# MEI STRUCTURED MATHEMATICS 4756

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2 Section A (54 marks) Answer all the questions (a) A curve has polar equation r acos 3q for p where a is a positive constant. (i) Sketch the curve, using a continuous line for sections where r 0 and a broken line for sections where r 0. [3] (ii) Find the area enclosed by one of the loops. [5] 3 4 (b) Find the exact value of Û Ù d x. [5] ı 3-4x 0 Û (c) Use a trigonometric substitution to find Ù [5] 3 3 d x. ı + x 0 ( ) q p, In this question, q is a real number with 0 and w e3jq. (i) State the modulus and argument of each of the complex numbers Illustrate these three complex numbers on an Argand diagram. [6] (ii) Show that ( w)( w*) 5 4 cos 3q. [4] Infinite series C and S are defined by C cos q cos 5q S sin q sin 5q q 6 p, w, w* and jw. 4 cos 8q 8 cos q 4 sin 8q 8 sin q (iii) Show that C 4 cos q cos q, and find a similar expression for S. [8] 5 4 cos 3q , 4756 January 006

3 3 Ê 3ˆ 3 The matrix M = Á Ë - 4 (i) Show that the characteristic equation for M is 6l 9l 4 0. [3] (ii) Show that is an eigenvalue of M, and find the other two eigenvalues. [3] (iii) Find an eigenvector corresponding to the eigenvalue. [3] Ê3ˆ Ê 0ˆ (iv) Verify that 0 and 3 are eigenvectors of M. [3] Á Á Ë Ë - (v) Write down a matrix P, and a diagonal matrix D, such that M 3 PDP. [3] (vi) Use the Cayley-Hamilton theorem to express M in the form am bm ci. [3] Section B (8 marks) Answer one question Option : Hyperbolic functions 4 (a) Solve the equation sinh x 4 cosh x 8, giving the answers in an exact logarithmic form. [6] 0 (b) Find the exact value of e x sinh x dx. [4] (c) (i) Differentiate arsinh ( 3 x ) with respect to x. [] 0 (ii) Use integration by parts to show that arsinh ( 3 x ) dx ln 3. [6] 4756 January 006 [Turn over

4 Option : Investigation of curves 4 This question requires the use of a graphical calculator. 5 A curve has equation y x3 k 3 where k is a positive constant and k. x 4, (i) Find the equations of the three asymptotes. [3] (ii) Use your graphical calculator to obtain rough sketches of the curve in the two separate cases k and k. [4] (iii) In the case k, your sketch may not show clearly the shape of the curve near x 0. Use calculus to show that the curve has a minimum point when x 0. [5] (iv) In the case k, your sketch may not show clearly how the curve approaches its asymptote as x Æ. Show algebraically that the curve crosses this asymptote. [] (v) Use the results of parts (iii) and (iv) to produce more accurate sketches of the curve in the two separate cases k and k. These sketches should indicate where the curve crosses the axes, and should show clearly how the curve approaches its asymptotes. The presence of stationary points should be clearly shown, but there is no need to find their coordinates. [4] 4756 January 006

6 Section A (54 marks) Answer all the questions (a) A curve has polar equation r = a + cosq for 3 4 p where a is a positive constant. ( ) q 3 4 p, (i) Sketch the curve. [] (ii) Find, in an exact form, the area of the region enclosed by the curve. [7] (b) (i) Find the Maclaurin series for the function f(x) tan ( 4 p x), up to the term in x. [6] (ii) Use the Maclaurin series to show that, when h is small, h Û Ù x tan( p + x) dx ª h + h. ı h [3] (a) (i) Given that z cos q j sin q, express z n and z n z n z n in simplified trigonometric form. [] (ii) By considering 4 Ê ˆ Ê ˆ z - z + Ë z Ë z, find A, B, C and D such that sin 4 q cos q A cos 6q B cos 4q C cos q D. [6] (b) (i) Find the modulus and argument of 4 4j. [] (ii) Find the fifth roots of 4 4j in the form re jq, where r 0 and p q Illustrate these fifth roots on an Argand diagram. [6] (iii) Find integers p and q such that (p qj) 5 4 4j. [] p June 006

