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Graded exercises in Advanced level mathematics

Graded exercises in pure mathematics

Edited by Barrie Hunt

PUBLISHED BY THE PRESS SYNDICATE OF THE UNIVERSITY OF CAMBRIDGE

The Pitt Building, Trumpington Street, Cambridge, United Kingdom

CAMBRIDGE UNIVERSITY PRESS

The Edinburgh Building, Cambridge CB2 2RU, UK 40 West 20th Street, New York, NY 10011-4211, USA 10 Stamford Road, Oakleigh, VIC 3166, Australia  Ruiz de Alarcon 13, 28014 Madrid, Spain Dock House, The Waterfront, Cape Town 8001, South Africa http://www.cambridge.org # Cambridge University Press 2001 This book is in copyright. Subject to statutory exception and to the provisions of relevant collective licensing agreements, no reproduction of any part may take place without the written 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

ACKNOWLEDGEMENTS

The authors gratefully acknowledge the following for permission to use questions from past papers: AQA Edexcel OCR AEB Cambridge London MEI NEAB O&C SMP WJEC Authors Assessment and Quali®cations Alliance Oxford, Cambridge and RSA Examinations Associated Examining Board The University of Cambridge Local Examinations Syndicate The University of London Examinations and Assessment Council The Oxford and Cambridge Schools Examination Board Northern Examination and Assessment Board The Oxford and Cambridge Schools Examination Board The Oxford and Cambridge Schools Examination Board Welsh Joint Education Committee Robin Bevan Bob Carter Andy Hall Barrie Hunt Lorna Lyons Lucy Norman Sarah Payne Caroline Petryszak Rachel Williams Barrie Hunt

Edited by

Contents

How to use this book 0 Background knowledge 0.1 Basic arithmetic ± highest common factor; lowest common multiple; fractions 0.2 Laws of indices 0.3 Similar ®gures 0.4 Basic algebra ± multiplying brackets; factorising quadratics; solution of simultaneous equations 0.5 Solving equations; changing the subject of a formula 0.6 The straight line y mx c; gradient and intercept 0.7 The distance between two points 0.8 Trigonometry ± right-angled triangles; sine and cosine rules 0.9 The cone and sphere 0.10 Properties of a circle Algebra 1.1 Surds; laws of indices 1.2 Arithmetic of polynomials; factor theorem; remainder theorem; graphs of y kxn (n 1 or an integer) 2 1.3 Modulus sign; linear and quadratic equations and inequalities (including graphical methods); sum and product of roots of quadratic equations; simultaneous equations ± one linear and one quadratic 1.4 Partial fractions 1.5 Complex numbers Coordinate geometry 2.1 Equation of a straight line in the forms y À y1 mx À x1 and ax by c 0; ®nding the equation of a linear graph; parallel and perpendicular lines; distance between two points in two and three dimensions; mid-point of two points; equation of a circle

1 3 3 4 5 7 8 9 11 11 14 16 20 20 25

1

34 40 48 57

2

57

iv

Contents

3

Vector geometry 3.1 Addition of vectors; length of vectors; scalar product; angle between two vectors 3.2 2D vectors; position vectors; ratio theorem; vector equation of a line; intersection of lines 3.3 3D vectors; equations of lines and planes; intersection of lines and planes Functions 4.1 De®nitions of functions involving formulae and domains; ranges of functions; graphical representation of functions; composition of functions; inverse functions 4.2 Algebraic and geometric properties of simple transformations including fx a, fx a, afx, fax; composition of transformations up to and including afbx c d Sequences 5.1 Inductive de®nition and formula for the nth term; recognition of periodicity, oscillation, convergence and divergence; formulae for Æik k 1; 2; 3 5.2 Arithmetic and geometric series; sum to in®nity of a convergent series n 5.3 n3; notation; binomial expansion of 1 xn , n P Z r 5.4 Binomial series Trigonometry 6.1 Radian measure; s r, A 1 r2 2 6.2 Sine, cosine and tangent functions; their reciprocals, inverses and graphs 6.3 Trigonometric identities including sin2 cos2 1 etc.; addition formulae; double angle formulae and R sin 6.4 Solution of trigonometric equations 6.5 Sum and product formulae Exponential and logarithmic functions 7.1 Exponential growth and decay; laws of functions logarithms; ex and ln x; solution of ax b 7.2 Reduction of laws to linear form Differentiation 8.1 Differentiation of polynomials, trigonometric, exponential and logarithmic functions; product and quotient rules; composite functions

64 64 73 81 91 91 99 111 111 120 127 133 139 139 148 156 163 171 177 177 183 196 196

