Cambridge International Examinations Cambridge Ordinary Level

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1 Cambridge International Examinations Cambridge Ordinary Level * * ADDITIONAL MATHEMATICS 4037/11 Paper 1 May/June hours Candidates answer on the Question Paper. No Additional Materials are required. READ THESE INSTRUCTIONS FIRST Write your Centre number, candidate number and name on all the work you hand in. Write in dark blue or black pen. You may use an HB pencil for any diagrams or graphs. Do not use staples, paper clips, glue or correction fluid. DO NOT WRITE IN ANY BARCODES. Answer all the questions. Give non-exact numerical answers correct to 3 significant figures, or 1 decimal place in the case of angles in degrees, unless a different level of accuracy is specified in the question. The use of an electronic calculator is expected, where appropriate. You are reminded of the need for clear presentation in your answers. At the end of the examination, fasten all your work securely together. The number of marks is given in brackets [ ] at the end of each question or part question. The total number of marks for this paper is 80. This document consists of 15 printed pages and 1 blank page. DC (KN/SW) /1 [Turn over

2 2 Mathematical Formulae 1. ALGEBRA Quadratic Equation For the equation ax 2 + bx + c = 0, b! b 4ac x = - - 2a 2 Binomial Theorem (a + b) n = a n + ( n 1 ) an 1 b + ( n 2 ) an 2 b ( n r ) an r b r + + b n, where n is a positive integer and ( n r ) = n! (n r)!r! 2. TRIGONOMETRY Identities sin 2 A + cos 2 A = 1 sec 2 A = 1 + tan 2 A cosec 2 A = 1 + cot 2 A Formulae for ABC a sin A = b sin B = c sin C a 2 = b 2 + c 2 2bc cos A = 1 bc sin A /11/M/J/15

3 3 1 (i) State the period of sin 2 x. [1] (ii) State the amplitude of cos 3x. [1] (iii) On the axes below, sketch the graph of (a) (b) y = sin 2x for 0 G x G 180, [1] y = cos 3x for 0 G x G 180. [2] y 4 2 O x 2 4 (iv) State the number of solutions of sin 2x - 2 cos 3x = 1 for 0 G x G 180. [1] 4037/11/M/J/15 [Turn over

4 4 2 Do not use a calculator in this question. A i rad ^ h cm B ^ hcm C The diagram shows the triangle ABC where angle B is a right angle, AB = ^ h cm, BC = ^ h cm and angle BAC = i radians. Showing all your working, find (i) tan i in the form a + b 2, where a and b are integers, [2] (ii) sec 2 i in the form c + d 2, where c and d are integers. [3] 4037/11/M/J/15

5 5 3 (i) Find the first 4 terms in the expansion of ^2 + x h in ascending powers of x. [3] 2 6 (ii) Find the term independent of x in the expansion of ^2 + x h c1-2m. [3] x 4037/11/M/J/15 [Turn over

6 6 4 J2-4N (a) Given that the matrix X = K O, find X 2 in terms of the constant k. L k 0 P [2] J Ja 1N 5 1N K O - (b) Given that the matrix A = K O and the matrix A 1 = L b 5 2 1, find the value of each of the P K - O L 3 3 integers a and b. P [3] 4037/11/M/J/15

7 7 2 5 The curve y = xy + x - 4 intersects the line y = 3x - 1 at the points A and B. Find the equation of the perpendicular bisector of the line AB. [8] 4037/11/M/J/15 [Turn over

8 The polynomial f^xh = ax - 15x + bx - 2 has a factor of 2x - 1 and a remainder of 5 when divided by x - 1. (i) Show that b = 8 and find the value of a. [4] (ii) Using the values of a and b from part (i), express f^xh in the form ^2x - 1h g^xh, where g^xh is a quadratic factor to be found. [2] (iii) Show that the equation f ^xh = 0 has only one real root. [2] 4037/11/M/J/15

9 9 7 The point A, where x = 0, lies on the curve x-axis at the point B. 2 ln^4x + 3h y =. The normal to the curve at A meets the x - 1 (i) Find the equation of this normal. [7] (ii) Find the area of the triangle AOB, where O is the origin. [2] 4037/11/M/J/15 [Turn over