7 Ê4 kˆ 3 (i) Find the inverse of the matrix 3 5 where k 5. [6] Á, Ë (ii) Solve the simultaneous equations 4x + y + 7z = 3x + y + 5z = m 8x + 5y + 3z = 0 giving x, y and z in terms of m. [5] (iii) Find the value of p for which the simultaneous equations 3 4x + y + 5z = 3x + y + 5z = p 8x + 5y + 3z = 0 have solutions, and find the general solution in this case. [7] Section B (8 marks) Answer one question Option : Hyperbolic functions 4 (i) Starting from the definitions of sinh x and cosh x in terms of exponentials, prove that sinh x cosh x. [3] (ii) Solve the equation cosh x sinh x 5, giving the answers in an exact logarithmic form. [6] ln 3 0 (iii) Show that sinh x dx 0 9 ln 3. [5] 5 Û (iv) Find the exact value of x Ù - 9 d. x [4] ı 3 [Question 5 is printed overleaf.] 4756 June 006

8 Option : Investigation of curves 4 This question requires the use of a graphical calculator. 5 A curve has parametric equations x where k is a positive constant. q k sin q, y cos q, (i) For the case k, use your graphical calculator to sketch the curve. Describe its main features. [4] (ii) Sketch the curve for a value of k between 0 and. Describe briefly how the main features differ from those for the case k. [3] (iii) For the case k : (A) sketch the curve; [] dy (B) find in terms of q ; [] dx (C) show that the width of each loop, measured parallel to the x-axis, is 3 - p. 3 [5] (iv) Use your calculator to find, correct to one decimal place, the value of k for which successive loops just touch each other. [] 4756 June 006

10 Section A (54 marks) Answer all the questions (a) A curve has polar equation r ae kq for 0 p, where a and k are positive constants. The points A and B on the curve correspond to and respectively. (i) Sketch the curve. [] (ii) Find the area of the region enclosed by the curve and the line AB. [4] (b) Find the exact value of Û Ù [5] ı 3 + 4x d. x 0 q q 0 q p (c) (i) Find the Maclaurin series for tan x, up to the term in x 3. [4] tan x (ii) Use this Maclaurin series to show that, when h is small, dx 3h 7h 3. [3] x Ú 4h h (a) You are given the complex numbers w = 3e and - p j z = - 3j. w (i) Find the modulus and argument of each of the complex numbers w, z and [5] z. w (ii) Hence write in the form a bj, giving the exact values of a and b. [] z (b) In this part of the question, n is a positive integer and q is a real number with 0 p q. n (i) Express that e - jq + e jq in simplified trigonometric form, and hence, or otherwise, show jq jq +e = e cos q. [4] Series C and S are defined by n n n n C = + Ê ˆ + Ê ˆ + Ê ˆ +º+ Ê ˆ cosq cosq cos3q cos nq, Ë Ë Ë 3 Ën n n n n S = Ê ˆ + Ê ˆ + Ê ˆ +º+ Ê ˆ sinq sinq sin3q sin nq. Ë Ë Ë 3 Ën S (ii) Find C and S, and show that [7] C tan nq. OCR /0 Jan 07

11 3 Ê4 kˆ Ê - -6ˆ 3 Let P = 3 (where k 4) and M = - 3. Á Ë 0 - Á Ë - - Ê - 4ˆ (i) Find in terms of k, and show that, when P - P k, = [6] Á Ë - (ii) Verify that Ê4ˆ Êˆ Ê ˆ, and 3 Á Á Á Ë Ë0 Ë - are eigenvectors of M, and find the corresponding eigenvalues. [4] Ê4-6 -0ˆ Ê - 4 4ˆ (iii) Show that M n = - - n [8] Á Á Ë Ë - - Option : Hyperbolic functions Section B (8 marks) Answer one question 4 (i) Show that arcosh x = ln x + x -. [5] 39. ( ) Û (ii) Find Ù dx, ı 5. 4x - 9 giving your answer in the form a ln b, where a and b are rational numbers. [5] (iii) There are two points on the curve y cosh x sinh x ( ( ) ) at which the gradient is Show that one of these points is ln +, 3, and find the coordinates of the other point, in a similar form. [8] 9. OCR /0 Jan 07 [Turn over

