
Remember that, for example,
1 In each part of this question, find the values of x and y. the relationship 23 = 8 can
be written as the equation
log28 = 3.

a) 4x = 16, y = log4 16 b) x4 = 81, y = log3 81 c) x = 3-2, y = log3 x

2 Using your knowledge of indices, and without using your calculator, find the values of:

a) log2 32 b) log3 9 c) log5 215

3 Use the rules for manipulating logarithms to write each of the following as a single logarithm.

a) log 6 + log 2 d) 12log 9 - log 3 For example, log 4 + log 3
can be written as log(4 × 3)
= log 12

b) log 36 - log 9 e) 12log 25 + 13log 27

c) 5 log 2 f) log 5 + log 4 - log 2

38 Photocopying prohibited Cambridge IGCSE™ and O Level Additional Mathematics Workbook

Questions 1–6

4 Express each of the following in terms of log x. c) 2 log x + log x
a) log x3 - log x2

b) 2 log x3 + 3 log x2 d) 6 log 3 x - 4 log  x

5 a) Express log5 x12 + log5 x4 as a multiple of log5 x.
b) Hence solve the equation log5 x12 + log5 x4 = 2 without using your calculator.

6 Draw each of the following graphs on the axes below.
In each case show the vertical asymptote and the coordinates
of the points where the graph crosses the x axes.

a) y = log10 x b) y = log10 (x + 5) c) y = log10 (x - 10)

y 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 x

2
1

−5 −4 −3 −2 −1−10
−2

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7 Logarithmic and exponential functions

7 For each of the following graphs:

a) Starting with the curve y = lg x state the transformation (in order when Notice that, as in this
more than one is needed) required to sketch the curve: question, log10 x is
often written as lgx.

(i) y = 2lg x (ii) y = lg 2x.

b) Sketch the curves on the axes below, together with the curve y = lgx in each case.

yy

xx



c) Would the results be the same, or different, if logarithms to a different base were used?

40 Photocopying prohibited Cambridge IGCSE™ and O Level Additional Mathematics Workbook

Questions 7–9

8 Match the correct equation to each graph.

Equation Graph Equation Graph Notice that, as in this
y = ln (x + 2) y = ln x + 2 question, loge x is
y = ln 2x y = ln (2 - x) usually written as lnx.
y = ln x - 2
y = ln (x - 2)

a) c) e)

y yy

4 42

3 31

2 2 −1−10 1 2 3 4 5 6 7 8 x
1 1

−1−10 1 2 3 4 5 6 x −4 −3 −2 −1−10 1 2 3 x −2
−3

−2 −2 −4

−3 −3 −5

b) d) f)

yy y

44 5

33 4

22 3

11 2

−3 −2 −−110 1 2 3 4 5x −1−10 1 2 3 4 5 6 7 8 9x 1
−2 −2 −1−10
1 2 3 4 5 6 7x

−3 −3 −2

9 Solve the following equations for x, given that lg p = 5.

a) p = 10x b) p2x - 6px + 8 = 0

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7 Logarithmic and exponential functions

10 Use logarithms to solve the equation 32x - 1 = 23x + 1 giving your answer to 3 s.f.

( )11 The formula for compound interest is A = P1R n where A represents the final amount,
100
+

P the principal (the amount invested), R the rate of interest and n the number of years.

a) For how long, to the nearest month, would $10 000 need to be invested to produce a final
amount of $15 000 if the rate of interest is 2.8%?

b) $10 000 is invested for 5 years. What was the rate of interest if the final amount is $12 000?
Answer to 1 decimal place.

42 Photocopying prohibited Cambridge IGCSE™ and O Level Additional Mathematics Workbook

Questions 10–14

12 Sketch the curves of the given functions on the grids below.
Show any asymptotes and give the coordinates of any points of intersection with the axes.

a) y = ex, y = ex - 1 and y = ex - 1 b) y = ex, y = 3ex and y = e3x

yy

xx



13 Sketch each of the following curves. In each case write the equation of the asymptote
and the coordinates of the point where they cross the y-axis.

a) y = ex + 1 b) y = e-x + 1

yy

x x

 

14 Solve the following equations giving your answers to 3 s.f.

a) 4e5t = 30 c) et + 3 = 30

b) 5e4t = 30 d) et - 3 = 30

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7 Logarithmic and exponential functions