10 e 2x 10 8 It is given that f^xh = 3 for x H 0, 2 g^xh = ^x + 2h + 5 for x H 0. (i) Write down the range of f and of g. [2] (ii) Find g - 1, stating its domain. [3] (iii) Find the exact solution of gf ^xh = 41. [4] 4037/11/M/J/15

11 11 (iv) Evaluate f l^ln 4h. [2] 4037/11/M/J/15 [Turn over

12 12 9 y D y = 3x + 10 A B O C y = x 3 5x 2 + 3x + 10 x 3 2 The diagram shows parts of the line y = 3x + 10 and the curve y = x - 5x + 3x The line and the curve both pass through the point A on the y-axis. The curve has a maximum at the point B and a minimum at the point C. The line through C, parallel to the y-axis, intersects the line y = 3x + 10 at the point D. (i) Show that the line AD is a tangent to the curve at A. [2] (ii) Find the x-coordinate of B and of C. [3] 4037/11/M/J/15

13 13 (iii) Find the area of the shaded region ABCD, showing all your working. [5] 4037/11/M/J/15 [Turn over

14 14 10 (a) Solve 4 sin x = cosec x for 0 G x G 360. [3] 2 (b) Solve tan 3y - 2 sec 3y - 2 = 0 for 0 G y G 180. [6] 4037/11/M/J/15

15 15 (c) Solve r tan`z - j = 3 for 0 G z G 2r radians. [3] /11/M/J/15

16 16 BLANK PAGE Permission to reproduce items where third-party owned material protected by copyright is included has been sought and cleared where possible. Every reasonable effort has been made by the publisher (UCLES) to trace copyright holders, but if any items requiring clearance have unwittingly been included, the publisher will be pleased to make amends at the earliest possible opportunity. To avoid the issue of disclosure of answer-related information to candidates, all copyright acknowledgements are reproduced online in the Cambridge International Examinations Copyright Acknowledgements Booklet. This is produced for each series of examinations and is freely available to download at after the live examination series. Cambridge International Examinations is part of the Cambridge Assessment Group. Cambridge Assessment is the brand name of University of Cambridge Local Examinations Syndicate (UCLES), which is itself a department of the University of Cambridge. 4037/11/M/J/15

17 Cambridge International Examinations CambridgeOrdinaryLevel * * ADDITIONAL MATHEMATICS 4037/12 Paper1 May/June hours CandidatesanswerontheQuestionPaper. NoAdditionalMaterialsarerequired. READ THESE INSTRUCTIONS FIRST WriteyourCentrenumber,candidatenumberandnameonalltheworkyouhandin. Writeindarkblueorblackpen. YoumayuseanHBpencilforanydiagramsorgraphs. Donotusestaples,paperclips,glueorcorrectionfluid. DONOTWRITEINANYBARCODES. Answerallthequestions. Givenon-exactnumericalanswerscorrectto3significantfigures,or1decimalplaceinthecaseof anglesindegrees,unlessadifferentlevelofaccuracyisspecifiedinthequestion. Theuseofanelectroniccalculatorisexpected,whereappropriate. Youareremindedoftheneedforclearpresentationinyouranswers. Attheendoftheexamination,fastenallyourworksecurelytogether. Thenumberofmarksisgiveninbrackets[ ]attheendofeachquestionorpartquestion. Thetotalnumberofmarksforthispaperis80. Thisdocumentconsistsof16printedpages. DC(LEG/JG)107046/1 UCLES2015 [Turn over

18 2 Mathematical Formulae 1. ALGEBRA Quadratic Equation For the equation ax 2 + bx + c = 0, b b ac x = 4 2 a 2 Binomial Theorem (a + b) n = a n + ( n 1 ) an 1 b + ( n 2 ) an 2 b ( n r ) an r b r + + b n, where n is a positive integer and ( n r ) = n! (n r)!r! 2. TRIGONOMETRY Identities sin 2 A + cos 2 A = 1 sec 2 A = 1 + tan 2 A cosec 2 A = 1 + cot 2 A Formulae for ABC a sin A = b sin B = c sin C a 2 = b 2 + c 2 2bc cos A = 1 bc sin A /12/M/J/15