12 Option : Investigation of curves 4 This question requires the use of a graphical calculator. 5 Cartesian coordinates (x, y) and polar coordinates (r, q) are set up in the usual way, with the pole at the origin and the initial line along the positive x-axis, so that x r cos q and y r sin q. A curve has polar equation r k cos q, where k is a constant with k. (i) Use your graphical calculator to obtain sketches of the curve in the three cases k, k.5 and k 4. [5] (ii) Name the special feature which the curve has when k. [] (iii) For each of the three cases, state the number of points on the curve at which the tangent is parallel to the y-axis. [] dx (iv) Express x in terms of k and q, and find. Hence find the range of values of k for which there dq are just two points on the curve where the tangent is parallel to the y-axis. [4] The distance between the point (r, q) on the curve and the point (, 0) on the x-axis is d. (v) Use the cosine rule to express d in terms of k and q, and deduce that k d k. [4] (vi) Hence show that, when k is large, the shape of the curve is very nearly circular. [] 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 (OCR) 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. OCR 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. OCR /0 Jan 07

14 Section A (54 marks) Answer all the questions (a) A curve has polar equation r a( cos q), where a is a positive constant. (i) Sketch the curve. [] (ii) Find the area of the region enclosed by the section of the curve for which 0 and the line [6] q p. Û (b) Use a trigonometric substitution to show that 3 d x =. [4] Ù ı 4 - x 4 3 (c) In this part of the question, f(x) arccos (x). (i) Find f (x). [] (ii) Use a standard series to expand f (x), and hence find the series for f(x) in ascending powers of x, up to the term in x 5. [4] 0 ( ) q p (a) Use de Moivre s theorem to show that sin 5q 5 sin q 0 sin 3 q 6 sin 5 q. [5] (b) (i) Find the cube roots of j in the form re jq where r 0 and p. [6] p q These cube roots are represented by points A, B and C in the Argand diagram, with A in the first quadrant and ABC going anticlockwise. The midpoint of AB is M, and M represents the complex number w. (ii) Draw an Argand diagram, showing the points A, B, C and M. [] (iii) Find the modulus and argument of w. [] (iv) Find w 6 in the form a bj. [3] OCR /0 June 07

15 3 3 Let Ê 3 5 ˆ M = Á Ë - -4 (i) Show that the characteristic equation for M is l3 l 48l 0. [4] Ê ˆ You are given that - is an eigenvector of M corresponding to the eigenvalue 0. Á Ë (ii) Find the other two eigenvalues of M, and corresponding eigenvectors. [8] (iii) Write down a matrix P, and a diagonal matrix D, such that P M P D. [3] (iv) Use the Cayley-Hamilton theorem to find integers a and b such that M 4 am bm. [3] Section B (8 marks) Answer one question Option : Hyperbolic functions Û 4 (a) Find Ù d, x giving your answer in an exact logarithmic form. [5] ı 0 9x + 6 (b) (i) Starting from the definitions of sinh x and cosh x in terms of exponentials, prove that sinh x sinh x cosh x. [] (ii) Show that one of the stationary points on the curve has coordinates (ln 3, 59 3 ), and find the coordinates of the other two stationary points. [7] ln 3 Û (iii) Show that Ù ( 0 cosh x - 3cosh x) dx = 40. [4] ı -ln 3 y 0 cosh x 3 cosh x [Question 5 is printed overleaf.] OCR /0 June 07 [Turn over

16 Option : Investigation of curves 4 This question requires the use of a graphical calculator. 5 The curve with equation y x kx k is to be investigated for different values of k. x k (i) Use your graphical calculator to obtain rough sketches of the curve in the cases k, k 0.5 and k. [6] (ii) Show that the equation of the curve may be written as y x k k (k ). x k Hence find the two values of k for which the curve is a straight line. [4] (iii) When the curve is not a straight line, it is a conic. (A) Name the type of conic. [] (B) Write down the equations of the asymptotes. [] (iv) Draw a sketch to show the shape of the curve when k 8. This sketch should show where the curve crosses the axes and how it approaches its asymptotes. Indicate the points A and B on the curve where x and x k respectively. [5] 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 (OCR) 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. OCR 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. OCR /0 June 07

18 Section A (54 marks) Answer all the questions (a) Fig. shows the curve with polar equation r = a( cos θ) for 0 θ π, wherea is a positive constant. O Fig. Find the area of the region enclosed by the curve. [7] (b) (i) Given that f(x) =arctan( 3 + x),findf (x) and f (x). [4] (ii) Hence find the Maclaurin series for arctan( 3 + x), as far as the term in x. [4] h (iii) Hence show that, if h is small, x arctan( 3 + x) dx 6 h3. [3] h (a) Find the 4th roots of 6j, in the form re jθ where r > 0and π < θ π. Illustrate the 4th roots on an Argand diagram. [6] (b) (i) Show that ( e jθ )( e jθ )=5 4cosθ. [3] Series C and S are defined by C = cosθ + 4cosθ + 8cos3θ n cos nθ, S = sinθ + 4sinθ + 8sin3θ n sin nθ. (ii) Show that C = cosθ 4 n+ cos(n + )θ + n+ cos nθ, and find a similar expression 5 4cosθ for S. [9] OCR /0 Jan08

