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NSW HSC 2009 : MATHEMATICS EXTENSION-2

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2009 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. Question 1 (15 marks) Use a SEPARATE writing booklet. (a) ln x Find . d x x 2 (b) . Find x e 2 x d x 2 (c) x2 d x Find . 1 + 4 x 2 3 (d) 5 x 6 Evaluate dx . 2 3 4 2 x + x 4 (e) 1 d x . Evaluate 1 x 2 1 + x 2 3 4 2 Question 2 (15 marks) Use a SEPARATE writing booklet. (a) Write i 9 in the form a + ib where a and b are real. (b) Write (c) The points P and Q on the Argand diagram represent the complex numbers z and w respectively. 1 2 + 3i in the form a + ib where a and b are real. 2+i 1 P (z) Q (w) Copy the diagram into your writing booklet, and mark on it the following points: (i) the point S representing w 1 (iii) (e) 1 (ii) (d) the point R representing i z the point T representing z + w . 1 Sketch the region in the complex plane where the inequalities z 1 2 and hold simultaneously. arg ( z 1) 4 4 2 Find all the 5th roots of 1 in modulus-argument form. 2 (ii) (f) (i) Sketch the 5th roots of 1 on an Argand diagram. 1 (i) Find the square roots of 3 + 4 i . 3 (ii) Hence, or otherwise, solve the equation 2 z 2 + iz 1 i = 0. 3 Question 3 (15 marks) Use a SEPARATE writing booklet. (a) The diagram shows the graph y = ( x ) . y 1 O x 4 3 Draw separate one-third page sketches of the graphs of the following: 1 (x) (i) y= (ii) y = ( x ) (iii) y = x2 . 2 2 2 () 2 (b) Find the coordinates of the points where the tangent to the curve x 2 + 2 xy + 3y 2 = 18 is horizontal. 3 (c) Let P ( x ) = x 3 + ax 2 + bx + 5, where a and b are real numbers. 3 Find the values of a and b given that ( x 1)2 is a factor of P ( x ). Question 3 continues on page 5 4 Question 3 (continued) The diagram shows the region enclosed by the curves y = x + 1 and y = ( x 1)2 . x+ 1 y y= (d) O y = ( x 1)2 1 x The region is rotated about the y-axis. Use the method of cylindrical shells to find the volume of the solid formed. End of Question 3 5 3 Question 4 (15 marks) Use a SEPARATE writing booklet. (a) x2 y2 = 1 has foci S ( ae , 0 ) and S ( ae , 0 ) where e is the a2 b2 a a eccentricity, with corresponding directrices x = and x = . The point e e The ellipse + P ( x 0 , y0 ) is on the ellipse. The points where the horizontal line through P meets the directrices are M and M , as shown in the diagram. x= y a e x= P M (i) M S N a e S x Show that the equation of the normal to the ellipse at the point P is y y0 = a 2 y0 b 2 x0 2 ( x x0 ) . (ii) The normal at P meets the x-axis at N. Show that N has coordinates ( e 2x 0 , 0 ) . 2 (iii) Using the focus-directrix definition of an ellipse, or otherwise, show that 2 PS NS . = PS NS (iv) Let = S PN and = NPS . By applying the sine rule to rS PN and to rNPS, show that = . Question 4 continues on page 7 6 2 Question 4 (continued) (b) A light string is attached to the vertex of a smooth vertical cone. A particle P of mass m is attached to the string as shown in the diagram. The particle remains in contact with the cone and rotates with constant angular velocity on a circle of radius r . The string and the surface of the cone make an angle of with the vertical, as shown. T N r P mg The forces acting on the particle are the tension, T , in the string, the normal reaction, N , to the cone and the gravitational force mg . (i) Resolve the forces on P in the horizontal and vertical directions. (ii) Show that T = m g cos + r 2 sin ( ) and find a similar expression 2 2 for N . (iii) Show that if T = N then 2 = (iv) 2 g tan 1 . r tan + 1 For which values of can the particle rotate so that T = N ? End of Question 4 7 1 Question 5 (15 marks) Use a SEPARATE writing booklet. (a) In the diagram AB is the diameter of the circle. The chords AC and BD intersect at X . The point Y lies on AB such that XY is perpendicular to AB . The point K is the intersection of AD produced and YX produced. K D X A Y C B Copy or trace the diagram into your writing booklet. (i) Show that AKY = ABD . 2 (ii) Show that CKDX is a cyclic quadrilateral. 2 (iii) Show that B, C and K are collinear. 2 Question 5 continues on page 9 8 Question 5 (continued) (b) For each integer n 0, let 1 2 I n = x 2 n +1 e x d x . 0 (i) Show that for n 1 2 In = (ii) (c) e nI n 1. 2 Hence, or otherwise, calculate I2 . Let ( x ) = 2 e x e x x . 2 (i) Show that ( x ) > 0 for all x > 0. 2 (ii) Hence, or otherwise, show that ( x ) > 0 for all x > 0. 2 (iii) Hence, or otherwise, show that e x e x > x for all x > 0. 2 End of Question 5 9 1 Question 6 (15 marks) Use a SEPARATE writing booklet. (a) The base of a solid is the region enclosed by the parabola x = 4 y 2 and the y-axis. The top of the solid is formed by a plane inclined at 45 to the x y -plane. Each vertical cross-section of the solid parallel to the y -axis is a rectangle. A typical cross-section is shown shaded in the diagram. 