19 2 3 1 Given that the graph of y = ( 2k + 5) x + kx + 1 does not meet the x-axis, find the possible values of k. [4] 2 Show that tan i + cot i = sec i. [4] cosec i 4037/12/M/J/15 [Turn over

20 4 J 3 Find the inverse of the matrix K 4 L 5 2N O and hence solve the simultaneous equations 3 P 4x + 2y - 8 = 0, 5x + 3y - 9 = 0. [5] 4037/12/M/J/15

21 5 4 B A 12 cm 1.7 rad 2.4 rad O C The diagram shows a circle, centre O, radius 12 cm. The points A, B and C lie on the circumference of this circle such that angle AOB is 1.7 radians and angle AOC is 2.4 radians. (i) Find the area of the shaded region. [4] (ii) Find the perimeter of the shaded region. [5] 4037/12/M/J/15 [Turn over

22 6 5 (a) A security code is to be chosen using 6 of the following: the letters A, B and C the numbers 2, 3 and 5 the symbols * and $. None of the above may be used more than once. Find the number of different security codes that may be chosen if (i) there are no restrictions, [1] (ii) the security code starts with a letter and finishes with a symbol, [2] (iii) the two symbols are next to each other in the security code. [3] 4037/12/M/J/15

23 7 (b) Two teams, each of 4 students, are to be selected from a class of 8 boys and 6 girls. Find the number of different ways the two teams may be selected if (i) there are no restrictions, [2] (ii) one team is to contain boys only and the other team is to contain girls only. [2] 4037/12/M/J/15 [Turn over

24 8 6 A particle moves in a straight line such that its displacement, x m, from a fixed point O after t s, is 2 given by x = 10 ln^t + 4h - 4t. (i) Find the initial displacement of the particle from O. [1] (ii) Find the values of t when the particle is instantaneously at rest. [4] 4037/12/M/J/15

25 9 (iii) Find the value of t when the acceleration of the particle is zero. [5] 4037/12/M/J/15 [Turn over

26 10 7 A 4a B X b D 7a C In the diagram AB = 4a, BC = b and DC = 7a. The lines AC and DB intersect at the point X. Find, in terms of a and b, (i) DA, [1] (ii) DB. [1] Given that AX = mac, find, in terms of a, b and m, (iii) AX, [1] (iv) DX. [2] 4037/12/M/J/15

27 11 Given that DX = ndb, (v) find the value of m and of n. [4] 4037/12/M/J/15 [Turn over

28 12 2x 2x 8 (i) Find y ^10e + e - hdx. [2] k 2x 2x (ii) Hence find y ( 10e + e - ) dx in terms of the constant k. [2] -k k 2x -2x 2k -2k (iii) Given that y ( 10e + e ) dx =-60, show that 11e - 11e = 0. [2] -k 4037/12/M/J/15

29 e 2k 13 (iv) Using a substitution of y = or otherwise, find the value of k in the form a ln b, where a and b are constants. [3] 4037/12/M/J/15 [Turn over

30 14 r 9 A curve has equation y = 4x + 3 cos 2x. The normal to the curve at the point where x = meets 4 the x- and y-axes at the points A and B respectively. Find the exact area of the triangle AOB, where O is the origin. [8] 4037/12/M/J/15

31 15 10 (a) Solve 2 cos 3x = sec 3x for 0 G x G 120. [3] 2 (b) Solve 3 cosec y + 5 cot y - 5 = 0 for 0 G y G 360. [5] Question 10(c) is printed on the next page. 4037/12/M/J/15 [Turn over