19 3 3 YouaregiventhematrixM = ( ). (i) Find the eigenvalues, and corresponding eigenvectors, of the matrix M. [8] (ii) Write down a matrix P and a diagonal matrix D such that P MP = D. [] (iii) Given that M n = ( a b c d ),showthata = + 3 5n, and find similar expressions for b, c and d. [8] Option : Hyperbolic functions Section B (8 marks) Answer one question 4 (i) Given that k andcoshx = k, showthatx =±ln(k + k ). [5] (ii) Find dx, giving the answer in an exact logarithmic form. [5] 4x (iii) Solve the equation 6 sinh x sinh x = 0, giving the answers in an exact form, using logarithms where appropriate. [4] (iv) Show that there is no point on the curve y = 6sinhx sinh x at which the gradient is 5. [4] [Question 5 is printed overleaf.] OCR /0 Jan08

20 4 Option : Investigation of curves This question requires the use of a graphical calculator. 5 A curve has parametric equations x = t + t, y = t3 λt, whereλ is a constant. (i) Use your calculator to obtain a sketch of the curve in each of the cases λ =, λ = 0 and λ =. Name any special features of these curves. [5] (ii) By considering the value of x when t is large, write down the equation of the asymptote. [] Fortheremainderofthisquestion,assumethatλ is positive. (iii) Find, in terms of λ, the coordinates of the point where the curve intersects itself. [3] (iv) Show that the two points on the curve where the tangent is parallel to the x-axis have coordinates ( λ 3 + λ, ± 4λ 3 ). [4] 7 Fig. 5 shows a curve which intersects itself at the point (, 0) and has asymptote x = 8. The stationary points A and B have y-coordinates and. y A O B 8 x Fig. 5 (v) For the curve sketched in Fig. 5, find parametric equations of the form x = at + t, y = b(t3 λt), where a, λ and b aretobedetermined. [5] 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 (OCR) 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. OCR 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. OCR /0 Jan08

22 Section A (54 marks) Answer all the questions (a) A curve has cartesian equation (x + y ) = 3xy. (i) Show that the polar equation of the curve is r = 3cosθ sin θ. [3] (ii) Hence sketch the curve. [3] (b) Find the exact value of 0 dx. [5] 4 3x (c) (i) Write down the series for ln( + x) and the series for ln( x), both as far as the term in x 5. [] (ii) Hence find the first three non-zero terms in the series for ln( + x ). [] x (iii) Use the series in part (ii) to show that r=0 (r + )4r = ln 3. [3] You are given the complex numbers = 3( + j) and w = 8(cos 7 π + jsin 7 π). (i) Find the modulus and argument of each of the complex numbers, *, w and w. [7] (ii) Express w in the form a + bj, giving the exact values of a and b. [] (iii) Find the cube roots of, intheformre jθ,wherer > 0and π < θ π. [4] (iv) Show that the cube roots of can be written as k w*, k * and k 3 jw, where k, k and k 3 are real numbers. State the values of k, k and k 3. [5] OCR /0 Jun08

23 k 3 (i) Given the matrix Q = ( 0 ) (where k 3), find Q in terms of k Show that, when k = 4, Q = ( 8 ). [6] 5 4 The matrix M has eigenvectors ( ), ( 0 ) and ( ), with corresponding eigenvalues, and3 3 respectively. (ii) Write down a matrix P and a diagonal matrix D such that P MP = D, and hence find the matrix M. [7] (iii) Write down the characteristic equation for M, and use the Cayley-Hamilton theorem to find integers a, b and c such that M 4 = am + bm + ci. [5] Section B (8 marks) Answer one question Option : Hyperbolic functions 4 (i) Starting from the definitions of sinh x and cosh x in terms of exponentials, prove that cosh x sinh x =. [3] (ii) Solve the equation 4 cosh x + 9sinhx = 3, giving the answers in exact logarithmic form. [6] (iii) Show that there is only one stationary point on the curve y = 4cosh x + 9sinhx, and find the y-coordinate of the stationary point. [4] ln (iv) Show that 0 (4cosh x + 9sinhx) dx = ln [5] [Question 5 is printed overleaf.] OCR /0 Jun08

