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NSW HSC 2010 : MATHEMATICS EXTENSION-2

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2010 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 3380 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. Question 1 (15 marks) Use a SEPARATE writing booklet. (a) x Find dx . 1 + 3x 2 (b) 4 Evaluate tan x dx . 0 3 (c) 1 dx . Find x x2 + 1 3 (d) x 2 dx Using the substitution t = tan , or otherwise, evaluate . 2 0 1 + sin x 4 (e) dx Find . 1+ x 3 2 ( ) 3 Question 2 (15 marks) Use a SEPARATE writing booklet. (a) Let z = 5 i. (i) Find z 2 in the form x + i y. 1 (ii) Find z + 2 z in the form x + i y. 1 (iii) Find i in the form x + i y. z 2 (c) (i) Express 3 i in modulus argument form. (ii) (b) Show that 3 i ( )6 is a real number. Sketch the region in the complex plane where the inequalities 1 z 2 and 0 z + z 3 hold simultaneously. Question 2 continues on page 5 4 2 2 2 Question 2 (continued) (d) Let z = cos + i sin where 0 < < . 2 On the Argand diagram the point A represents z, the point B represents z 2 and the point C represents z + z 2 . C B A O Copy or trace the diagram into your writing booklet. 1 (i) Explain why the parallelogram OACB is a rhombus. (ii) Show that arg z + z 2 = 3 . 2 1 (iii) Show that z + z 2 = 2 cos . 2 2 (iv) ( ) By considering the real part of z + z 2 , or otherwise, deduce that 3 cos + cos 2 = 2 cos cos . 2 2 End of Question 2 5 1 Question 3 (15 marks) Use a SEPARATE writing booklet. (a) (i) 1 Sketch the graph y = x 2 + 4x . (ii) Sketch the graph y = 1 x2 + 4 x 2 . (b) The region shaded in the diagram is bounded by the x-axis and the curve y = 2x x 2. 4 y y = 2x x 2 O 2 4 x The shaded region is rotated about the line x = 4. Find the volume generated. (c) Two identical biased coins are each more likely to land showing heads than showing tails. The two coins are tossed together, and the outcome is recorded. After a large number of trials it is observed that the probability that the two coins land showing a head and a tail is 0.48. What is the probability that both coins land showing heads? Question 3 continues on page 7 6 2 Question 3 (continued) (d) The diagram shows the rectangular hyperbola x y = c 2, with c > 0. The points , and are points on the hyperbola, with t 1. (i) The line l1 is the line through R perpendicular to QA. 2 Show that the equation of l1 is . (ii) The line l2 is the line through Q perpendicular to RA. 1 Write down the equation of l2 . (iii) Let P be the point of intersection of the lines l1 and l2 . Show that P is the point (iv) 2 . Give a geometric description of the locus of P. End of Question 3 7 1 Question 4 (15 marks) Use a SEPARATE writing booklet. (a) (i) Use implicit differentiation to find (ii) (b) dy . dx 2 x + y = 1. Sketch the curve (iii) 2 x + y = 1. A curve is defined implicitly by Sketch the curve x+ 1 y = 1. A bend in a highway is part of a circle of radius r, centre O. Around the bend the highway is banked at an angle to the horizontal. A car is travelling around the bend at a constant speed v. Assume that the car is represented by a point P of mass m. The forces acting on the car are a lateral force F, the gravitational force mg and a normal reaction N to the road, as shown in the diagram. N F P r O NOT TO SCALE mg (i) By resolving forces, show that F = m g s in mv 2 cos . r 3 (ii) Find an expression for v such that the lateral force F is zero. 1 Question 4 continues on page 9 8 Question 4 (continued) (c) Let k be a real number, k 4 . Show that, for every positive real number b, there is a positive real number a 11 k such that . += a b a+b (d) 3 A group of 12 people is to be divided into discussion groups. (i) In how many ways can the discussion groups be formed if there are 8 people in one group, and 4 people in another? 1 (ii) In how many ways can the discussion groups be formed if there are 3 groups containing 4 people each? 2 End of Question 4 9 Question 5 (15 marks) Use a SEPARATE writing booklet. (a) The diagram shows two circles, C1 and C2, centred at the origin with radii a and b, where a > b . The point A lies on C1 and has coordinates (a cos , a sin ). The point B is the intersection of OA and C2. The point P is the intersection of the horizontal line through B and the vertical line through A . y A (a cos , a sin ) B P O b a x C2 C1 (i) 1 Write down the coordinates of B . x2 + y2 1 = 1. (ii) Show that P lies on the ellipse (iii) Find the equation of the tangent to the ellipse (iv) Assume that A is not on the y-axis. a2 b2 x2 a2 + y2 b2 = 1 at P. Show that the tangent to the circle C1 at A, and the tangent to the ellipse x2 a2 + y2 b2 = 1 at P, intersect at a point on the x-axis. Question 5 continues on page 11 10 2 2 Question 5 (continued) (b) 2 Show that dy y = ln +c 1 y y (1 y ) for some constant c, where 0 < y < 1. (c) A TV channel has estimated that if it spends $x on advertising a particular program it will attract a proportion y (x) of the potential audience for the program, where dy = a y (1 y ) dx and a > 0 is a given constant. dy 1 has its maximum value when y = . dx 2 (i) Explain why (ii) Using part (b), or otherwise, deduce that y( x) = 1 3 1 ke ax +1 for some constant k > 0. (iii) The TV channel knows that if it spends no money on advertising the program then the audience will be one-tenth of the potential audience. 