Graded exercises in

Advanced level mathematics

Graded exercisesin pure mathematics

Edited by Barrie Hunt

PUBL I SHED BY THE PRE S S S YND I CATE OF THE UN I VER S I TY OF CAMBR IDGE

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# Cambridge University Press 2001

This book is in copyright. Subject to statutory exception and to the provisions of relevantcollective licensing agreements, no reproduction of any part may take place without thewritten permission of Cambridge University Press.

First published 2001

Printed in the United Kingdom at the University Press, Cambridge

Typeface Times System 3B2

A catalogue record for this book is available from the British Library

ISBN 0 521 63753 8 paperback

ACKNOWLEDGEMENT S

The authors gratefully acknowledge the following for permission to use questions from pastpapers:

AQA Assessment and Quali®cations AllianceEdexcelOCR Oxford, Cambridge and RSA Examinations

AEB Associated Examining BoardCambridge The University of Cambridge Local Examinations SyndicateLondon The University of London Examinations and Assessment CouncilMEI The Oxford and Cambridge Schools Examination BoardNEAB Northern Examination and Assessment BoardO&C The Oxford and Cambridge Schools Examination BoardSMP The Oxford and Cambridge Schools Examination BoardWJEC Welsh Joint Education Committee

Authors Robin BevanBob CarterAndy HallBarrie HuntLorna LyonsLucy NormanSarah PayneCaroline PetryszakRachel Williams

Edited by Barrie Hunt

Contents

How to use this book 1

0 Background knowledge 30.1 Basic arithmetic ± highest common factor; lowest common

multiple; fractions 30.2 Laws of indices 40.3 Similar ®gures 50.4 Basic algebra ± multiplying brackets; factorising quadratics;

solution of simultaneous equations 70.5 Solving equations; changing the subject of a formula 80.6 The straight line y � mx� c; gradient and intercept 90.7 The distance between two points 110.8 Trigonometry ± right-angled triangles; sine and cosine rules 110.9 The cone and sphere 140.10 Properties of a circle 16

1 Algebra 201.1 Surds; laws of indices 201.2 Arithmetic of polynomials; factor theorem; remainder theorem;

graphs of y � kxn(n � 12 or an integer) 25

1.3 Modulus sign; linear and quadratic equations and inequalities(including graphical methods); sum and product of roots ofquadratic equations; simultaneous equations ± one linearand one quadratic 34

1.4 Partial fractions 401.5 Complex numbers 48

2 Coordinate geometry 572.1 Equation of a straight line in the forms yÿ y1 � m�xÿ x1�

and ax� by� c � 0; ®nding the equation of a linear graph;parallel and perpendicular lines; distance between two pointsin two and three dimensions; mid-point of two points;equation of a circle 57

3 Vector geometry 643.1 Addition of vectors; length of vectors; scalar product; angle

between two vectors 643.2 2D vectors; position vectors; ratio theorem; vector equation

of a line; intersection of lines 733.3 3D vectors; equations of lines and planes; intersection of lines

and planes 81

4 Functions 914.1 De®nitions of functions involving formulae and domains;

ranges of functions; graphical representation of functions;composition of functions; inverse functions 91

4.2 Algebraic and geometric properties of simple transformationsincluding f�x� � a, f�x� a�, af�x�, f�ax�; composition oftransformations up to and including af�bx� c� � d 99

5 Sequences 1115.1 Inductive de®nition and formula for the nth term; recognition

of periodicity, oscillation, convergence and divergence;formulae for �ik�k � 1; 2; 3� 111

5.2 Arithmetic and geometric series; sum to in®nity of aconvergent series 120

5.3 n!;nr

� �notation; binomial expansion of �1� x�n, n 2 Z� 127

5.4 Binomial series 133

6 Trigonometry 1396.1 Radian measure; s � r�, A � 1

2 r2� 139

6.2 Sine, cosine and tangent functions; their reciprocals, inversesand graphs 148

6.3 Trigonometric identities including sin2 � � cos2 � � 1 etc.;addition formulae; double angle formulae and R sin�� � �� 156

6.4 Solution of trigonometric equations 1636.5 Sum and product formulae 171

7 Exponential and logarithmic functions 1777.1 Exponential growth and decay; laws of functions logarithms;

ex and ln x; solution of ax � b 1777.2 Reduction of laws to linear form 183

8 Differentiation 1968.1 Differentiation of polynomials, trigonometric, exponential

and logarithmic functions; product and quotient rules;composite functions 196

iv Contents

8.2 Increasing and decreasing functions; rates of change; tangentsand normals; maxima, minima and stationary points; pointsof in¯exion; optimisation problems 202

