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NSW HSC 2003 : MATHEMATICS EXTENSION-2

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2003 H I G H E R S C H O O L C E R T I F I C AT E E X A M I N AT I O N Mathematics Extension 2 General Instructions Reading time 5 minutes Working time 3 hours Write using black or blue pen Board-approved calculators may be used A table of standard integrals is provided at the back of this paper All necessary working should be shown in every question 412 Total marks 120 Attempt Questions 1 8 All questions are of equal value Total marks 120 Attempt Questions 1 8 All questions are of equal value Answer each question in a SEPARATE writing booklet. Extra writing booklets are available. Marks Question 1 (15 marks) Use a SEPARATE writing booklet. 1 (a) ex Evaluate 1 + ex 0 (b) Use integration by parts to find ( ) 2 dx . 2 3 3 x loge x dx . (c) By completing the square and using the table of standard integrals, find 2 dx . 2 x 2x + 5 (d) (i) Find the real numbers a and b such that 5 x 2 3 x + 13 ( x 1 )( x +4 ) a + x 1 bx 1 x2 + 4 . 5 x 2 3 x + 13 dx . Find x 1 x2 + 4 2 Use the substitution x = 3 sin to evaluate 4 (ii) (e) 2 2 ( )( ) 3 dx 2 2 0 9 x ( 2 ) 3 2 . Marks Question 2 (15 marks) Use a SEPARATE writing booklet. (a) Let z = 2 + i and w = 1 i. Find, in the form x + iy, (i) 1 (ii) (b) zw 4 . z 1 Let = 1 + i. (i) 2 (ii) Show that is a root of the equation z4 + 4 = 0. 1 (iii) (c) Express in modulus-argument form. Hence, or otherwise, find a real quadratic factor of the polynomial z4 + 4. 2 3 Sketch the region in the complex plane where the inequalities z 1 i < 2 and 0 < arg( z 1 i ) < 4 hold simultaneously. (d) By applying de Moivre s theorem and by also expanding (cos + i sin )5, express cos 5 as a polynomial in cos . (e) Suppose that the complex number z lies on the unit circle, and 0 arg( z ) Prove that 2 arg(z + 1) = arg(z). 3 . 2 3 2 Marks Question 3 (15 marks) Use a SEPARATE writing booklet. (a) The diagram shows the graph of y = f (x). y 2 1 O 1 x Draw separate one-third page sketches of the graphs of the following: 1 f ( x) (i) (ii) y = f ( x) + f ( x) 2 (iii) y = ( f (x))2 1 (iv) (b) y= y = e f (x). 2 2 Find the eccentricity, foci and the equations of the directrices of the ellipse x 2 y2 + =1. 9 4 Question 3 continues on page 5 4 3 Marks Question 3 (continued) (c) The region bounded by the curve y = (x 1)(3 x) and the x-axis is rotated about the line x = 3 to form a solid. When the region is rotated, the horizontal line segment l at height y sweeps out an annulus. y l 1 O 3 x (i) Show that the area of the annulus at height y is given by 4 1 y . 3 (ii) Find the volume of the solid. 2 End of Question 3 Please turn over 5 Marks Question 4 (15 marks) Use a SEPARATE writing booklet. (a) A particle P of mass m moves with constant angular velocity on a circle of radius r. Its position at time t is given by: x = r cos y = r sin , where = t. (i) Show that there is an inward radial force of magnitude mr 2 acting on P. 3 (ii) A telecommunications satellite, of mass m, orbits Earth with constant 1 angular velocity at a distance r from the centre of Earth. The Am gravitational force exerted by Earth on the satellite is , where A is a r2 constant. By considering all other forces on the satellite to be negligible, show that r=3 (b) (i) A 2 . Derive the equation of the tangent to the hyperbola x2 a2 y2 b2 2 =1 at the point P (a sec , b tan ). (ii) Show that the tangent intersects the asymptotes of the hyperbola at the points a cos , b cos A 1 sin 1 sin (iii) and a cos , b cos B . 1 + sin 1 + sin Prove that the area of the triangle OAB is ab. Question 4 continues on page 7 6 2 4 Marks Question 4 (continued) (c) A hall has n doors. Suppose that n people each choose any door at random to enter the hall. (i) In how many ways can this be done? 1 (ii) What is the probability that at least one door will not be chosen by any of the people? 