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NSW HSC 2008 : MATHEMATICS EXTENSION-2

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2008 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 BLANK PAGE 2 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) x2 Find 5 + x3 (b) Find (c) Evaluate tan 1 x dx . 0 (d) dx Evaluate . x 2x 1 1 4 (e) It can be shown that 4 ( ) 2 dx . 2 dx . 4x2 + 1 2 1 3 2 ( 8 (1 x ) )( ) 2 x 2 2 2 x + x 2 = 4 2x 2 2x + x2 2x 2 x2 . (Do NOT prove this.) 8 (1 x ) Use this result to evaluate dx . 2 2 0 2 x 2 2 x + x 1 ( )( 3 ) Marks Question 2 (15 marks) Use a SEPARATE writing booklet. (a) (b) Find real numbers a and b such that (1 + 2i)(1 3i) = a + ib . (i) Write 2 1+ i 3 in the form x + iy , where x and y are real. 1+ i 2 (ii) By expressing both 1 + i 3 and 1 + i in modulus-argument form, write 3 1+ i 3 in modulus-argument form. 1+ i (iii) Hence find cos in surd form. 12 1 12 1+ i 3 (iv) By using the result of part (ii), or otherwise, calculate . 1 + i (c) The point P on the Argand diagram represents the complex number z = x + iy which satisfies z2 + z 2 = 8. Find the equation of the locus of P in terms of x and y . What type of curve is the locus? Question 2 continues on page 5 4 1 2 Marks Question 2 (continued) (d) P Q O M R S The point P on the Argand diagram represents the complex number z . The points Q and R represent the points z and z respectively, where 2 2 = cos + i sin . The point M is the midpoint of QR . 3 3 (i) Find the complex number representing M in terms of z . 2 (ii) The point S is chosen so that PQSR is a parallelogram. 2 Find the complex number represented by S . End of Question 2 5 Marks Question 3 (15 marks) Use a SEPARATE writing booklet. (a) The following diagram shows the graph of y = g (x) . y O x 1 Draw separate one-third page sketches of the graphs of the following: (i) y = g(x) 1 (ii) y= 1 g(x) 2 (iii) y = ( x ) , where 2 g ( x ) (x) = g ( 2 x ) (b) for x 1 for x < 1. Let p (z) = 1 + z 2 + z4. (i) Show that p (z) has no real zeros. 1 Let be a zero of p (z). (ii) Show that 6 = 1. 1 (iii) Show that 2 is also a zero of p (z). 1 Question 3 continues on page 7 6 Marks Question 3 (continued) (c) For n 0, let 4 I n = tan 2 n d . 0 (i) Show that for n 1, 2 In = 1 I . 2n 1 n 1 Hence, or otherwise, calculate I3 . 2 A (ii) 3 (d) P O A particle P of mass m is attached by a string of length to a point A. The particle moves with constant angular velocity in a horizontal circle with centre O which lies directly below A. The angle the string makes with OA is . The forces acting on the particle are the tension, T, in the string and the force due to gravity, mg . By resolving the forces acting on the particle in the horizontal and vertical directions, show that 2 = g . cos End of Question 3 7 Marks Question 4 (15 marks) Use a SEPARATE writing booklet. (a) K m O r L k M The diagram shows a circle, centre O and radius r, which touches all three sides of KLM. Let LM = k , MK = , and KL = m . (i) Write down an expression for the area of LOM. 1 (ii) Let P be the perimeter of KLM. Show that A, the area of KLM, is given by 1 A = Pr . 2 1 Question 4 continues on page 9 8 Marks Question 4 (continued) 3 (iii) NOT TO SCALE 8 2 A wheel of radius 2 units rests against a fence of height 8 units. A thin straight board leans against the wheel with one end at the top of the fence and the other on the ground. Using the result of part (ii), or otherwise, find how far from the foot of the fence the board touches the ground. 3 (iv) NOT TO SCALE 8 2 9 A second wheel rests on the ground, touching the board. A second thin straight board leans against the top of the fence and this second wheel. This board touches the ground 9 units further from the foot of the fence than the first board. Find the radius of the second wheel. Question 4 continues on page 10 9 Marks Question 4 (continued) (b) y P ( x1, y1 ) O x T Q ( x 2 , y2 ) The points P ( x1, y1 ) and Q ( x2, y2 ) lie on the ellipse x 2 a 2 + y2 b2 = 1. The tangents at P and Q meet at T. (i) Show that the equation of the tangent at P is (ii) Show that T lies on the line ( x1 x2 ) a2 x+ x1 a 2 x+ ( y1 y2 ) b2 y1 b2 y = 1. y = 0. 2 2 3 (iii) Let M be the midpoint of PQ. Show that O, M and T are collinear. End of Question 4 10 BLANK PAGE Please turn over 11 Marks Question 5 (15 marks) Use a SEPARATE writing booklet. (a) A model for the population, P, of elephants in Serengeti National Park is 21 000 P= t 3 7 + 3e where t is the time in years from today. (i) Show that P satisfies the differential equation 2 dP 1 P = 1 P. dt 3 3000 (ii) 1 (iii) What does the model predict that the eventual population will be? 1 (iv) (b) What is the population today? What is the annual percentage rate of growth today? 1 Let p (x) = x n + 1 (n + 1) x + n where n is a positive integer. (i) Show that p (x) has a double zero at x = 1 . 2 (ii) By considering concavity, or otherwise, show that p (x) 0 for x 0 . 1 (iii) Factorise p (x) when n = 3 . 