32 16 (c) Solve J rn 2 sinkz + O = 1 for 0 G z G 2r radians. [4] L 3 P Permissiontoreproduceitemswherethird-partyownedmaterialprotectedbycopyrightisincludedhasbeensoughtandclearedwherepossible.Everyreasonable efforthasbeenmadebythepublisher(ucles)totracecopyrightholders,butifanyitemsrequiringclearancehaveunwittinglybeenincluded,thepublisherwill bepleasedtomakeamendsattheearliestpossibleopportunity. Toavoidtheissueofdisclosureofanswer-relatedinformationtocandidates,allcopyrightacknowledgementsarereproducedonlineintheCambridgeInternational ExaminationsCopyrightAcknowledgementsBooklet.Thisisproducedforeachseriesofexaminationsandisfreelyavailabletodownloadatwww.cie.org.ukafter theliveexaminationseries. CambridgeInternationalExaminationsispartoftheCambridgeAssessmentGroup.CambridgeAssessmentisthebrandnameofUniversityofCambridgeLocal ExaminationsSyndicate(UCLES),whichisitselfadepartmentoftheUniversityofCambridge. 4037/12/M/J/15

33 Cambridge International Examinations Cambridge Ordinary Level * * ADDITIONAL MATHEMATICS 4037/21 Paper 2 May/June hours Candidates answer on the Question Paper. No Additional Materials are required. READ THESE INSTRUCTIONS FIRST Write your Centre number, candidate number and name on all the work you hand in. Write in dark blue or black pen. You may use an HB pencil for any diagrams or graphs. Do not use staples, paper clips, glue or correction fluid. DO NOT WRITE IN ANY BARCODES. Answer all the questions. Give non-exact numerical answers correct to 3 significant figures, or 1 decimal place in the case of angles in degrees, unless a different level of accuracy is specified in the question. The use of an electronic calculator is expected, where appropriate. You are reminded of the need for clear presentation in your answers. At the end of the examination, fasten all your work securely together. The number of marks is given in brackets [ ] at the end of each question or part question. The total number of marks for this paper is 80. This document consists of 15 printed pages and 1 blank page. DC (KN/SW) /1 [Turn over

34 2 Mathematical Formulae 1. ALGEBRA Quadratic Equation For the equation ax 2 + bx + c = 0, b! b 4ac x = - - 2a 2 Binomial Theorem (a + b) n = a n + ( n 1 ) an 1 b + ( n 2 ) an 2 b ( n r ) an r b r + + b n, where n is a positive integer and ( n r ) = n! (n r)!r! 2. TRIGONOMETRY Identities sin 2 A + cos 2 A = 1 sec 2 A = 1 + tan 2 A cosec 2 A = 1 + cot 2 A Formulae for ABC a sin A = b sin B = c sin C a 2 = b 2 + c 2 2bc cos A = 1 bc sin A /21/M/J/15

35 1 (a) Write log x 27 3 as a logarithm to base 3. [2] (b) Given that log y = 3^log 15 - log 3h + 1, express y in terms of a. [3] a a a 4037/21/M/J/15 [Turn over

36 4 2 (a) y 4 O 2 x The diagram shows the graph of y = f( x) passing through ^0, 4h and touching the x-axis at ^2, 0h. Given that the graph of y = f( x) is a straight line, write down the two possible expressions for f( x ). [2] (b) On the axes below, sketch the graph of y = e + 3, stating the coordinates of any point of intersection with the coordinate axes. [3] y -x O x 4037/21/M/J/15

37 3 (a) Find the matrix A if 4A + 5c m = c m. [2] (b) P = f p Q = ^ h The matrix P represents the number of 4 different televisions that are on sale in each of 3 shops. The matrix Q represents the value of each television in dollars. (i) State, without evaluation, what is represented by the matrix QP. [1] 1 (ii) Given that the matrix R = f1p, state, without evaluation, what is represented by the matrix QPR. 1 [1] 4037/21/M/J/15 [Turn over

38 6 4 P 8 cm O 3r rad 4 T Q The diagram shows a circle, centre O, radius 8 cm. The points P and Q lie on the circle. The lines PT and QT 3r are tangents to the circle and angle POQ = radians. 4 (i) Find the length of PT. [2] (ii) Find the area of the shaded region. [3] (iii) Find the perimeter of the shaded region. [2] 4037/21/M/J/15