24 4 Option : Investigation of curves This question requires the use of a graphical calculator. 5 A curve has parametric equations x = λ cos θ sin θ, y = cos θ + sin θ, whereλ is a positive λ constant. (i) Use your calculator to obtain a sketch of the curve in each of the cases λ = 0.5, λ = 3 and λ = 5. [3] (ii) Given that the curve is a conic, name the type of conic. [] (iii) Show that y has a maximum value of whenθ = π. [] 4 (iv) Show that x + y =( + λ )+( λ λ ) sin θ, and deduce that the distance from the origin of any point on the curve is between + λ and + λ. [6] (v) For the case λ =, show that the curve is a circle, and find its radius. [] (vi) For the case λ =,drawasketchofthecurve,andlabelthepointsa,b,c,d,e,f,g,honthe curve corresponding to θ = 0, 4 π, π, 3 4 π, π, 5 4 π, 3 π, 7 π respectively. You should make clear what 4 is special about each of these points. [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 (OCR) 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. OCR 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. OCR /0 Jun08

26 Section A (54 marks) Answer all the questions (a) (i) By considering the derivatives of cos x, show that the Maclaurin expansion of cos x begins (ii) The Maclaurin expansion of sec x begins x + 4 x4. [4] + ax + bx 4, where a and b are constants. Explain why, for sufficiently small x, ( x + 4 x4 )( + ax + bx 4 ). Hence find the values of a and b. [5] (b) (i) Given that y = arctan( x dy ), show that a dx = a a + x. [4] (ii) Find the exact values of the following integrals. (A) (B) 4 + x dx 4 + 4x dx [3] [3] (i) Write down the modulus and argument of the complex number e jπ/3. [] (ii) The triangle OAB in an Argand diagram is equilateral. O is the origin; A corresponds to the complex number a = ( + j); B corresponds to the complex number b. Show A and the two possible positions for B in a sketch. Express a in the form re jθ. Find the two possibilities for b in the form re jθ. [5] (iii) Given that = e jπ/3, show that 6 = 8. Write down, in the form rejθ, the other five complex numbers such that 6 = 8. Sketch all six complex numbers in a new Argand diagram. [6] Let w = e jπ/. (iv) Find w in the form x + jy, and mark this complex number on your Argand diagram. [3] (v) Find w 6, expressing your answer in as simple a form as possible. [] OCR Jan09

27 3 (a) A curve has polar equation r = a tan θ for 0 θ π, where a is a positive constant. 3 3 (i) Sketch the curve. [3] (ii) Find the area of the region between the curve and the line θ = π. Indicate this region on 4 your sketch. [5] (b) (i) Find the eigenvalues and corresponding eigenvectors for the matrix M where M = ( ). [6] (ii) Give a matrix Q and a diagonal matrix D such that M = QDQ. [3] Option : Hyperbolic functions Section B (8 marks) Answer one question 4 (a) (i) Prove, from definitions involving exponentials, that (ii) Given that sinh x = tan y, where π < y < π, show that (A) tanh x = sin y, cosh x sinh x =. [] (B) x = ln(tan y + sec y). [6] (b) (i) Given that y = artanh x, find dy in terms of x. dx Hence show that x dx = artanh. [4] (ii) Express x in partial fractions and hence find an expression for dx in terms of x logarithms. [4] (iii) Use the results in parts (i) and (ii) to show that artanh = ln 3. [] OCR Jan09 Turn over

28 4 Option : Investigation of curves This question requires the use of a graphical calculator. 5 The limaçon of Pascal has polar equation r = + a cos θ, where a is a constant. (i) Use your calculator to sketch the curve when a =. (You need not distinguish between parts of the curve where r is positive and negative.) [3] (ii) By using your calculator to investigate the shape of the curve for different values of a, positive and negative, (A) state the set of values of a for which the curve has a loop within a loop, (B) state, with a reason, the shape of the curve when a = 0, (C) state what happens to the shape of the curve as a ±, (D) name the feature of the curve that is evident when a = 0.5, and find another value of a for which the curve has this feature. [7] (iii) Given that a > 0 and that a is such that the curve has a loop within a loop, write down an equation for the values of θ at which r = 0. Hence show that the angle at which the curve crosses itself is arccos( a ). Obtain the cartesian equations of the tangents at the point where the curve crosses itself. Explain briefly how these equations relate to the answer to part (ii)(a). [8] 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 (OCR) 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. OCR 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. OCR Jan09