8.3 Parametric curves, including the parabola, circle and ellipse;implicit differentiation; logarithmic differentiation 212

8.4 Curve sketching 2208.5 Formation of simple differential equations; small angle

approximations; Maclaurin's series for simple functions 229

9 Integration 2389.1 Integration as the reverse of differentiation; integrals of

xn, ex,1

x, sin x etc.; area under curve; de®nite integrals 238

9.2 Integration by inspection, substitution, partial fractions andparts 247

9.3 Choice of method of integration 2559.4 Problems involving differential equations; solution of

simple differential equations of the form dy=dx � f�x�;separation of variables 262

9.5 Further applications of integration; volumes of revolution;Ry dx � lim

Py �x 272

10 Numerical methods 28210.1 Absolute and relative errors;�y � �dy=dx��x 28210.2 Locating roots of equations by sign changes; simple iterative

methods including Newton±Raphson, bisection method,xn � g�xnÿ1�; failure of iterative methods; cobweb andstaircase diagrams 292

10.3 Numerical integration; solution of differential equationsusing numerical methods 302

11 Proof 31311.1 Use of mathematical symbols and language ± ), (, ,, if

and only if, converses, necessary and suf®cient conditions;construction of mathematical arguments; proof bycontradiction and disproof by counter-example 313

Answers and solutions 3210 Background knowledge 3211 Algebra 3252 Coordinate geometry 3413 Vector geometry 3444 Functions 3525 Sequences 3646 Trigonometry 372

Contents v

7 Exponential and logarithmic functions 3888 Differentiation 3959 Integration 426

10 Numerical methods 44211 Proof 453

vi Contents

0Background knowledge

0.1 Basic arithmetic ± highest common factor;lowest common multiple; fractions

1 Express as a product of prime factors:

(a) 30 (b) 49 (c) 53 (d) 84

(e) 108 (f) 693 (g) 1144 (h) 14 553

2 Find the highest common factor (HCF) of:

(a) 6, 10 (b) 7, 14 (c) 30, 42

(d) 24, 40, 64 (e) 42, 70, 182 (f) 169, 234, 299

(g) 252, 378, 567 (h) 51, 527, 1343

3 Find the lowest common multiple (LCM) of:

(a) 6, 10 (b) 7, 14 (c) 30, 42 (d) 2, 3, 4

(e) 5, 25 (f) 5, 7, 11 (g) 4, 21, 22 (h) 14, 18, 21

4 Express each fraction in its lowest terms, without using a calculator:

(a) 735 (b) 15

125 (c) 2639 (d) 16

80

(e) 81108 (f)

3a

12a(g)

42a2

56a(h)

22ab2

121b

5 Complete:

(a)3

4�

24(b)

4

5�

20(c)

4

7�

21(d)

7

8�

64

(e)7

4�

20(f)

2a

3�

9(g)

a

b�

bx(h)

2

a�

a2

6 Simplify, without using a calculator:

(a)3

4� 2

3(b)

2

7ÿ 1

5(c)

4

13� 2

7(d)

5

12ÿ 3

8

3

a

b� c

d� ad � bc

bd

a

b� c

d� ac

bd

(e) 13

4� 2

7

8(f) 5

2

3ÿ 3

1

9(g) 2

1

7� 3

4(h) 3

2

5� 2

2

3

7 Express as a single fraction:

(a)3a

4� 2a

3(b)

2a

7ÿ a

5(c)

3

a� 2

a(d)

3

a� 2

b

(e)1

u� 1

v(f)

5

aÿ 2

a2(g) pÿ 2

q(h)

3

abÿ 5

ac

8 Without using a calculator, simplify and express each fraction in itslowest terms:

(a) 6� 23 (b) 1

2 � 34 (c) 3

5 � 47 (d) 3

5 � 49

(e)3a

7� 2

5a(f)

4a2

11� 3

2ab(g) x� 1

x(h) x2

3

x� 2

x2

� �9 Without using a calculator, simplify and express each fraction in its

lowest terms:

(a) 6� 23 (b) 1

2 � 34 (c) 3

5 � 625 (d) 3

5 � 49

(e)3a

7� 2a

5(f)

4

11a2� 2

3ab(g) x� 1

x(h)

1

x2� 1

x

10 Which is larger, 7778 or

7879 ?

11 (a) The fraction 2091 is written as 1

7 � 1a. Find a.