39 7 5 (a) A lock can be opened using only the number State whether this is a permutation or a combination of digits, giving a reason for your answer. [1] (b) There are twenty numbered balls in a bag. Two of the balls are numbered 0, six are numbered 1, five are numbered 2 and seven are numbered 3, as shown in the table below. Number on ball Frequency Four of these balls are chosen at random, without replacement. Calculate the number of ways this can be done so that (i) the four balls all have the same number, [2] (ii) the four balls all have different numbers, [2] (iii) the four balls have numbers that total 3. [3] 4037/21/M/J/15 [Turn over

40 8 6 A particle P is projected from the origin O so that it moves in a straight line. At time t seconds after projection, the velocity of the particle, v ms 1 2, is given by v = 2t - 14t (i) Find the time at which P first comes to instantaneous rest. [2] (ii) Find an expression for the displacement of P from O at time t seconds. [3] (iii) Find the acceleration of P when t = 3. [2] 4037/21/M/J/15

41 9 7 (a) The four points O, A, B and C are such that OA = 5a, OB = 15b, OC = 24b - 3a. Show that B lies on the line AC. [3] (b) Relative to an origin O, the position vector of the point P is i 4j and the position vector of the point Q is 3i + 7j. Find (i) PQ, [2] (ii) the unit vector in the direction PQ, [1] (iii) the position vector of M, the mid-point of PQ. [2] 4037/21/M/J/15 [Turn over

42 10 8 (a) (i) Find e 4 x + y 3 dx. [2] 3 4x + 3 (ii) Hence evaluate y e dx. [2] 2. 5 (b) (i) Find J cos x N y K Odx. [2] L 3 P r J (ii) Hence evaluate cos x N 6 y K Odx. [2] 0 L 3 P 4037/21/M/J/15

43 (c) Find y ^x + xh dx. [4] 4037/21/M/J/15 [Turn over

44 (a) Find the set of values of x for which 4x + 19x - 5 G 0. [3] 2 (b) (i) Express x 2 + 8x - 9 in the form ^ x + a h + b, where a and b are integers. [2] (ii) Use your answer to part (i) to find the greatest value of 9-8x - x and the value of x at which this occurs. [2] /21/M/J/15

45 2 13 (iii) Sketch the graph of y = 9-8x - x, indicating the coordinates of any points of intersection with the coordinate axes. [2] y O x 4037/21/M/J/15 [Turn over

46 14 10 The relationship between experimental values of two variables, x and y, is given by y Ab x =, where A and b are constants. (i) By transforming the relationship y = Ab x, show that plotting ln y against x should produce a straight line graph. [2] (ii) The diagram below shows the results of plotting ln y against x for 7 different pairs of values of variables, x and y. A line of best fit has been drawn. ln y x By taking readings from the diagram, find the value of A and of b, giving each value correct to 1 significant figure. [4] (iii) Estimate the value of y when x = 2.5. [2] 4037/21/M/J/15

47 15 11 A B C The Venn diagram above shows the sets A, B and C. It is given that n^a, B, Ch = 48, n^ah = 30, n( B) = 25, n( C) = 15, n( A + B) = 7, n^b + Ch = 6, n^al + B + Clh = 16. (i) Find the value of x, where x = n^a + B + Ch. [3] (ii) Find the value of y, where y = n^a + Bl + Ch. [3] (iii) Hence show that Al + Bl + C = Q. [1] 4037/21/M/J/15

48 16 BLANK PAGE Permission to reproduce items where third-party owned material protected by copyright is included has been sought and cleared where possible. Every reasonable effort has been made by the publisher (UCLES) to trace copyright holders, but if any items requiring clearance have unwittingly been included, the publisher will be pleased to make amends at the earliest possible opportunity. To avoid the issue of disclosure of answer-related information to candidates, all copyright acknowledgements are reproduced online in the Cambridge International Examinations Copyright Acknowledgements Booklet. This is produced for each series of examinations and is freely available to download at after the live examination series. Cambridge International Examinations is part of the Cambridge Assessment Group. Cambridge Assessment is the brand name of University of Cambridge Local Examinations Syndicate (UCLES), which is itself a department of the University of Cambridge. 4037/21/M/J/15

