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NSW HSC 2007 : MATHEMATICS EXTENSION-1

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2007 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. Marks Question 1 (12 marks) Use a SEPARATE writing booklet. ( )3 in the form a + b (a) Write 1 + 5 (b) The interval AB, where A is (4, 5) and B is (19, 5), is divided internally in the ratio 2 : 3 by the point P (x, y). Find the values of x and y. 2 (c) Differentiate tan 1 ( x4 ) with respect to x. 2 (d) The graphs of the line x 2y + 3 = 0 and the curve y = x 3 + 1 intersect at (1, 2). Find the exact value, in radians, of the acute angle between the line and the tangent to the curve at the point of intersection. 3 5 , where a and b are integers. 4 (e) Use the substitution u = 25 x to evaluate 3 2 2 2x 25 x dx . 2 2 3 Marks Question 2 (12 marks) Use a SEPARATE writing booklet. (a) 1 cos = tan . By using the substitution t = tan , or otherwise, show that 2 sin 2 2 (b) Let ( x ) = 2 cos 1 x . (i) Sketch the graph of y = ( x ) , indicating clearly the coordinates of the endpoints of the graph. (ii) State the range of ( x ) . (c) 2 1 The polynomial P (x) = x 2 + ax + b has a zero at x = 2. When P (x) is divided by x + 1, the remainder is 18. 3 Find the values of a and b. (d) A skydiver jumps from a hot air balloon which is 2000 metres above the ground. The velocity, v metres per second, at which she is falling t seconds after jumping is given by v = 50 (1 e 0.2t ) . (i) Find her acceleration ten seconds after she jumps. Give your answer correct to one decimal place. 2 (ii) Find the distance that she has fallen in the first ten seconds. Give your answer correct to the nearest metre. 2 3 Marks Question 3 (12 marks) Use a SEPARATE writing booklet. (a) Find the volume of the solid of revolution formed when the region bounded by the curve y = 1 9+ x 2 3 , the x-axis, the y-axis and the line x = 3, is rotated about the x-axis. (b) (i) (ii) (c) x 2 Find the vertical and horizontal asymptotes of the hyperbola y = x 4 x 2 and hence sketch the graph of y = . x 4 Hence, or otherwise, find the values of x for which x 2 3 . x 4 3 2 A particle is moving in a straight line with its acceleration as a function of x given by x = e 2 x . It is initially at the origin and is travelling with a velocity of 1 metre per second. (i) Show that x = e x . 2 (ii) Hence show that x = loge ( t + 1). 2 4 Marks Question 4 (12 marks) Use a SEPARATE writing booklet. (a) In a large city, 10% of the population has green eyes. (i) What is the probability that two randomly chosen people both have green eyes? 1 (ii) What is the probability that exactly two of a group of 20 randomly chosen people have green eyes? Give your answer correct to three decimal places. 1 (iii) What is the probability that more than two of a group of 20 randomly chosen people have green eyes? Give your answer correct to two decimal places. 2 (b) Use mathematical induction to prove that 72n 1 + 5 is divisible by 12, for all integers n 1. 3 (c) A B P X NOT TO SCALE Q C D The diagram shows points A, B, C and D on a circle. The lines AC and BD are perpendicular and intersect at X. The perpendicular to AD through X meets AD at P and BC at Q . Copy or trace this diagram into your writing booklet. (i) Prove that QXB = QBX . 3 (ii) Prove that Q bisects BC . 2 5 Marks Question 5 (12 marks) Use a SEPARATE writing booklet. (a) O Q T r P The points P and Q lie on the circle with centre O and radius r. The arc PQ subtends an angle at O . The tangent at P and the line OQ intersect at T, as shown in the diagram. (i) The arc PQ divides triangle TPO into two regions of equal area. 2 Show that tan = 2 . (ii) A first approximation to the solution of the equation 2 tan = 0 is = 1.15 radians. Use one application of Newton s method to find a better approximation. Give your answer correct to four decimal places. 