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NSW HSC 2009 : MATHEMATICS EXTENSION-1

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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 1 General Instructions Reading time 5 minutes Working time 2 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 411 Total marks 84 Attempt Questions 1 7 All questions are of equal value Total marks 84 Attempt Questions 1 7 All questions are of equal value Answer each question in a SEPARATE writing booklet. Extra writing booklets are available. Question 1 (12 marks) Use a SEPARATE writing booklet. (a) Factorise 8x 3 + 27. 2 (b) Let ( x ) = ln ( x 3) . What is the domain of ( x ) ? 1 (c) Find lim (d) Solve the inequality (e) Differentiate x cos2 x . (f) 3 Using the substitution u = x + 1, or otherwise, evaluate x 2 e x +1 d x . 0 x 0 sin 2 x . x 1 x + 3 > 1 . 2 x 3 2 2 3 2 3 Question 2 (12 marks) Use a SEPARATE writing booklet. (a) The polynomial p ( x ) = x 3 ax + b has a remainder of 2 when divided by ( x 1) and a remainder of 5 when divided by ( x + 2) . 3 Find the values of a and b . (b) (i) Express 3 sin x + 4 cos x in the form A sin( x + ) where 0 . 2 (ii) Hence, or otherwise, solve 3 sin x + 4 cos x = 5 for 0 x 2 . Give your answer, or answers, correct to two decimal places. (c) 2 2 The diagram shows points P ( 2 t , t 2 ) and Q ( 4 t , 4 t 2 ) which move along the parabola x 2 = 4y . The tangents to the parabola at P and Q meet at R . y NOT TO SCALE x 2 = 4y Q (4 t , 4 t 2 ) R P (2 t , t 2 ) x (i) Show that the equation of the tangent at P is y = t x t 2 . 2 (ii) Write down the equation of the tangent at Q , and find the coordinates of the point R in terms of t . 2 (iii) Find the Cartesian equation of the locus of R . 1 3 Question 3 (12 marks) Use a SEPARATE writing booklet. (a) Let ( x ) = 3 + e2 x . 4 (i) Find the inverse function 1( x ) . 2 (i) On the same set of axes, sketch the graphs of y = cos 2x and y = for x . (ii) Use your graph to determine how many solutions there are to the equation 2 cos 2x = x + 1 for x . 1 (iii) (c) 1 (ii) (b) Find the range of ( x ) . One solution of the equation 2 cos 2x = x + 1 is close to x = 0.4. Use one application of Newton s method to find another approximation to this solution. Give your answer correct to three decimal places. 3 1 cos 2 provided that cos 2 1. 1 + cos 2 (i) Prove that tan 2 = (ii) Hence find the exact value of tan . 8 4 x +1 , 2 2 2 1 Question 4 (12 marks) Use a SEPARATE writing booklet. (a) A test consists of five multiple-choice questions. Each question has four alternative answers. For each question only one of the alternative answers is correct. Huong randomly selects an answer to each of the five questions. (i) What is the probability that Huong selects three correct and two incorrect answers? 2 (ii) What is the probability that Huong selects three or more correct answers? 2 (iii) What is the probability that Huong selects at least one incorrect answer? 1 (b) Consider the function ( x ) = x 4 + 3 x 2 x4 + 3 . (i) Show that ( x ) is an even function. (ii) What is the equation of the horizontal asymptote to the graph y = (x)? 1 1 (iii) Find the x-coordinates of all stationary points for the graph y = ( x ) . 3 (iv) Sketch the graph y = ( x ) . You are not required to find any points of inflexion. 2 5 Question 5 (12 marks) Use a SEPARATE writing booklet. (a) The equation of motion for a particle moving in simple harmonic motion is given by d 2x dt 2 = n2 x where n is a positive constant, x is the displacement of the particle and t is time. (i) Show that the square of the velocity of the particle is given by 3 v 2 = n 2( a 2 x 2) where v = dx and a is the amplitude of the motion. dt (ii) Find the maximum speed of the particle. 1 (iii) Find the maximum acceleration of the particle. 1 (iv) The particle is initially at the origin. Write down a formula for x as a function of t, and hence find the first time that the particle s speed is half its maximum speed. 