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NSW HSC 2004 : MATHEMATICS EXTENSION-2

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2004 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. (a) Use integration by parts to find xe3 x dx . 2 4 (b) sin x Evaluate 3 dx . 0 cos x 3 (c) dx By completing the square, find . 5 + 4x x2 2 (d) (i) 2 Find real numbers a and b such that x2 7x + 4 a b 1 . + 2 ( x + 1)( x 1) x + 1 x 1 ( x 1)2 (ii) x2 7x + 4 Hence find 2 dx . ( x + 1)( x 1) 2 1 (e) Use the substitution x = 2sin to find 0 2 x2 4 x2 dx . 4 Marks Question 2 (15 marks) Use a SEPARATE writing booklet. (a) Let z = 1 + 2i and w = 3 i. Find, in the form x + iy, (i) 1 (ii) (b) zw 10 . z 1 Let = 1 + i 3 and = 1 + i. (i) Find , in the form x + iy. 1 (ii) Express in modulus-argument form. 2 (iii) Given that has the modulus-argument form 1 = 2 cos + i sin 4 4 (iv) (c) find the modulus-argument form of . Hence find the exact value of sin . 12 Sketch the region in the complex plane where the inequalities z + z 1 and z i 1 hold simultaneously. Question 2 continues on page 4 3 1 3 Marks Question 2 (continued) (d) The diagram shows two distinct points A and B that represent the complex numbers z and w respectively. The points A and B lie on the circle of radius r centred at O. The point C representing the complex number z + w also lies on this circle. C B A O Copy the diagram into your writing booklet. (i) Using the fact that C lies on the circle, show geometrically that AOB = 2 2 . 3 (ii) Hence show that z3 = w3. 2 (iii) Show that z 2 + w 2 + zw = 0. 1 End of Question 2 4 Marks Question 3 (15 marks) Use a SEPARATE writing booklet. 4x2 showing all asymptotes. x2 9 (a) Sketch the curve y = (b) The diagram shows the graph of y = f (x). 3 y 4 y = f(x) 5 5 x 2 Draw separate one-third page sketches of the graphs of the following: (i) (ii) y = ( f ( x )) (iii) (c) y = f ( x) y= 2 2 1 f ( x) 2 2 . Find the equation of the tangent to the curve defined by x 2 xy + y3 = 5 at the point (2, 1). Question 3 continues on page 6 5 3 Marks Question 3 (continued) (d) The base of a solid is the region in the xy plane enclosed by the curves y = x4, y = x4 and the line x = 2. Each cross-section perpendicular to the x-axis is an equilateral triangle. y y = x4 O 2 x y = x 4 (i) Show that the area of the triangular cross-section at x = h is (ii) Hence find the volume of the solid. End of Question 3 6 3 h8 . 1 2 Marks Question 4 (15 marks) Use a SEPARATE writing booklet. (a) Let , , and be the zeros of the polynomial p(x) = 3x 3 + 7x 2 + 11x + 51. (i) 1 (ii) Find 2 + 2 + 2. 2 (iii) (b) Find 2 + 2 + 2. Using part (ii), or otherwise, determine how many of the zeros of p(x) are real. Justify your answer. 1 The vertices of an acute-angled triangle ABC lie on a circle. The perpendiculars from A, B and C meet BC, AC and AB at D, E and F respectively. These perpendiculars meet at H. The perpendiculars AD, BE and CF are produced to meet the circle at K, L and M respectively. A L E M H F B C D K (i) Prove that AHE = DCE. 2 (ii) Deduce that AH = AL. 1 (iii) State a similar result for triangle AMH. 1 (iv) Show that the length of the arc BKC is half the length of the arc MKL. 2 Question 4 continues on page 8 7 Marks Question 4 (continued) (c) y P S x O T Q x 2 y2 + = 1 . The chord through P and the focus a2 b2 S(ae, 0) meets the ellipse at Q. The tangents to the ellipse at P and Q meet at the xx yy point T(x0, y0), so the equation of PQ is 20 + 20 = 1 . (Do NOT prove this.) a b The point P lies on the ellipse (i) Using the equation of PQ, show that T lies on the directrix. 1 The point P is now chosen so that T also lies on the x-axis. PS ? ST 2 (ii) What is the value of the ratio (iii) Show that PTQ is less than a right angle. 1 (iv) 1 Show that the area of triangle PQT is b 2 e . e 1 End of Question 4 8 Marks Question 5 (15 marks) Use a SEPARATE writing booklet. Let a > 0. Find the points where the line y = ax and the curve y = x(x a) intersect. 1 Let R be the region in the plane for which x(x a) y ax. Sketch R. 1 (iii) (b) (i) (ii) (a) A solid is formed by rotating the region R about the line x = 2a. Use the method of cylindrical shells to find the volume of the solid. 4 (i) In how many ways can n students be placed in two distinct rooms so that neither room is empty? 1 (ii) In how many ways can five students be placed in three distinct rooms so that no room is empty? 