`Graded exercises in Advanced level mathematicsGraded exercises in pure mathematicsEdited by Barrie HuntPUBLISHED BY THE PRESS SYNDICATE OF THE UNIVERSITY OF CAMBRIDGEThe Pitt Building, Trumpington Street, Cambridge, United KingdomCAMBRIDGE UNIVERSITY PRESSThe Edinburgh Building, Cambridge CB2 2RU, UK 40 West 20th Street, New York, NY 10011-4211, USA 10 Stamford Road, Oakleigh, VIC 3166, Australia Â 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 paperbackACKNOWLEDGEMENTSThe authors gratefully acknowledge the following for permission to use questions from past papers: AQA Edexcel OCR AEB Cambridge London MEI NEAB O&amp;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 HuntEdited byContentsHow 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 circle1 3 3 4 5 7 8 9 11 11 14 16 20 20 25134 40 48 57257ivContents3Vector 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 functions64 64 73 81 91 91 99 111 111 120 127 133 139 139 148 156 163 171 177 177 183 196 19645678Contentsv8.2 8.3 8.4 8.5 9Increasing 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 functions202 212 220 229 238 238 247 255 262 272 282 282Integration 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 Trigonometry10292 302 31311313 321 321 325 341 344 352 364 372Answers 0 1 2 3 4 5 6viContents7 8 9 10 11Exponential and logarithmic functions Differentiation Integration Numerical methods Proof388 395 426 442 4530Background knowledgeBasic arithmetic ± highest common factor; lowest common multiple; fractionsa c ad  bc   bd b d a c ac Â  b d bd0.11Express 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, 2992Find 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, 2226 393Find 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, 214Express each fraction in its lowest terms, without using a calculator: (a) (e)7 35 81 108(c) (g)(d) (h)16 803a 12a42a2 56a22ab2 121b 7  8 64 2  (h) a a2 (d) 5 3 À 12 835Complete: 3  (a) 4 24 7  (e) 4 20(b)4  5 20 2a  (f) 3 94  7 21 a  (g) b bx (c)6Simplify, without using a calculator: 3 2 2 1 4 2 (b) À (c)  (a)  4 3 7 5 13 7(d)4Background knowledge73 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 a1 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 ac8Without 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 2Â3 4(c)3 5Â4 7 1 x(d)4a2 3 Â 11 2ab(g) x ÂÂ4 9   2 2 3 (h) x  x x23 59Without using a calculator, simplify and express each fraction in its lowest terms: (a) 6 Ä 2 3 (e) (b) (f)77 78 20 913a 2a Ä 7 51 24 2 Ä 2 3ab 11a78 79 ?Ä3 4(c)3 56 Ä 25(d) (h)(g) x Ä1 x3 51 1 Ä x2 xÄ4 910 Which is larger, 11 (a) The fractionoris 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, leavesremainders of 15 and 23 respectively.0.2Laws of indicesam Â an  amn am  amÀn an am n  amn1Simplify: (a) a3 Â a4 (d) 2a3 Â 3a2 (b) a7 Â a6 (e) 5a2 Â a7 p4 p3 x12 x (c) a Â a3 (f) 2 a3 Â 6a4 3 12a7 4a2 12a5 8a3 2a2 b 6ab22Simplify: x9 (a) 2 x(b)(c)(d)(e)(f)0.3 Similar figures53Simplify: (a) a5 3 (e) À2a2 4 (b) 2a4 (f) 3a2 b3 3 p (b) p x6 (f) 9a10 b42 2(c) 5a3 2(d) 5a3 24Simplify: p (a) p x2 3 (e) Àx6 Expand: (a) 1  x (c)p a2 b2(d)p 4a25(b) 3 À a 3 2  1 2 2 (c) x À 2 x6Simplify: x2  x5 (a) x (c) 3x2  5x2 À3x3 x3x8  2x4 x4 2 10x y  6xy2 À 8x2 y2 (d) 2xy (b)0.31Similar ®guresFind the sides marked x and/or y in each of the following pairs of similar triangles. (a)(b)(c)6Background knowledge(d)(e)(f)(g)2OAB is the cross-section of a cone, radius r, height h. Express y in terms of r, h and x.3The coordinates of Q are (4, 0). What are the coordinates of P?4A 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 algebra75A 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.60.4Basic algebra ± multiplying brackets, factorising quadratics, solution of simultaneous equationsa  bc  d  ac  ad  bc  bd1Expand: (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 5q2Multiply 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  33Factorise: (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 x44Factorise: (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 x15Simplify: 3x  6 (a) 3 (a) x  y  4 x À y  À6(b)(c)(d)6Solve the simultaneous equations: (b) x  2y  8 x  5y  17 (c) 2x  3y  2 x À 2y  88Background knowledge(d) 3x À 2y  1 À5x  4y  3 (g) 4x À 3y  0 6x  15y  137(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  À17Multiply out the brackets: (a) x À 1x2  xp1  p (d) x  2x À 2 (b) a  b3 (c) a  b48Simplify: (a) a  b2 À a À b2 (b) x3  2x2  x x2  x (c) x4 À 13x2  36 x À 2x2 À 99Solve the pairs of simultaneous equations below, explaining your results graphically. (a) 2x  3y  8 6x  9y  12 (b) 2x  3y  8 6x  9y  240.51Solving equations; changing the subject of a formulaSolve 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  02x3  27  0 4 (l) x À  0 x (n) x  3x À 7  0Rearrange 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 2d0.6 The straight line93Rearrange 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 rV  1 r2 h r 3 I  1 mv2 À u2  2 s ` T  2 ` g 1 x y xÀa vp a2  b2 a4In 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 r5The 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.0.6The straight line y  mx  c; gradient and interceptThe line y  mx  c has gradient m, intercept c1Plot 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).10Background knowledge2Complete 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 Intercept2 6 7 1 2y  4x  1 5y  2x5 À2 01 23 4Sketch 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 Trigonometry115 6Find 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.0.71The 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.23 4 5Show 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.0.8Trigonometry ± right-angled triangles; sine and cosine rulesIn 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 ccosine rule a2  b2  c2 À 2bc cos A12Background knowledge1Find the angles marked x. (a) (b) (c)(d)2Find the sides marked x. (a) (b)(c)(d)(e)(f)(g)(h)0.8 Trigonometry133(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 458.4Use the sine rule to ®nd the value of x. (a) (b)(c)(d)(e)(f)14Background knowledge5Use the cosine rule to ®nd the value of x. (a) (b)(c)(d)(e)(f)6Use appropriate methods to ®nd all sides and angles for: (a) (b)0.9The cone and sphereVolume 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 sphere151Find the volumes of the following solid objects, giving your answers as multiples of . (a) (b)(c)(d)(e)2A 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?3416Background knowledge5 6A 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 80.10Properties of a circleAngle 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 circle171Find 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.18Background 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 circle19(c) In the diagram, OA is parallel to PQ. Find the angle QPR in terms of .4Two 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.5The 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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