4

5

6

7

8

Contents

v

8.2 8.3 8.4 8.5 9

Increasing and decreasing functions; rates of change; tangents and normals; maxima, minima and stationary points; points of in¯exion; optimisation problems Parametric curves, including the parabola, circle and ellipse; implicit differentiation; logarithmic differentiation Curve sketching Formation of simple differential equations; small angle approximations; Maclaurin's series for simple functions

202 212 220 229 238 238 247 255 262 272 282 282

Integration 9.1 Integration as the reverse of differentiation; integrals of 1 xn , ex , , sin x etc.; area under curve; de®nite integrals x 9.2 Integration by inspection, substitution, partial fractions and parts 9.3 Choice of method of integration 9.4 Problems involving differential equations; solution of simple differential equations of the form dy=dx fx; separation of variables 9.5 R Further applications of integration; volumes of revolution; P y dx lim y x Numerical methods 10.1 Absolute and relative errors;y % dy=dxx 10.2 Locating roots of equations by sign changes; simple iterative methods including Newton±Raphson, bisection method, xn gxnÀ1 ; failure of iterative methods; cobweb and staircase diagrams 10.3 Numerical integration; solution of differential equations using numerical methods Proof 11.1 Use of mathematical symbols and language ± A, @, D, if and only if, converses, necessary and suf®cient conditions; construction of mathematical arguments; proof by contradiction and disproof by counter-example and solutions Background knowledge Algebra Coordinate geometry Vector geometry Functions Sequences Trigonometry

10

292 302 313

11

313 321 321 325 341 344 352 364 372

Answers 0 1 2 3 4 5 6

vi

Contents

7 8 9 10 11

Exponential and logarithmic functions Differentiation Integration Numerical methods Proof

388 395 426 442 453

0

Background knowledge

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

a c ad bc bd b d a c ac  b d bd

0.1

1

Express as a product of prime factors: (a) 30 (e) 108 (b) 49 (f) 693 (c) 53 (g) 1144 (d) 84 (h) 14 553 (c) 30, 42 (f) 169, 234, 299

2

Find the highest common factor (HCF) of: (a) 6, 10 (d) 24, 40, 64 (g) 252, 378, 567 (b) 7, 14 (e) 42, 70, 182 (h) 51, 527, 1343 (c) 30, 42 (g) 4, 21, 22

26 39

3

Find the lowest common multiple (LCM) of: (a) 6, 10 (e) 5, 25 (b) 7, 14 (f) 5, 7, 11 (b) (f)

15 125

(d) 2, 3, 4 (h) 14, 18, 21

4

Express each fraction in its lowest terms, without using a calculator: (a) (e)

7 35 81 108

(c) (g)

(d) (h)

16 80

3a 12a

42a2 56a

22ab2 121b 7 8 64 2 (h) a a2 (d) 5 3 À 12 8

3

5

Complete: 3 (a) 4 24 7 (e) 4 20

(b)

4 5 20 2a (f) 3 9

4 7 21 a (g) b bx (c)

6

Simplify, without using a calculator: 3 2 2 1 4 2 (b) À (c) (a) 4 3 7 5 13 7

(d)

4

Background knowledge

7

3 7 2 1 (f) 5 À 3 (e) 1 2 4 8 3 9 Express as a single fraction: 3a 2a 2a a (a) (b) À 4 3 7 5 1 1 5 2 (e) (f) À 2 u v a a

1 3 (g) 2 7 4 3 2 a a 2 (g) p À q (c) (d)

2 2 (h) 3 2 5 3 3 2 a b 3 5 (h) À ab ac

8

Without using a calculator, simplify and express each fraction in its lowest terms: (a) 6 Â 2 3 (e) 3a 2 Â 7 5a (b) (f)

1 2

Â3 4

(c)

3 5

Â4 7 1 x

(d)

4a2 3 Â 11 2ab

(g) x Â

Â4 9 2 2 3 (h) x x x2

3 5

9

Without using a calculator, simplify and express each fraction in its lowest terms: (a) 6 Ä 2 3 (e) (b) (f)

77 78 20 91

3a 2a Ä 7 5

1 2

4 2 Ä 2 3ab 11a

78 79 ?

Ä3 4

(c)

3 5

6 Ä 25

(d) (h)

(g) x Ä

1 x

3 5

1 1 Ä x2 x

Ä4 9

10 Which is larger, 11 (a) The fraction

or

is written as 1 1. Find a. 7 a (ii) 1 À 11 À 11 À 1 . . . 1 À 1 2 3 4 n (i) 1 À 11 À 11 À 1 2 3 4

(b) Calculate:

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

remainders of 15 and 23 respectively.