3 y 4 2 45 O 2 4 x x = 4 y2 Find the volume of the solid. (b) Let P ( x ) = x 3 + qx 2 + qx + 1, where q is real. One zero of P ( x ) is 1. 1 is a zero of P ( x ) . (i) Show that if is a zero of P ( x ) then (ii) Suppose that is a zero of P ( x ) and is not real. (1) Show that = 1. (2) Show that Re ( ) = 1 2 1 q . 2 Question 6 continues on page 11 10 2 Question 6 (continued) (c) The diagram shows a circle of radius r , centred at the origin, O . The line PQ is tangent to the circle at Q , the line PR is horizontal, and R lies on the line x = c . y R P (x, y) Q r c O x (i) Find the length of PQ in terms of x , y and r . 1 (ii) The point P moves such that PQ = PR . 2 Show that the equation of the locus of P is y 2 = r 2 + c 2 2 cx . (iii) Find the focus, S , of the parabola in part (ii). 2 (iv) Show that the difference between the length PS and the length PQ is independent of x . 2 End of Question 6 11 Question 7 (15 marks) Use a SEPARATE writing booklet. (a) A bungee jumper of height 2 m falls from a bridge which is 125 m above the surface of the water, as shown in the diagram. The jumper s feet are tied to an elastic cord of length L m. The displacement of the jumper s feet, measured downwards from the bridge, is x m. x=0 Lm x=L 2m 125 125 m NOT TO SCALE x = 125 x The jumper s fall can be examined in two stages. In the first stage of the fall, where 0 x L , the jumper falls a distance of L m subject to air resistance, and the cord does not provide resistance to the motion. In the second stage of the fall, where x > L , the cord stretches and provides additional resistance to the downward motion. (i) The equation of motion for the jumper in the first stage of the fall is xx x = g rv where g is the acceleration due to gravity, r is a positive constant, and v is the velocity of the jumper. (1) Given that x = 0 and v = 0 initially, show that x= 3 v g g ln r . 2 r g r v (2) Given that g = 9.8 m s 2 and r = 0.2 s 1, find the length, L , of the cord such that the jumper s velocity is 30 m s 1 when x = L . Give your answer to two significant figures. 1 (ii) In the second stage of the fall, where x > L , the displacement x is given by 4 x=e t 10 ( 29sin t 10 cos t ) + 92 where t is the time in seconds after the jumper s feet pass x = L . Determine whether or not the jumper s head stays out of the water. Question 7 continues on page 13 12 Question 7 (continued) (b) Let z = cos + i sin . (i) Show that z n + z n = 2 cos n , where n is a positive integer. 2 (ii) Let m be a positive integer. Show that 3 ( 2 cos ) (iii) 2m 2m 2m = 2 cos 2m + cos ( 2m 2 ) + cos ( 2m 4 ) 1 2 2m 2m +.+ cos 2 + m . m 1 Hence, or otherwise, prove that 2 2m 2m cos d = 2 m +1 m 2 0 where m is a positive integer. End of Question 7 13 2 Question 8 (15 marks) Use a SEPARATE writing booklet. (a) (i) Using the substitution t = tan , or otherwise, show that 2 cot + (ii) r =1 (iii) 1 2 r 1 tan x 2 r = 1 2 n 1 cot x 2n 3 2 cot x . Show that 2 n n lim r =1 (iv) 1 1 tan = cot . 2 22 2 Use mathematical induction to prove that, for integers n 1, n 2 1 2r 1 tan x 2r = 2 x 2 cot x . Hence find the exact value of tan 2 1 1 + tan + tan + .. 42 84 16 Question 8 continues on page 15 14 Question 8 (continued) (b) 2 y y= O n 1 1 x n x Let n be a positive integer greater than 1. 1 from x = n 1 to x = n is x between the areas of two rectangles, as shown in the diagram. The area of the region under the curve y = Show that e (c) n n 1 < 1 1 n n < e 1 . A game is being played by n people, A1, A2, ..., An , sitting around a table. Each person has a card with their own name on it and all the cards are placed in a box in the middle of the table. Each person in turn, starting with A1, draws a card at random from the box. If the person draws their own card, they win the game and the game ends. Otherwise, the card is returned to the box and the next person draws a card at random. The game continues until someone wins. Let W be the probability that A1 wins the game. Let p = 1 1 and q = 1 . n n (i) Show that W = p + q n W . 1 (ii) Let m be a fixed positive integer and let Wm be the probability that A1 wins in no more than m attempts. 3 Use part (b) to show that, if n is large, to 1 e m . End of paper 15 Wm W is approximately equal 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 sin ax , a 0 a sin ax d x 1 = cos ax , a 0 a sec 2 a x d x = 1 tan a x , a 0 a sec ax tan a x d x = 1 sec ax , a 0 a 1 a +x 2 2 1 x +a x = sin 1 , a > 0 , a < x < a a = ln x + x 2 a 2 , x > a > 0 = ln x + x 2 + a 2 2 1 2 dx dx 1 x a 1 x tan 1 , a 0 a a dx a2 x 2 2 = dx 1 n +1 x , n 1; x 0 , if n < 0 n +1 2 ( ) ( ) NOTE : ln x = loge x , x>0 16 Board of Studies NSW 2009

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Additional Info : New South Wales Higher School Certificate Mathematics Extension-2 2009
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