2 Question 5 continues on page 10 9 Marks Question 5 (continued) (c) A smooth sphere with centre O and radius R is rotating about its vertical diameter at a uniform angular velocity, radians per second. A marble is free to roll around the inside of the sphere. R O N r P mg Assume that the marble can be considered as a point P which is acted upon by gravity and the normal reaction force N from the sphere. The marble describes a horizontal circle of radius r with the same uniform angular velocity, radians per second. Let the angle between OP and the vertical diameter be . (i) Explain why mr 2 = N sin and mg = N cos . (ii) Show that either cos = (iii) Hence, or otherwise, show that if 0 then > 2 g or = 0. R 2 End of Question 5 10 3 g . R 1 Marks Question 6 (15 marks) Use a SEPARATE writing booklet. (a) (i) 2 Show that sin x dx = . 2 2 0 1 + cos x (ii) By making the substitution x = u, find 3 x sin x dx . 0 1 + cos2 x (b) A particle is released from the origin O with an initial velocity of A ms 1 directed vertically downward. The particle is subject to a constant gravitational force and a resistance which is proportional to the velocity, v ms 1, of the particle. Let x be the displacement in metres of the particle below O at time t seconds after the release of the particle, so that the equation of motion is x = g kv , where g ms 2 is the acceleration due to gravity. (i) The terminal velocity of the particle is B ms 1. Show that k = (ii) Verify that v satisfies the equation (iii) Hence show that the velocity of the particle is given by () d ve kt = ge kt . dt v = B ( B A)e (iv) 1 2 2 gt B. gt B Deduce that x = Bt ( B A) 1 e B . g Question 6 continues on page 12 11 g . B 2 Marks Question 6 (continued) At the same time as the particle is released from O, an identical particle is released from the point P which is h metres below O. The second particle has an initial velocity of A ms 1 directed vertically upward. gt B Its displacement below O is given by x = h + Bt ( B + A) 1 e B . g (Do NOT prove this.) (v) Suppose that the two particles meet after T seconds. Show that T= (vi) 2 B 2 AB loge . g 2 AB gh The value of A can be varied. What condition must A satisfy so that the two particles can meet? End of Question 6 12 1 Marks Question 7 (15 marks) Use a SEPARATE writing booklet. 1 2. a Let a be a positive real number. Show that a + Let n be a positive integer and a1, a2, , an be n positive real numbers. 1 1 1 + K + n2 . Prove by induction that ( a1 + a2 + K + an ) + an a1 a2 4 (iii) (b) (i) (ii) (a) Hence show that cosec2 + sec2 + cot 2 9 cos2 . 1 2 Let be a real number and suppose that z is a complex number such that z+ (i) By reducing the above equation to a quadratic equation in z, solve for z and use de Moivre s theorem to show that zn + (ii) 1 = 2 cos . z 3 1 = 2 cos n . zn 1 Let w = z + . Prove that z 2 1 1 1 w3 + w 2 2 w 2 = z + + z 2 + 2 + z3 + 3 . z z z (iii) Hence, or otherwise, find all solutions of cos + cos2 + cos3 = 0, in the range 0 2 . 13 3 Marks Question 8 (15 marks) Use a SEPARATE writing booklet. (a) 1 1 1 Let P p, and Q q, be points on the hyperbola y = with p > q > 0. Let x p q P be the point ( p, 0) and Q be the point (q, 0). The shaded region OPQ in Figure 1 is bounded by the lines OP, OQ and the hyperbola. The shaded region Q QPP in Figure 2 is bounded by the lines QQ , PP , P Q and the hyperbola. y y Figure 1 Q P O Figure 2 Q P P x O Q P x (i) Find the area of triangle OPP . 1 (ii) Prove that the area of the shaded region OPQ is equal to the area of the shaded region Q QPP . 1 1 Let M be the midpoint of the chord PQ and R r, be the intersection of the r line OM with the hyperbola. Let R be the point (r, 0), as shown in Figure 3. y Q Figure 3 M R P O R x (iii) By using similar triangles, or otherwise, prove that r 2 = pq. 2 (iv) By using integration, or otherwise, show that the line RR divides the shaded region Q QPP into two pieces of equal area. 2 (v) Deduce that the line OR divides the shaded region OPQ into two pieces of equal area. 1 Question 8 continues on page 15 14 Marks Question 8 (continued) (b) 4 Let In = tan n x dx and let Jn = ( 1)n I2 n for n = 0, 1, 2, K 0 1 . n +1 (i) Show that In + In + 2 = (ii) Deduce that Jn Jn 1 = (iii) Show that Jm = (iv) un Use the substitution u = tan x to show that In = 2 du . 0 1 + u (v) Deduce that 0 In 2 ( 1)n 2n 1 for n 1 . 1 m ( 1)n + . 4 n =1 2 n 1 2 1 1 and conclude that Jn 0 as n . n +1 End of paper 15 1 2 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 2004

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Additional Info : New South Wales Higher School Certificate Mathematics Extension-2 2004
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