0.2

Laws of indices

am  an amn am amÀn an am n amn

1

Simplify: (a) a3  a4 (d) 2a3  3a2 (b) a7  a6 (e) 5a2  a7 p4 p3 x12 x (c) a  a3 (f) 2 a3  6a4 3 12a7 4a2 12a5 8a3 2a2 b 6ab2

2

Simplify: x9 (a) 2 x

(b)

(c)

(d)

(e)

(f)

0.3 Similar figures

5

3

Simplify: (a) a5 3 (e) À2a2 4 (b) 2a4 (f) 3a2 b3 3 p (b) p x6 (f) 9a10 b4

2 2

(c) 5a3 2

(d) 5a3 2

4

Simplify: p (a) p x2 3 (e) Àx6 Expand: (a) 1 x

(c)

p a2 b2

(d)

p 4a2

5

(b) 3 À a

3 2

1 2 2 (c) x À 2 x

6

Simplify: x2 x5 (a) x (c) 3x2 5x2 À

3x3 x

3x8 2x4 x4 2 10x y 6xy2 À 8x2 y2 (d) 2xy (b)

0.3

1

Similar ®gures

Find the sides marked x and/or y in each of the following pairs of similar triangles. (a)

(b)

(c)

6

Background knowledge

(d)

(e)

(f)

(g)

2

OAB is the cross-section of a cone, 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?

0.4 Basic algebra

7

5

A solid metal cylinder of radius 6 cm and height 12 cm weighs 6 kg. A second cylinder is made from the same material and has radius 8 cm and height 16 cm. How much does this cylinder weigh? A liquid is poured into a hollow cone, which is placed with its vertex down. 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 poured in? Plot the graph to show how depth varies with volume.

6

0.4

Basic algebra ± multiplying brackets, factorising quadratics, solution of simultaneous equations

a bc d ac ad bc bd

1

Expand: (a) 34 a (d) a2a 3b (b) 62 À 3a (e) 3a5a À 2b (c) aa 3 3 (f) x 2 x (c) 2x 13x 5 (f) p 3q2p À 5q

2

Multiply out the brackets: (a) x 2x 5 (d) 5x À 25x 2 2 2 (g) x x

(b) x À 3x 4 (e) 3a 22 (h) 2x2 1x 3

3

Factorise: (a) 4x 8y (d) 2r2 2rh (b) x2 À 3x (e) ut 1 at2 2 (b) (e) (h) (k) (c) 5x2 2xy (f) 2x3 3x4 (c) (f) (i) (l) a2 À 6a 9 2a2 7a 3 p2 4pq À 12q2 10a2 31a À 14 16 À x2 x4

4

Factorise: (a) (d) (g) (j) x2 4x 3 x2 7x 10 6y2 À 7y À 5 15p2 À 34pq À 16q2 x2 2x x x2 2x À 3 p2 p À 30 p2 À 4q2 9x2 30xy 25y2 x2 3x 2 x1

5

Simplify: 3x 6 (a) 3 (a) x y 4 x À y À6

(b)

(c)

(d)

6

Solve the simultaneous equations: (b) x 2y 8 x 5y 17 (c) 2x 3y 2 x À 2y 8

8

Background knowledge

(d) 3x À 2y 1 À5x 4y 3 (g) 4x À 3y 0 6x 15y 13

7

(e) 2x 5y À14 3x 2y 1 (h) 2x 3y 4 0 5x À y À 7 0

(f) 5x À 3y 23 7x 4y À17

Multiply out the brackets: (a) x À 1x2 xp1 p (d) x 2x À 2 (b) a b3 (c) a b4

8

Simplify: (a) a b2 À a À b2 (b) x3 2x2 x x2 x (c) x4 À 13x2 36 x À 2x2 À 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

1

Solving equations; changing the subject of a formula

Solve the following equations. (a) (c) (e) (g) (i) (k) 2x 1 7 5x 2 3x À 5 3x 2 9x x2 81 16 x x 2 x 7x (b) (d) (f) (h) (j) 2 À 3x 8 6x 3 8 À 2x 42x À 7 35x 1 x2 À 25 0

(m) xx À 4 0 (o) 2x À 3x 43x 2 0

2

x3 27 0 4 (l) x À 0 x (n) x 3x À 7 0

Rearrange to make the given variable the subject of the formula: (a) (c) (e) (g) (i) Q CV C F 9 C 32 C 5 P 2` w ` v2 u2 2as a u a n À 1d d (b) (d) (f) (h) (j) C 2r r y mx c m S 1 na ` a 2 s ut 1 at2 a 2 s n f2a n À 1dg 2

d

0.6 The straight line

9

3

Rearrange to make the given variable the subject of the formula: (a) (c) (e) (g) (i) E mc2 c (b) V 4 r3 r 3 4 (d) y 2 x x p (f) y 2 x 3 x (h) A r2 À r2 1 (j) c r

V 1 r2 h r 3 I 1 mv2 À u2 2 s ` T 2 ` g 1 x y xÀa v

p a2 b2 a

4

In each case, show clearly how the second formula may be obtained from the ®rst. (a) I (b) (c) (d) (e) (f) iR ; Rr x2 y2 1; a2 b2 xÀ2 ; y x 3x 2 y ; 5Àx Er I ; Rr 1 1 1 ; R u v i À IR I q b a2 À x2 y a 2 x 1Ày 5y À 2 x y3 IR r EÀI Ru v uÀR r

5

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

0.6

The straight line y mx c; gradient and intercept

The line y mx c has gradient m, intercept c

1

Plot the graph of y 4x 2 for À3 x 3. Calculate the gradient of the line. Write down where it crosses the y-axis (the y-intercept).