30 Section A (54 marks) Answer all the questions (a) (i) Use the Maclaurin series for ln( + x) and ln( x) to obtain the first three non-zero terms in the Maclaurin series for ln( + x ). State the range of validity of this series. [4] x (ii) Find the value of x for which + x = 3. Hence find an approximation to ln 3, giving your x answer to three decimal places. [4] a (b) A curve has polar equation r = for 0 θ π, where a is a positive constant. The points + sin θ on the curve have cartesian coordinates x and y. (i) By plotting suitable points, or otherwise, sketch the curve. [3] (ii) Show that, for this curve, r + y = a and hence find the cartesian equation of the curve. [5] (i) Obtain the characteristic equation for the matrix M where 3 M = ( 0 0 ). 0 Hence or otherwise obtain the value of det(m). [3] (ii) Show that is an eigenvalue of M, and show that the other two eigenvalues are not real. Find an eigenvector corresponding to the eigenvalue. Hence or otherwise write down the solution to the following system of equations. [9] 3x + y = 0. y = 0.6 x + = 0. (iii) State the Cayley-Hamilton theorem and use it to show that M 3 = 3M 3M 7I. Obtain an expression for M in terms of M, M and I. [4] (iv) Find the numerical values of the elements of M, showing your working. [3] OCR Jun09

31 3 3 (a) (i) Sketch the graph of y = arcsin x for x. [] Find dy, justifying the sign of your answer by reference to your sketch. [4] dx (ii) Find the exact value of the integral 0 dx. [3] x (b) The infinite series C and S are defined as follows. By considering C + js, show that C = cos θ + 3 cos 3θ + cos 5θ S = sin θ + 3 sin 3θ + sin 5θ C = 3 cos θ 5 3 cos θ, and find a similar expression for S. [] Section B (8 marks) Answer one question Option : Hyperbolic functions 4 (i) Prove, from definitions involving exponentials, that cosh u = cosh u. [3] (ii) Prove that arsinh y = ln(y + y + ). [4] (iii) Use the substitution x = sinh u to show that x + 4 dx = arsinh x + x x c, where c is an arbitrary constant. [6] (iv) By first expressing t + t + 5 in completed square form, show that t + t + 5 dt = (ln( + ) + ). [5] [Question 5 is printed overleaf.] OCR Jun09

34 Section A (54 marks) Answer all the questions (a) Given that y = arctan x, find dy, giving your answer in terms of x. Hence show that dx 0 x(x + ) dx = π. [6] (b) A curve has cartesian equation x + y = xy +. (i) Show that the polar equation of the curve is r = sin θ. [4] (ii) Determine the greatest and least positive values of r and the values of θ between 0 and π for which they occur. [6] (iii) Sketch the curve. [] (a) Use de Moivre s theorem to find the constants a, b, c in the identity cos 5θ a cos 5 θ + b cos 3 θ + c cos θ. [6] (b) Let C = cos θ + cos(θ + π n ) + cos(θ + 4π n and S = sin θ + sin(θ + π n ) + sin(θ + 4π n where n is an integer greater than. (n )π ) cos(θ + ), n (n )π ) sin(θ + ), n By considering C + js, show that C = 0 and S = 0. [7] (c) Write down the Maclaurin series for e t as far as the term in t. Hence show that, for t close to zero, t e t t. [5] OCR Jan0

35 3 3 (i) Find the inverse of the matrix where a 4. a 3 Show that when a = the inverse is [6] (ii) Solve, in terms of b, the following system of equations. [5] x + y = x y + = b 3x y + = (iii) Find the value of b for which the equations x + y + 4 = x y + = b 3x y + = have solutions. Give a geometrical interpretation of the solutions in this case. [7] Section B (8 marks) Answer one question Option : Hyperbolic functions 4 (i) Prove, using exponential functions, that cosh x = + sinh x. Differentiate this result to obtain a formula for sinh x. [4] (ii) Solve the equation cosh x + 3 sinh x = 3, expressing your answers in exact logarithmic form. [7] (iii) Given that cosh t = 5 4, show by using exponential functions that t = ± ln. Find the exact value of the integral 4 5 dx. [7] x 6 OCR Jan0 Turn over