49 Cambridge International Examinations Cambridge Ordinary Level * * ADDITIONAL MATHEMATICS 4037/22 Paper 2 May/June hours Candidates answer on the Question Paper. No Additional Materials are required. READ THESE INSTRUCTIONS FIRST Write your Centre number, candidate number and name on all the work you hand in. Write in dark blue or black pen. You may use an HB pencil for any diagrams or graphs. Do not use staples, paper clips, glue or correction fluid. DO NOT WRITE IN ANY BARCODES. Answer all the questions. Give non-exact numerical answers correct to 3 significant figures, or 1 decimal place in the case of angles in degrees, unless a different level of accuracy is specified in the question. The use of an electronic calculator is expected, where appropriate. You are reminded of the need for clear presentation in your answers. At the end of the examination, fasten all your work securely together. The number of marks is given in brackets [ ] at the end of each question or part question. The total number of marks for this paper is 80. This document consists of 16 printed pages. DC (KN/SW) /1 [Turn over

50 2 Mathematical Formulae 1. ALGEBRA Quadratic Equation For the equation ax 2 + bx + c = 0, b! b 4ac x = - - 2a 2 Binomial Theorem (a + b) n = a n + ( n 1 ) an 1 b + ( n 2 ) an 2 b ( n r ) an r b r + + b n, where n is a positive integer and ( n r ) = n! (n r)!r! 2. TRIGONOMETRY Identities sin 2 A + cos 2 A = 1 sec 2 A = 1 + tan 2 A cosec 2 A = 1 + cot 2 A Formulae for ABC a sin A = b sin B = c sin C a 2 = b 2 + c 2 2bc cos A = 1 bc sin A /22/M/J/15

51 3 1 The universal set contains all the integers from 0 to 12 inclusive. Given that A = {1, 2, 3, 8, 12}, B = {0, 2, 3, 4, 6} and C = {1, 2, 4, 6, 7, 9, 10}, (i) complete the Venn diagram, [3] A B C (ii) state the value of n( Al k Bl k C), [1] (iii) write down the elements of the set Al k B k C. [1] 4037/22/M/J/15 [Turn over

52 4 2 The table shows the number of passengers in Economy class and in Business class on 3 flights from London to Paris. The table also shows the departure times for the 3 flights and the cost of a single ticket in each class. Departure time Number of passengers in Economy class Number of passengers in Business class Single ticket price ( ) (i) Write down a matrix, P, for the numbers of passengers and a matrix, Q, of single ticket prices, such that the matrix product QP can be found. [2] (ii) Find the matrix product QP. [2] (iii) Given that J1N K O R = 1, explain what information is found by evaluating the matrix product QPR. [1] K O L1P 4037/22/M/J/15

53 5 3 Do not use a calculator in this question. C B ( ) cm A The diagram shows the right-angled triangle ABC, where AB = ^ hcm and angle B = 90. The J N area of this triangle is cm 2 K O. L 2 P (i) Find the length of the side BC in the form ^a + b 5h cm, where a and b are integers. [3] (ii) Find ^ACh 2 in the form ^c + d 5h cm 2, where c and d are integers. [2] 4037/22/M/J/15 [Turn over

54 6 4 A river, which is 80 m wide, flows at 2 ms 1 between parallel, straight banks. A man wants to row his boat straight across the river and land on the other bank directly opposite his starting point. He is able to row his boat in still water at 3 ms 1. Find (i) the direction in which he must row his boat, [2] (ii) the time it takes him to cross the river. [3] 4037/22/M/J/15

55 7 5 Solve the simultaneous equations 2 2 2x + 3y = 7 xy, x + y = 4. [5] 6 (a) Solve 6 x =. [2] 4 2 a a a a a (b) Solve log 2y + log 8 + log 16y - log 64y = 2 log 4. [4] 4037/22/M/J/15 [Turn over