2 (b) Mr and Mrs Roberts and their four children go to the theatre. They are randomly allocated six adjacent seats in a single row. 2 What is the probability that the four children are allocated seats next to each other? (c) Find the exact values of x and y which satisfy the simultaneous equations sin 1 x + 3sin 1 x 1 cos 1 y = 2 3 and 1 2 cos 1 y = . 2 3 Question 5 continues on page 7 6 3 Marks Question 5 (continued) (d) y Q (2aq, aq 2 ) x 2 = 4 ay P (2ap, ap2 ) O x The diagram shows a point P (2ap, ap2 ) on the parabola x 2 = 4 ay. The normal to the parabola at P intersects the parabola again at Q (2aq, aq2 ) . The equation of PQ is x + py 2ap ap3 = 0 . (Do NOT prove this.) (i) Prove that p2 + pq + 2 = 0 . 1 (ii) If the chords OP and OQ are perpendicular, show that p2 = 2 . 2 End of Question 5 7 BLANK PAGE 8 Marks Question 6 (12 marks) Use a SEPARATE writing booklet. (a) A particle moves in a straight line. Its displacement, x metres, after t seconds is given by x = 3 sin 2t cos 2t + 3. (i) Prove that the particle is moving in simple harmonic motion about x = 3 by showing that x = 4 ( x 3) . 2 (ii) What is the period of the motion? 1 (iii) Express the velocity of the particle in the form x = A cos ( 2t ) , where is in radians. 2 (iv) Hence, or otherwise, find all times within the first seconds when the particle is moving at 2 metres per second in either direction. 2 (b) Consider the function ( x ) = e x e x . (i) Show that ( x ) is increasing for all values of x. 1 (ii) Show that the inverse function is given by 3 x + x2 + 4 1( x ) = loge . 2 (iii) Hence, or otherwise, solve e x e x = 5. Give your answer correct to two decimal places. 9 1 Marks Question 7 (12 marks) Use a SEPARATE writing booklet. (a) y y = kxn nt nge ta mon com O y = loge x a x The graphs of the functions y = k x n and y = loge x have a common tangent at x = a , as shown in the diagram. (i) By considering gradients, show that a n = 1 . nk 1 (ii) Express k as a function of n by eliminating a . 2 Question 7 continues on page 11 10 Marks Question 7 (continued) (b) A small paintball is fired from the origin with initial velocity 14 metres per second towards an eight-metre high barrier. The origin is at ground level, 10 metres from the base of the barrier. The equations of motion are x = 14 t cos y = 14 t sin 4.9 t 2 where is the angle to the horizontal at which the paintball is fired and t is the time in seconds. (Do NOT prove these equations of motion.) y 8 metres e O (i) 10 10 metres x Show that the equation of trajectory of the paintball is 1 + m2 2 y = mx x , 40 (ii) 2 where m = tan . Show that the paintball hits the barrier at height h metres when 2 m = 2 3 0 .4 h . Hence determine the maximum value of h. (iii) There is a large hole in the barrier. The bottom of the hole is 3.9 metres above the ground and the top of the hole is 5.9 metres above the ground. The paintball passes through the hole if m is in one of two intervals. One interval is 2.8 m 3.2. 2 Find the other interval. (iv) Show that, if the paintball passes through the hole, the range is 40 m 1 + m2 metres. Hence find the widths of the two intervals in which the paintball can land at ground level on the other side of the barrier. End of paper 11 3 STANDARD INTEGRALS x n dx = 1 dx x = ln x , x > 0 e ax d x = 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 ax d x = 1 tan ax , a 0 a sec ax tan ax d x = 1 sec 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 2 x 2 + a ( ) ( ) NOTE : ln x = loge x , x > 0 12 Board of Studies NSW 2007

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Additional Info : New South Wales Higher School Certificate Mathematics Extension-1 2007
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