(b) Calculate: (i) �1ÿ 12��1ÿ 1

3��1ÿ 14�

(ii) �1ÿ 12��1ÿ 1

3��1ÿ 14� . . . �1ÿ 1

n�12 Find the greatest number which, when divided into 1407 and 2140, leaves

remainders of 15 and 23 respectively.

0.2 Laws of indices

1 Simplify:

(a) a3 � a4 (b) a7 � a6 (c) a� a3

(d) 2a3 � 3a2 (e) 5a2 � a7 (f) 23 a

3 � 6a4

2 Simplify:

(a)x9

x2(b)

p4

p3(c)

x12

x(d)

12a7

4a2(e)

12a5

8a3(f)

2a2b

6ab2

4 Background knowledge

am � an � am�n am

an� amÿn �am�n � amn

0.3 Similar figures 5

3 Simplify:

(a) �a5�3 (b) �2a�4 (c) �5a3�2 (d) 5�a3�2(e) �ÿ2a2�4 (f) �3a2b3�3

4 Simplify:

(a)�����x2

p(b)

�����x6

p(c)

���������a2b2

p(d)

�������4a2

p

(e) 3���������ÿx6

p(f)

�������������9a10b4

p

5 Expand:

(a) �1� x2�2 (b) �3ÿ a3�2 (c) x2 ÿ 1

x2

� �2

6 Simplify:

(a)x2 � x5

x(b)

3x8 � 2x4

x4

(c) 3x2 � �5x�2 ÿ 3x3

x(d)

10x2y� 6xy2 ÿ 8x2y2

2xy

0.3 Similar ®gures

1 Find the sides marked x and/or y in each of the following pairs ofsimilar triangles.

(a)

(b)

(c)

(d)

(e) (f)

(g)

2 OAB is the cross-section of acone, radius r, height h.

Express y in terms of r, h and x.

3 The coordinates of Q are (4, 0).

What are the coordinates of P?

4 A sphere has radius 8 cm and a second sphere has radius 12 cm.What is the ratio of their (a) areas, (b) volumes?

6 Background knowledge

5 A solid metal cylinder of radius 6 cm and height 12 cm weighs 6 kg. Asecond cylinder is made from the same material and has radius 8 cm andheight 16 cm. How much does this cylinder weigh?

6 A liquid is poured into a hollow cone, which is placed with its vertexdown. When 400 cm3 has been poured in, the depth of water is 100 cm.What is the depth of water after (a) 1000 cm3, (b) x cm3 has been pouredin? Plot the graph to show how depth varies with volume.

0.4 Basic algebra ± multiplying brackets, factorisingquadratics, solution of simultaneous equations

1 Expand:

(a) 3�4� a� (b) 6�2ÿ 3a� (c) a�a� 3�(d) a�2a� 3b� (e) 3a�5aÿ 2b� (f) x 2� 3

x

� �2 Multiply out the brackets:

(a) �x� 2��x� 5� (b) �xÿ 3��x� 4� (c) �2x� 1��3x� 5�(d) �5xÿ 2��5x� 2� (e) �3a� 2�2 (f) �p� 3q��2pÿ 5q�

(g) x� 2

x

� �2

(h) �2x2 � 1��x� 3�

3 Factorise:

(a) 4x� 8y (b) x2 ÿ 3x (c) 5x2 � 2xy

(d) 2�r2 � 2�rh (e) ut� 12 at

2 (f) 2x3 � 3x4

4 Factorise:

(a) x2 � 4x� 3 (b) x2 � 2xÿ 3 (c) a2 ÿ 6a� 9

(d) x2 � 7x� 10 (e) p2 � pÿ 30 (f) 2a2 � 7a� 3

(g) 6y2 ÿ 7yÿ 5 (h) p2 ÿ 4q2 (i) p2 � 4pqÿ 12q2

(j) 15p2 ÿ 34pqÿ 16q2 (k) 9x2 � 30xy� 25y2 (l) 10a2 � 31aÿ 14

5 Simplify:

(a)3x� 6

3(b)

x2 � 2x

x(c)

x2 � 3x� 2

x� 1(d)

16ÿ x2

x� 4

6 Solve the simultaneous equations:

(a) x� y � 4

xÿ y � ÿ6

(b) x� 2y � 8

x� 5y � 17

(c) 2x� 3y � 2

xÿ 2y � 8

0.4 Basic algebra 7

�a� b��c� d� � ac� ad � bc� bd

(d) 3xÿ 2y � 1

ÿ5x� 4y � 3

(e) 2x� 5y � ÿ14

3x� 2y � 1

(f) 5xÿ 3y � 23

7x� 4y � ÿ17

(g) 4xÿ 3y � 0

6x� 15y � 13

(h) 2x� 3y� 4 � 0

5xÿ yÿ 7 � 0

7 Multiply out the brackets:

(a) �xÿ 1��x2 � x� 1� (b) �a� b�3 (c) �a� b�4(d) �x� ���

2p ��xÿ ���

2p �

8 Simplify:

(a) �a� b�2 ÿ �aÿ b�2 (b)x3 � 2x2 � x

x2 � x(c)

x4 ÿ 13x2 � 36

�xÿ 2��x2 ÿ 9�9 Solve the pairs of simultaneous equations below, explaining your results

graphically.