10

Background knowledge

2

Complete the table. (a) (b) (c) (d) (e) (f) (g) (h) (i) (j) Equation y 5x À 2 y 1 À 3x y 1x 2 y À4 À 3x Gradient Intercept

2 6 7 1 2y 4x 1 5y 2x

5 À2 0

1 2

3 4

Sketch the following lines. (a) y 2x 5 (b) y 1 x 2 2 (b) (c) y Àx (d) y Àx 1 Write down the equation of each of the lines shown. (a)

(c)

(d)

(e)

0.8 Trigonometry

11

5 6

Find the equation of the line perpendicular to y 2x À 1 which passes through (0, 3). y x State the coordinates of the point where the line 1 crosses 4 6 (a) the x-axis, (b) the y-axis.

0.7

1

The distance between two points

(a) P and Q are two points with coordinates (2, 3) and (5, 7) respectively. By applying Pythagoras' theorem to triangle PQR, ®nd the distance PQ. (b) By drawing a suitable diagram, ®nd a formula for the distance PQ where P and Q have coordinates x1 ; y1 , x2 ; y2 respectively. Find the distance between the following pairs of points. (a) (1, 2), (6, 14) (d) (4, 2), (1, À3) (b) (3, 2), (6, 3) (c) (À1, 4), (2, 7) p 3) is equilateral.

2

3 4 5

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

Which of the points (6, 4), (À3, 6), (2, À4) is nearest to (1, 2)? Find the distance of the point Px; y from (i) O0; 0 (ii) R4; 3. If P is equidistant from O and R, ®nd the equation of the locus of P.

0.8

Trigonometry ± right-angled triangles; sine and cosine rules

In right-angled triangles: Pythagoras' theorem a2 b2 c2 opp a ; hyp c opp a tan A adj b sin A In all triangles: sine rule a b c sin A sin B sin C cos A adj b ; hyp c

cosine rule a2 b2 c2 À 2bc cos A

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 pBC, (ii) AB giving your (i) answer in the form a. (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)

14

Background knowledge

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

Volume of cone = 1 r2 h; Volume of sphere = 4 r3 3 3 Surface area of cone = r`; Surface area of sphere = 4r2

0.9 The cone and sphere

15

1

Find the volumes of the following solid objects, giving your answers as multiples of . (a) (b)

(c)

(d)

(e)

2

A child's toy is formed by attaching a cone to a hemisphere as shown. The radius of the hemisphere is 6 cm and the height of the toy is 14 cm. Find (a) its volume, (b) its surface area. The earth may be treated as a sphere of radius 6370 km. Find (a) its surface area, (b) its volume. 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 in the water level?

3

4

16

Background knowledge

5 6

A solid metal cube of side 4 cm is melted down and recast as a sphere. p Show that its radius is 3 48=. A gas balloon, in the shape of a sphere, is made from 1000 m2 of material. Estimate the volume of gas in the balloon. What assumptions have you made? A hollow sphere has internal diameter 10 cm and external diameter 12 cm. What is the volume of the material used to make the sphere? A bucket is in the shape of the frustrum of a cone. The radius of the base is 15 cm and the radius of the top is 20 cm. Find the volume of the bucket, given that its height is 30 cm.

7 8

0.10

Properties of a circle

Angle facts:

The angle in a semicircle is a right angle.

The perpendicular from the centre to a chord bisects the chord.

The radius is perpendicular to the tangent.

0.10 Properties of a circle

17

1

Find the value of x in each of the following. (a) (b)

(c)

2

(a) AB is a chord of a circle, radius 5 cm, at a distance of 3 cm from the centre O. Find (i) the length AB, (ii) the angle .

(b) Find the angle subtended by the chord AB in the diagram.

18

Background knowledge

(c) Find the area of triangle AOB and hence ®nd the area of the minor segment 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) OP1 P2 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 the circle with centre O2 .

0.10 Properties of a circle

19

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

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 Â 108 km. The diameter of the sun is 1:39 Â 106 km. Find the angle subtended by the sun from a point on the Earth. What assumptions have you made?

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