38 Section A (54 marks) Answer all the questions (a) (i) Given that f(t) = arcsin t, write down an expression for f (t) and show that f t (t) =. [3] ( t ) 3 (ii) Show that the Maclaurin expansion of the function arcsin(x + ) begins π 6 + x, 3 and find the term in x. (b) Sketch the curve with polar equation r = πa, where a > 0, for 0 θ < π. π + θ [5] Find, in terms of a, the area of the region bounded by the part of the curve for which 0 θ π and the lines θ = 0 and θ = π. [6] (c) Find the exact value of the integral 0 3 dx. [5] 9 + 4x (a) Given that = cos θ + j sin θ, express n + n and n n in simplified trigonometric form. Hence find the constants A, B, C in the identity sin 5 θ A sin θ + B sin 3θ + C sin 5θ. [5] (b) (i) Find the 4th roots of 9j in the form re jθ, where r > 0 and 0 < θ < π. Illustrate the roots on an Argand diagram. [6] (ii) Let the points representing these roots, taken in order of increasing θ, be P, Q, R, S. The mid-points of the sides of PQRS represent the 4th roots of a complex number w. Find the modulus and argument of w. Mark the point representing w on your Argand diagram. [5] OCR Jun0

39 3 (a) (i) A 3 3 matrix M has characteristic equation 3 λ 3 + λ 3λ + 6 = 0. Show that λ = is an eigenvalue of M. Find the other eigenvalues. [4] 3 (ii) An eigenvector corresponding to λ = is ( 3 ). 3 Evaluate M( 3 ) and M ( ). 3 x Solve the equation M( y ) = ( (iii) Find constants A, B, C such that 3 ). [5] 3 M 4 = AM + BM + CI. [4] (b) A matrix N has eigenvalues and, with eigenvectors ( ) and ( ) respectively. Find N. [6] Section B (8 marks) Answer one question Option : Hyperbolic functions 4 (i) Prove, using exponential functions, that sinh x = sinh x cosh x. Differentiate this result to obtain a formula for cosh x. [4] (ii) Sketch the curve with equation y = cosh x. The region bounded by this curve, the x-axis, and the line x = is rotated through π radians about the x-axis. Find, correct to 3 decimal places, the volume generated. (You must show your working; numerical integration by calculator will receive no credit.) [7] (iii) Show that the curve with equation has exactly one stationary point. y = cosh x + sinh x Determine, in exact logarithmic form, the x-coordinate of the stationary point. [7] OCR Jun0 Turn over

42 Section A (54 marks) Answer all the questions (a) A curve has polar equation r = (cos θ + sin θ) for 4 π θ 3 4 π. (i) Show that a cartesian equation of the curve is x + y = x + y. Hence or otherwise sketch the curve. [5] (ii) Find, by integration, the area of the region bounded by the curve and the lines θ = 0 and θ = π. Give your answer in terms of π. [7] (b) (i) Given that f(x) = arctan( x), find f (x). [] (ii) Expand f (x) in ascending powers of x as far as the term in x 4. Hence obtain an expression for f(x) in ascending powers of x as far as the term in x 5. [5] (a) (i) Given that = cos θ + j sin θ, express n + n and n n in simplified trigonometrical form. [] (ii) By considering ( + ) 6, show that cos 6 θ = (cos 6θ + 6 cos 4θ + 5 cos θ + 0). [3] 3 (iii) Obtain an expression for cos 6 θ sin 6 θ in terms of cos θ and cos 6θ. [5] (b) The complex number w is 8e jπ/3. You are given that is a square root of w and that is a cube root of w. The points representing and in the Argand diagram both lie in the third quadrant. (i) Find and in the form re jθ. Draw an Argand diagram showing w, and. [6] (ii) Find the product, and determine the quadrant of the Argand diagram in which it lies. [3] 3 (i) Show that the characteristic equation of the matrix M = is λ 3 5λ + 8λ 66 = 0. [4] (ii) Show that λ = 3 is an eigenvalue of M, and determine whether or not M has any other real eigenvalues. [4] (iii) Find an eigenvector, v, of unit length corresponding to λ = 3. State the magnitude of the vector M n v, where n is an integer. [5] (iv) Using the Cayley-Hamilton theorem, obtain an equation for M in terms of M, M and I. [3] OCR Jan

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