56 8 7 In the expansion of ^1 + 2xh n, the coefficient of x 4 is ten times the coefficient of x 2. Find the value of the positive integer, n. [6] 4037/22/M/J/15

57 8 y = x 2 6x y y = x + 10 B A O C x 2 The graph of y = x - 6x + 10 cuts the y-axis at A. The graphs of y = x - 6x + 10 and y = x + 10 cut one another at A and B. The line BC is perpendicular to the x-axis. Calculate the area of the shaded region enclosed by the curve and the line AB, showing all your working. [8] /22/M/J/15 [Turn over

58 10 9 Solutions by accurate drawing will not be accepted. y y = mx + 4 P (2, 10) R 4 Q 1 O 2 x The line y = mx + 4 meets the lines x = 2 and x =- 1 at the points P and Q respectively. The point R is such that QR is parallel to the y-axis and the gradient of RP is 1. The point P has coordinates (2, 10). (i) Find the value of m. [2] (ii) Find the y-coordinate of Q. [1] (iii) Find the coordinates of R. [2] 4037/22/M/J/15

59 11 (iv) Find the equation of the line through P, perpendicular to PQ, giving your answer in the form ax + by = c, where a, b and c are integers. [3] (v) Find the coordinates of the midpoint, M, of the line PQ. [2] (vi) Find the area of triangle QRM. [2] 4037/22/M/J/15 [Turn over

60 12 10 (a) The function f is defined by f: x 7 sin x for 0 G x G 360. On the axes below, sketch the graph of y = f( x). [2] y O x (b) The functions g and hg are defined, for x H 1, by (i) Show that g^xh = ln^4x - 3 h, hg^xh = x. 3 h^xh = +. [2] 4 e x 4037/22/M/J/15

61 13 (ii) y y = g(x) O 1 x The diagram shows the graph of y = g^xh. Given that g and h are inverse functions, sketch, on the same diagram, the graph of y = h^xh. Give the coordinates of any point where your graph meets the coordinate axes. [2] (iii) State the domain of h. [1] (iv) State the range of h. [1] 4037/22/M/J/15 [Turn over

62 14 11 O B C 8 A D Q R h P 4 S The diagram shows a cuboid of height h units inside a right pyramid OPQRS of height 8 units and with square base of side 4 units. The base of the cuboid sits on the square base PQRS of the pyramid. The points A, B, C and D are corners of the cuboid and lie on the edges OP, OQ, OR and OS, respectively, of the pyramid OPQRS. The pyramids OPQRS and OABCD are similar. (i) Find an expression for AD in terms of h and hence show that the volume V of the cuboid is given 3 h 2 by V = - 4h + 16h units 4 3. [4] 4037/22/M/J/15

63 15 (ii) Given that h can vary, find the value of h for which V is a maximum. [4] Question 12 is printed on the next page. 4037/22/M/J/15 [Turn over

64 (i) Show that x = 2 is a root of the polynomial equation 15x + 26x - 11x - 6 = 0. [1] 3 2 (ii) Find the remainder when 15x + 26x - 11x - 6 is divided by x 3. [2] 3 2 (iii) Find the value of p and of q such that 15x + 26x - 11x - 6 is a factor of x + px - 37x + qx + 6. [4] Permission to reproduce items where third-party owned material protected by copyright is included has been sought and cleared where possible. Every reasonable effort has been made by the publisher (UCLES) to trace copyright holders, but if any items requiring clearance have unwittingly been included, the publisher will be pleased to make amends at the earliest possible opportunity. To avoid the issue of disclosure of answer-related information to candidates, all copyright acknowledgements are reproduced online in the Cambridge International Examinations Copyright Acknowledgements Booklet. This is produced for each series of examinations and is freely available to download at after the live examination series. Cambridge International Examinations is part of the Cambridge Assessment Group. Cambridge Assessment is the brand name of University of Cambridge Local Examinations Syndicate (UCLES), which is itself a department of the University of Cambridge. 4037/22/M/J/15