2 End of Question 4 Please turn over 7 Marks Question 5 (15 marks) Use a SEPARATE writing booklet. (a) Let , and be the three roots of x3 + px + q = 0, and define sn by sn = n + n + n for n = 1, 2, 3, (i) Explain why s1 = 0, and show that s2 = 2p and s3 = 3q. 3 (ii) Prove that for n > 3 2 sn = psn 2 qsn 3 . (iii) 2 Deduce that 5 + 5 + 5 2 + 2 + 2 3 + 3 + 3 . = 5 2 3 Question 5 continues on page 9 8 Marks Question 5 (continued) (b) A particle of mass m is thrown from the top, O, of a very tall building with an initial velocity u at an angle to the horizontal. The particle experiences the effect of gravity, and a resistance proportional to its velocity in both the horizontal and vertical directions. The equations of motion in the horizontal and vertical directions are given respectively by x = kx and = ky g , y where k is a constant and the acceleration due to gravity is g. (You are NOT required to show these.) y u x O (i) (ii) Derive the result x = ue kt cos from the relevant equation of motion. ( ) 1 (ku sin + g)e kt g satisfies the appropriate equation k of motion and initial condition. Verify that y = 2 2 (iii) Find the value of t when the particle reaches its maximum height. 2 (iv) What is the limiting value of the horizontal displacement of the particle? 2 End of Question 5 Please turn over 9 BLANK PAGE 10 Marks Question 6 (15 marks) Use a SEPARATE writing booklet. (b) (i) Prove the identity cos ( a + b) x + cos ( a b) x = 2 cos ax cos bx . 1 (ii) (a) Hence find cos 3 x cos 2 x dx . 2 A sequence sn is defined by s1 = 1, s2 = 2 and, for n > 2, sn = sn 1 + (n 1)sn 2 . (i) (ii) Prove that (iii) (c) Find s3 and s4. Prove by induction that sn n! for all integers n 1. 3 Let x and y be real numbers such that x 0 and y 0. 1 (i) Prove that (ii) 1 x + x x ( x + 1) for all real numbers x 0. 2 x+y xy . 2 Suppose that a, b, c are real numbers. 2 Prove that a4 + b4 + c4 a2b2 + a2c2 + b2c2. (iii) Show that a2b2 + a2c2 + b2c2 a2bc + b2ac + c2ab. 2 (iv) Deduce that if a + b + c = d, then a4 + b4 + c4 abcd. 1 11 Marks Question 7 (15 marks) Use a SEPARATE writing booklet. (a) ( ) The region bounded by 0 x 3 , 0 y x 3 x 2 is rotated about the y-axis to form a solid. 3 Use the method of cylindrical shells to find the volume of the solid. (b) S 1 A B P 2 T Two circles 1 and 2 intersect at the points A and B. Let P be a point on AB produced and let PS and PT be tangents to 1 and 2 respectively, as shown in the diagram. Copy or trace the diagram into your writing booklet. (i) Prove that ASP SBP . (ii) Hence, prove that SP 2 = AP BP and deduce that PT = PS. 2 (iii) The perpendicular to SP drawn from S meets the bisector of SPT at D. Prove that DT passes through the centre of 2 . 3 2 Question 7 continues on page 13 12 Marks Question 7 (continued) (c) Suppose that is a real number with 0 < < . Let Pn = cos cos cos K cos n . 2 4 8 2 (i) 1 Show that Pn sin n = Pn 1 sin n 1 . 2 2 2 (ii) Deduce that Pn = (iii) Given that sin x < x for x > 0, show that 2 sin . n 2 sin n 2 sin cos cos cos K cos n 2 4 8 2 End of Question 7 Please turn over 13 1 2 < . Marks Question 8 (15 marks) Use a SEPARATE writing booklet. (a) Suppose that 3 = 1, 1, and k is a positive integer. (i) Find the two possible values of 1 + k + 2k. 2 (ii) Use the binomial theorem to expand (1 + )n and (1 + 2)n, where n is a positive integer. 1 (iii) Let l be the largest integer such that 3l n. 2 Deduce that n n n n 1 n n 2 n + + + K + = 2 + (1 + ) + 1 + . 0 3 6 3l 3 ( (iv) If n is a multiple of 6, prove that 2 n n n n 1 n + + +K+ = 2 + 2 . 0 3 6 n 3 ( Question 8 continues on page 15 14 ) ) Marks Question 8 (continued) (b) Suppose that could be written in the form p ,where p and q are positive integers. q Define the family of integrals In for n = 0, 1, 2, by n 2 q 2 In = x 2 cos x dx . n! 4 2 2n You are given that I0 = 2 and I1 = 4q 2. (Do NOT prove this.) (i) Use integration by parts twice to show that, for n 2, 2 2q2n 2 In = x2 (n 1)! 4 2 (ii) n 1 2 2 4q2n 2 x2 cos x dx x (n 2)! 4 3 n 2 cos x dx . 2 2 2 x 2 where appropriate, deduce that By writing x as 4 4 2 1 In = ( 4 n 2)q 2 In 1 p2 q 2 In 2 , for n 2. (iii) Explain briefly why In is an integer for n = 0, 1, 2, 1 (iv) Prove that 2 p p 2n 1 0 < In < for n = 0, 1, 2, q 2 n! (v) Given that p p 2n 1 < 1 , if n is sufficiently large, deduce that is q 2 n! irrational. End of paper 15 1 STANDARD INTEGRALS n x dx = 1 x dx = ln x, x > 0 ax e dx = 1 ax e , a 0 a cos ax dx = 1 sin ax, a 0 a sin ax dx 1 = cos ax, a 0 a 2 sec ax dx = 1 tan ax, a 0 a sec ax tan ax dx = 1 sec ax, a 0 a 1 dx 2 a + x2 = 1 x tan 1 , a 0 a a 1 dx 2 a x2 x = sin 1 , a > 0, a < x < a a 1 dx 2 x a2 = ln x + x 2 a 2 , x > a > 0 1 dx 2 x + a2 = ln x + x 2 + a 2 1 n +1 x , n 1; x 0, if n < 0 n +1 ( ) ( ) NOTE : ln x = loge x, x>0 16 Board of Studies NSW 2003

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Additional Info : New South Wales Higher School Certificate Mathematics Extension-2 2003
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