15 Match the correct equation to each of the graphs.

Equation Graph
y = ex - 1
y = ex + 1 c)
y = ex + 1
y = ex - 1 y

a) 6
5
y 4
3
6 2
1
5

4

3

2

1

–3 –2 –1–10 1 2 3 x –3 –2 –1–10 1 2 3 x

b) d)

yy

56

45

34

23

12

–3 –2 –1–10 123 x 1 x
–2 –1–10
compound –3 –2 1 23 formula r
general, continuous interest on given A = × t,
16 In an investment is by the Pe100
where $P is the amount invested for t years at a rate of r%, giving a final amount of $A.

a) To the nearest $, how much will an amount of $20 000 invested at a rate of 4% be worth in
5 years’ time?

b) How long would $20 000 need to be invested at 4% to be worth $30 000?
Give your answer to the nearest month.

44 Photocopying prohibited Cambridge IGCSE™ and O Level Additional Mathematics Workbook

Questions 15–17

17 A radioactive substance of mass 250 g is decaying so that the mass M left after t days
is given by the formula M = 250e-0.0035t.
a) On the axes below, sketch the graph of M against t.

M

t

b) How much, to the nearest gram, is left after one year (i.e. 365 days)?

c) What is the rate of decay after 365 days?

d) After how long, to the nearest day, will there be less than 10 g remaining?

Photocopying prohibited Cambridge IGCSE™ and O Level Additional Mathematics Workbook 45

8 Straight line graphs

1 A(2, 3), B(6, 4) and C(5, 0) are the vertices of a triangle.
a) Draw the triangle ABC on the axes provided.

y

7
6
5
4
3
2
1

–1–10 1 2 3 4 5 6 7 x



b) Show by calculation that ABC is an isosceles triangle and write down the two equal sides.

c) Find the coordinates of D, the midpoint of AC.

d) Prove that BD is perpendicular to AC.

e) Find the area of the triangle.

46 Photocopying prohibited Cambridge IGCSE™ and O Level Additional Mathematics Workbook

Questions 1–3

2 The graph shows a quadrilateral PQRS.

a) Prove, using suitable calculations, that PQRS is a trapezium.

y

8

7 R

6
5Q

4

3

2 S

1 1 2 3 4 5 6 7 8 9 10 x
P
–5 –4 –3 –2 –1–10



b) PQRT is a parallelogram. Find the coordinates of T.

c) Prove that PQRT is not a rhombus.

3 In each of parts (a) and (b) you are given the equation of a line and the coordinates of a point.
Find the equation of the line through the given point that is

(i) parallel to the given line (ii) perpendicular to the given line.
a) y = 3x - 2; (3,1) b) 2x + y = 2; (-1,-2)

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8 Straight line graphs

4 Find the equation of the perpendicular bisector of the line joining each pair of points.

a) (1, 4) and (5, 2) b) (-3, -4) and (2, 4)

5 A median of a triangle is a line joining one of the vertices to the midpoint of the opposite side.
In a triangle ABC, A is the point (0, 6), B is (4, 8) and C is the point (2, -2).

a) Sketch the triangle on the axes provided. b) Find the equations of the three medians of
the triangle.

y

8
7
6
5
4
3
2
1

–2 –1–10 1 2 3 4 5 x

–2


c) Show that the three medians are concurrent (i.e. all three intersect at the same point).

6 Match the equivalent relationships: c) log y = log x + b log a
a) log y = log b + x log a d) log y = log a + x log b
b) log y = log a + b log x
The relationship
Matches to p = qrn can be written
using logarithms as
y = axb p = log q + log rn and
y = xab so is equivalent to
y = abx log p = log q + n log r.
y = bax
Cambridge IGCSE™ and O Level Additional Mathematics Workbook
48 Photocopying prohibited

Questions 4–7

7 In this question k and a are constants. In each of the following cases:

i) write the equation with ln y as subject;

ii) given that ln y is on the vertical axis and the graph is a straight line,
state the quantity on the horizontal axis;

iii) state the gradient of the line and its intercept with the vertical axis.

a) y = axk c) y = kax

i) i)

ii) ii)

iii) iii)

b) y = kxa d) y = akx
i) i)

ii) ii)

iii) iii)

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