1 Find the value of the constant k referred to in part (c) (ii). (iv) What feature of the graph y = 1 ke ax part (c) (i)? (v) Sketch the graph y = 1 ke ax +1 +1 . End of Question 5 11 is determined by the result in 1 1 Question 6 (15 marks) Use a SEPARATE writing booklet. (a) The diagram shows the frustum of a right square pyramid. (A frustum of a pyramid is a pyramid with its top cut off.) The height of the frustum is h m. Its base is a square of side a m, and its top is a square of side b m (with a > b > 0 ). b h s x a A horizontal cross-section of the frustum, taken at height x m, is a square of side s m, shown shaded in the diagram. (a b) x 2 (i) (ii) (b) Show that s = a Find the volume of the frustum. h . 2 3 A sequence an is defined by an = 2 a n 1 + a n 2 , for n 2 , with a0 = a1 = 2 . Use mathematical induction to prove that ( an = 1 + 2 ) + (1 2 ) n n for all n 0 . Question 6 continues on page 13 12 Question 6 (continued) (c) (i) Expand ( cos + i s in ) (ii) Expand ( cos + i s in ) show that 5 5 using the binomial theorem. 1 using de Moivre s theorem, and hence 3 s in 5 = 1 6 s in5 2 0 sin3 + 5 s in . (iii) Deduce that x = sin is one of the solutions to 10 1 16 x 5 20 x 3 + 5x 1 = 0 . (iv) Find the polynomial p (x) such that (x 1) p (x) = 16 x 5 20 x 3 + 5x 1 . 1 (v) Find the value of a such that p (x) = (4x2 + ax 1)2. 1 (vi) Hence find an exact value for s in . 10 1 End of Question 6 13 Question 7 (15 marks) Use a SEPARATE writing booklet. (a) In the diagram ABCD is a cyclic quadrilateral. The point K is on AC such that ADK = CDB, and hence ADK is similar to BDC. A D K B C Copy or trace the diagram into your writing booklet. (i) Show that ADB is similar to KDC . 2 (ii) Using the fact that AC = AK + KC, show that BD AC = AD BC + AB DC . 2 (iii) A regular pentagon of side length 1 is inscribed in a circle, as shown in the diagram. 2 1 x Let x be the length of a chord in the pentagon. Use the result in part (ii) to show that x = 1+ 5 . 2 Question 7 continues on page 15 14 Question 7 (continued) (b) The graphs of y = 3x 1 and y = 2 x intersect at (1, 2) and at (3, 8) . 1 Using these graphs, or otherwise, show that 2 x 3x 1 for x 3. (c) Let P ( x ) = ( n 1) x n nx n 1 + 1, where n is an odd integer, n 3. (i) Show that P ( x ) has exactly two stationary points. 1 (ii) Show that P ( x ) has a double zero at x = 1. 1 (iii) Use the graph y = P ( x ) to explain why P ( x ) has exactly one real zero other than 1. 2 (iv) Let be the real zero of P ( x ) other than 1. 2 1 Using part (b), or otherwise, show that 1 < . 2 (v) Deduce that each of the zeros of 4 x 5 5 x 4 + 1 has modulus less than or equal to 1. End of Question 7 15 2 Question 8 (15 marks) Use a SEPARATE writing booklet. Let 2 2 An = cos2 n x d x and Bn = x 2 cos 2n x d x , 0 0 where n is an integer, n 0 . (Note that An > 0 , Bn > 0 .) 2n 1 An 1 for n 1. 2 2 (a) Show that n An = (b) Using integration by parts on An , or otherwise, show that 1 2 An = 2n x s in x cos2 n 1 x d x for n 1. 0 (c) Use integration by parts on the integral in part (b) to show that An = n2 (d) n2 n Show that k =1 (f) n 1 n 2 Bn for n 1. 1 Use parts (a) and (c) to show that 1 (e) ( 2n 1) B 1 k2 = B B = 2 n 1 n for n 1. An An 1 B 2 2 n. 6 An Use the fact that s in x 3 2 2 x for 0 x to show that 2 n 2 4x2 2 Bn x 1 2 d x . 0 Question 8 continues on page 17 16 1 Question 8 (continued) (g) n n +1 2 4x2 4x2 2 2 2 1 dx . Show that x 1 2 d x = 8 ( n + 1) 2 0 0 1 (h) From parts (f) and (g) it follows that 2 2 Bn 8 ( n + 1) 0 2 Use the substitution x = 4x2 1 2 n +1 dx . s in t in this inequality to show that 2 2 3 3 Bn cos 2 n + 3 t dt A. 1 6 ( n + 1) 0 16 ( n + 1) n (i) 1 Use part (e) to deduce that 2 3 6 8 ( n + 1) n (j) What is lim n k =1 1 k2 n k =1 1 k2 < 2 . 6 1 ? End of paper 17 BLANK PAGE 18 BLANK PAGE 19 STANDARD INTEGRALS x n dx = 1 dx x = ln x , x > 0 e ax dx = 1 ax e , a 0 a cos ax d x = 1 s in ax , a 0 a s in a x d x 1 = cos ax , a 0 a s ec 2 a x d x = 1 tan ax , a 0 a s ec ax tan ax d x = 1 s ec ax , a 0 a 1 a +x 2 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 x 2 + a2 ( ) ( ) NOTE : ln x = loge x , x>0 20 Board of Studies NSW 2010

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Additional Info : New South Wales Higher School Certificate Mathematics Extension-2 2010
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