2 Question 5 continues on page 7 6 Question 5 (continued) (b) The cross-section of a 10 metre long tank is an isosceles triangle, as shown in the diagram. The top of the tank is horizontal. 10 m 3m NOT TO SCALE hm 120 120 When the tank is full, the depth of water is 3 m. The depth of water at time t days is h metres. (i) Find the volume, V , of water in the tank when the depth of water is h metres. 1 (ii) Show that the area, A, of the top surface of the water is given by 1 A = 20 3 h . (iii) The rate of evaporation of the water is given by 2 dV = kA , dt where k is a positive constant. Find the rate at which the depth of water is changing at time t . (iv) It takes 100 days for the depth to fall from 3 m to 2 m. Find the time taken for the depth to fall from 2 m to 1 m. End of Question 5 7 1 Question 6 (12 marks) Use a SEPARATE writing booklet. (a) Two points, A and B, are on cliff tops on either side of a deep valley. Let h and R be the vertical and horizontal distances between A and B as shown in the h diagram. The angle of elevation of B from A is , so that = tan 1 . R y B h A x R At time t = 0 , projectiles are fired simultaneously from A and B . The projectile from A is aimed at B , and has initial speed U at an angle above the horizontal. The projectile from B is aimed at A and has initial speed V at an angle below the horizontal. The equations for the motion of the projectile from A are x l = Ut cos y1 = Ut sin and 12 gt , 2 and the equations for the motion of the projectile from B are x 2 = R Vt cos and y2 = h Vt sin 12 gt . 2 (Do NOT prove these equations.) (i) Let T be the time at which x1 = x 2 . Show that T = 1 R . (U + V ) cos (ii) Show that the projectiles collide. 2 (iii) If the projectiles collide on the line x = R , where 0 < < 1, show that 1 1 V = 1 U . Question 6 continues on page 9 8 Question 6 (continued) (b) (i) Sum the geometric series 3 (1 + x ) r + (1 + x )r + 1 + + (1 + x )n and hence show that r r + 1 n n + 1 . + + + = r r r r + 1 (ii) Consider a square grid with n rows and n columns of equally spaced points. y x The diagram illustrates such a grid. Several intervals of gradient 1, whose endpoints are a pair of points in the grid, are shown. (1) Explain why the number of such intervals on the line y = x is 1 n equal to . 2 (2) Explain why the total number, Sn , of such intervals in the grid is given by 1 2 3 n 1 n n 1 3 2 Sn = + + + + 2 + 2 + + 2 + 2 . 2 2 2 (iii) Using the result in part (i), show that Sn = n ( n 1) ( 2n 1) . 6 End of Question 6 9 3 Question 7 (12 marks) Use a SEPARATE writing booklet. (a) d ( x ) = 1. dx (i) Use differentiation from first principles to show that (ii) Use mathematical induction and the product rule for differentiation dn to prove that x = nx n 1 for all positive integers n. dx 1 2 () (b) A billboard of height a metres is mounted on the side of a building, with its bottom edge h metres above street level. The billboard subtends an angle at the point P , x metres from the building. a P (i) h x Use the identity tan ( A B ) = tan A tan B to show that 1 + tan A tan B 2 ax = tan 1 . 2 x + h (a + h) (ii) The maximum value of occurs when d = 0 and x is positive. dx Find the value of x for which is a maximum. Question 7 continues on page 11 10 3 Question 7 (continued) (c) Consider the billboard in part (b). There is a unique circle that passes through the top and bottom of the billboard (points Q and R respectively) and is tangent to the street at T . Let be the angle subtended by the billboard at S , the point where PQ intersects the circle. Q a S R P h T Copy the diagram into your writing booklet. (i) Show that < when P and T are different points, and hence show that is a maximum when P and T are the same point. 3 (ii) Using circle properties, find the distance of T from the building. 1 End of paper 11 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 12 Board of Studies NSW 2009

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Additional Info : New South Wales Higher School Certificate Mathematics Extension-1 2009
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