(a) 2x� 3y � 8

6x� 9y � 12

(b) 2x� 3y � 8

6x� 9y � 24

0.5 Solving equations; changing the subject of a formula

1 Solve the following equations.

(a) 2x� 1 � 7 (b) 2ÿ 3x � 8

(c) 5x� 2 � 3xÿ 5 (d) 6x� 3 � 8ÿ 2x

(e) 3�x� 2� � 9x (f) 4�2xÿ 7� � 3�5x� 1�(g) x2 � 81 (h) x2 ÿ 25 � 0

(i) x � 16

x(j) x3 � 27 � 0

(k) x2 � 7x (l) xÿ 4

x� 0

(m) x�xÿ 4� � 0 (n) �x� 3��xÿ 7� � 0

(o) �2xÿ 3��x� 4��3x� 2� � 0

2 Rearrange to make the given variable the subject of the formula:

(a) Q � CV �C� (b) C � 2�r �r�(c) F � 9

5C � 32 �C� (d) y � mx� c �m�(e) P � 2�`� w� �`� (f) S � 1

2 n�a� `� �a�(g) v2 � u2 � 2as �a� (h) s � ut� 1

2 at2 �a�

(i) u � a� �nÿ 1�d �d� (j) s � n2 f2a� �nÿ 1�dg �d�

8 Background knowledge

3 Rearrange to make the given variable the subject of the formula:

(a) E � mc2 �c� (b) V � 43�r

3 �r�(c) V � 1

3�r2h �r� (d) y � 4

x2�x�

(e) I � 12m�v2 ÿ u2� �v� (f) y � 2

���x

p � 3 �x�

(g) T � 2�

���̀g

s�`� (h) A � ��r2 ÿ r21� �r�

(i) y � 1

xÿ a�x� (j) c �

���������������a2 � b2

p�a�

4 In each case, show clearly how the second formula may be obtained fromthe ®rst.

(a) I � iR

R� r; r � �i ÿ I�R

I

(b)x2

a2� y2

b2� 1; y � b

a

�������������������a2 ÿ x2�

q(c) y � xÿ 2

x; x � 2

1ÿ y

(d) y � 3x� 2

5ÿ x; x � 5yÿ 2

y� 3

(e) I � Er

R� r; r � IR

E ÿ I

(f)1

R� 1

u� 1

v; v � Ru

uÿ R

5 The surface area, S, of a cylinder is given by S � 2�r2 � 2�rh.Its volume, V, is given by V � �r2h. Express V in terms of S and r only.

0.6 The straight line y � mx � c; gradient and intercept

1 Plot the graph of y � 4x� 2 for ÿ3 � x � 3. Calculate the gradient ofthe line.Write down where it crosses the y-axis (the y-intercept).

0.6 The straight line 9

The line y � mx� c has gradient m, intercept c

2 Complete the table.

3 Sketch the following lines.

(a) y � 2x� 5 (b) y � 12x� 2 (c) y � ÿx (d) y � ÿx� 1

4 Write down the equation of each of the lines shown.

(a) (b)

(c) (d)

(e)

10 Background knowledge

Equation Gradient Intercept

y � 5xÿ 2

y � 1ÿ 3x

y � 12x

y � ÿ4ÿ 3x

2 5

6 ÿ2

7 12

1 0

2y � 4x� 1

5y � 2x

(a)

(b)

(c)

(d)

(e)

(f)

(g)

(h)

(i)

(j)

5 Find the equation of the line perpendicular to y � 2xÿ 1 which passesthrough (0, 3).

6 State the coordinates of the point where the liney

4� x

6� 1 crosses

(a) the x-axis, (b) the y-axis.

0.7 The distance between two points

1 (a) P and Q are two points withcoordinates (2, 3) and (5, 7)respectively. By applyingPythagoras' theorem to trianglePQR, ®nd the distance PQ.

(b) By drawing a suitable diagram,®nd a formula for the distancePQ where P and Q havecoordinates �x1; y1�, �x2; y2�respectively.