2 Question 5 continues on page 13 12 Marks Question 5 (continued) (c) Let a and b be constants, with a > b > 0 . A torus is formed by rotating the circle (x a)2 + y 2 = b 2 about the y-axis. y h O x1 x2 x The cross-section at y = h , where b h b, is an annulus. The annulus has inner radius x1 and outer radius x 2 where x1 and x 2 are the roots of (x a)2 = b2 h 2 . (i) Find x1 and x 2 in terms of h . 1 (ii) Find the area of the cross-section at height h , in terms of h . 2 (iii) Find the volume of the torus. 2 End of Question 5 13 Marks Question 6 (15 marks) Use a SEPARATE writing booklet. (a) Let be the complex number satisfying 3 = 1 and Im ( ) > 0. The cubic polynomial, p (z) = z 3 + az 2 + bz + c , has zeros 1, and . 3 Find p (z) . (b) y P ( a sec , b tan ) R O S ( a e, 0 ) S ( a e, 0 ) x R Let P (a sec , b tan ) be a point on the hyperbola x2 y2 = 1 where a > 0 a 2 b 2 and b > 0 as shown in the diagram. The foci of the hyperbola are S and S , and is the tangent at the point P. The points R and R lie on so that SR and S R are perpendicular to . (i) Show that the line has equation 2 bx sec ay tan ab = 0. (ii) Show that S R = (iii) ab ( e sec 1) . 1 a 2 tan 2 + b 2 sec 2 Show that SR S R = b2. 3 Question 6 continues on page 15 14 Marks Question 6 (continued) (c) Suppose n and r are integers with 1 < r n . (i) 3 Show that 1 r 1 1 = . n r 1 n 1 n r 1 r r 1 (ii) Hence show that, if m is an integer with m r , then 2 1 1 1 r 1 . + + + = 1 r r 1 m r + 1 m r r r r 1 (iii) What is the limiting value of the sum m n=r 1 n r as m increases without bound? End of Question 6 15 1 Marks Question 7 (15 marks) Use a SEPARATE writing booklet. (a) An urn contains n red balls, n white balls and n blue balls. Three balls are drawn at random from the urn, one at a time, without replacement. (i) What is the probability, ps , that the three balls are all the same colour? 2 (ii) What is the probability, pd , that the three balls are all of different colours? 1 (iii) What is the probability, pm , that two balls are of one colour and the third is of a different colour? 1 (iv) If n is large, what is the approximate ratio ps : pd : pm ? 1 (b) Q b S a R P c T In the diagram, the points P, Q and R lie on a circle. The tangent at P and the secant QR intersect at T. The bisector of QPR meets QR at S so that QPS = RPS = . The intervals RS, SQ and PT have lengths a, b and c respectively. (i) Show that TSP = TPS . (ii) Hence show that 2 111 =+. abc 2 Question 7 continues on page 17 16 Marks Question 7 (continued) (c) A fishing boat drifts with a current in a straight line across a fishing ground. The boat s velocity v, at time t after the start of this drift is given by ( ) v = b b v0 e t , where v0 , b and are positive constants, and v0 < b . (i) Show that dv = (b v) . dt 1 (ii) The physical significance of v0 is that it represents the initial velocity of the boat. 1 What is the physical significance of b ? (iii) Let x be the distance travelled by the boat from the start of the drift. 3 Find x as a function of t . Hence show that x= v0 v b b v0 l o ge + . b v (iv) The initial velocity of the boat is b . 10 How far has the boat drifted when v = 1 b ? 2 End of Question 7 17 Marks Question 8 (15 marks) Use a SEPARATE writing booklet. (a) It is given that 2 cos A sin B = sin (A + B) sin (A B) . (Do NOT prove this.) 3 Prove by induction that, for integers n 1 , cos + cos3 + + cos ( 2n 1) = (b) sin 2n . 2 sin y Pn Pk Pk 1 Sk R O P0 x In the diagram, the points P0 , P1 , , Pn , are equally spaced points in the first quadrant on the circular arc of radius R and centre O. The point P0 is (R, 0), Pn is (0, R) and Pk 1 OPk = for k = 1, , n . Each of the intervals Pk 1 Pk is rotated about the y-axis to form Sk , a part of a cone. The area, Ak , of Sk is given by Ak = 2 R 2 sin cos ( 2 k 1) . 2 (Do NOT prove this.) Let S be the surface formed by all of the Sk . (i) Write down an expression for the area, A, of S. 4 By using the result of part (a), or otherwise, find an expression for A in terms of n and R only. (ii) Find the limiting value of A as n increases without bound. Question 8 continues on page 19 18 1 Marks Question 8 (continued) (c) Let ( t ) = sin ( a + nt ) sin b sin a sin ( b nt ) , where a, b and n are constants with a > 0, b > 0, a + b < and n 0. (i) Show that 3 ( t ) = n 2 ( t ) and ( 0 ) = 0. (ii) Hence, or otherwise, show that 2 ( t ) = sin ( a + b ) sin nt. (iii) Find all values of t for which 2 sin ( a + n t ) sin a . = sin ( b n t ) sin b End of paper 19 STANDARD INTEGRALS x n dx = 1 dx x = ln x , x > 0 e ax d x = 1 ax e , a 0 a cos ax dx = 1 sin ax , a 0 a sin ax dx 1 = cos ax , a 0 a sec 2 ax dx = 1 tan ax , a 0 a sec ax tan ax dx = 1 sec ax , a 0 a 1 a2 + x 2 1 1 x a 1 x tan 1 , a 0 a a dx x = sin 1 , a > 0 , a < x < a a dx a2 x 2 2 = dx 1 n +1 x , n 1; x 0 , if n < 0 n +1 = ln x + x 2 a 2 , x > a > 0 dx = ln x + x 2 + a 2 2 1 2 x 2 + a ( ) ( ) NOTE : ln x = loge x , x > 0 20 Board of Studies NSW 2008

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Additional Info : New South Wales Higher School Certificate Mathematics Extension-2 2008
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