2 Find the distance between the following pairs of points.

(a) (1, 2), (6, 14) (b) (3, 2), (6, 3) (c) (ÿ1, 4), (2, 7)

(d) (4, 2), (1, ÿ3)

3 Show that the triangle with vertices at (1, 0), (3, 0), (2,���3

p) is equilateral.

4 Which of the points (6, 4), (ÿ3, 6), (2, ÿ4) is nearest to (1, 2)?

5 Find the distance of the point P�x; y� from (i) O�0; 0� (ii) R�4; 3�.If P is equidistant from O and R, ®nd the equation of the locus of P.

0.8 Trigonometry ± right-angled triangles; sine andcosine rules

0.8 Trigonometry 11

In right-angled triangles:

Pythagoras' theorem a2 � b2 � c2

sinA � opp

hyp� a

c; cosA � adj

hyp� b

c;

tanA � opp

adj� a

b

In all triangles:

sine rulea

sinA� b

sinB� c

sinC

cosine rule a2 � b2 � c2 ÿ 2bc cosA

12 Background knowledge

1 Find the angles marked x.

(a) (b) (c)

(d)

2 Find the sides marked x.

(a) (b)

(c) (d)

(e) (f)

(g) (h)

0.8 Trigonometry 13

3 (a) Find the lengths of (i) BC, (ii) AB giving youranswer in the form

���a

p.

(b) Write down exact values for(i) sin 458, (ii) cos 458, (iii) tan 458.

4 Use the sine rule to ®nd the value of x.

(a) (b)

(c) (d)

(e) (f)

5 Use the cosine rule to ®nd the value of x.

(a) (b)

(c) (d)

(e) (f)

6 Use appropriate methods to ®nd all sides and angles for:

(a) (b)

0.9 The cone and sphere

14 Background knowledge

Volume of cone = 13�r

2h; Volume of sphere = 43�r

3

Surface area of cone = �r`; Surface area of sphere = 4�r2

1 Find the volumes of the following solid objects, giving your answers asmultiples of �.

(a) (b)

(c) (d)

(e)

2 A child's toy is formed by attaching a cone to ahemisphere as shown. The radius of the hemisphere is6 cm and the height of the toy is 14 cm. Find (a) itsvolume, (b) its surface area.

3 The earth may be treated as a sphere of radius 6370 km. Find (a) itssurface area, (b) its volume.

4 Twelve balls, each of radius 3 cm, are immersed in a cylinder of water,radius 10 cm, so that they are each fully submerged. What is the rise inthe water level?

0.9 The cone and sphere 15

5 A solid metal cube of side 4 cm is melted down and recast as a sphere.Show that its radius is

����������48=�3

p.

6 A gas balloon, in the shape of a sphere, is made from 1000m2 ofmaterial. Estimate the volume of gas in the balloon.What assumptions have you made?

7 A hollow sphere has internal diameter 10 cm and external diameter12 cm. What is the volume of the material used to make the sphere?

8 A bucket is in the shape of thefrustrum of a cone. The radius ofthe base is 15 cm and the radiusof the top is 20 cm. Find thevolume of the bucket, given thatits height is 30 cm.

0.10 Properties of a circle

16 Background knowledge

Angle facts:

The angle in a semi- The perpendicular The radius iscircle is a right angle. from the centre to perpendicular

a chord bisects the to the tangent.chord.

1 Find the value of x in each of the following.

(a) (b)

(c)

2 (a) AB is a chord of a circle, radius5 cm, at a distance of 3 cm from thecentre O. Find (i) the length AB,(ii) the angle �.

(b) Find the angle � subtended by the chordAB in the diagram.

0.10 Properties of a circle 17

(c) Find the area of triangle AOB andhence ®nd the area of the minorsegment cut off by AB.

3 (a) AP and BP are tangents to the circle with centre O and radius 5 cm.OP � 13 cm. Find (i) AP, (ii) �.

(b) OP1P2 is a tangent to two circles with centres O1, O2. OP1 � 12 cm.The radius of the circle with centre O1 is 5 cm. Find the radius of thecircle with centre O2.

18 Background knowledge

(c) In the diagram, OA is parallel to PQ. Find the angle QPR in termsof �.

4 Two circles, radii 3 cm and 5 cm, have centres P, Q respectively,PQ � 7 cm. If the circles intersect at A and B, ®nd the length AB.

5 The distance from the Earth to the sun is 1:50� 108 km. The diameter ofthe sun is 1:39� 106 km.Find the angle subtended by the sun from a point on the Earth.What assumptions have you made?

0.10 Properties of a circle 19