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NSW HSC 2005 : MATHEMATICS EXTENSION-1

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2005 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 BLANK PAGE 2 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. (a) 1 dx . Find 2 x + 49 1 (b) Sketch the region in the plane defined by y 2 x + 3 . 2 (c) x State the domain and range of y = cos 1 . 4 2 (d) 5 x 2 x 2 + 1 4 dx . Using the substitution u = 2 x + 1, or otherwise, find ( 2 ) 3 (e) The point P (1, 4) divides the line segment joining A ( 1, 8) and B (x, y) internally in the ratio 2 : 3. Find the coordinates of the point B . 2 (f) The acute angle between the lines y = 3x + 5 and y = mx + 4 is 45 . Find the two possible values of m . 2 3 Marks Question 2 (12 marks) Use a SEPARATE writing booklet. ( ) d 2 sin 1 5 x . dx 2 (a) Find (b) Use the binomial theorem to find the term independent of x in the expansion 3 1 12 of 2 x . x2 (d) (i) Differentiate e3x ( cos x 3sin x ) . 2 (ii) (c) 3x Hence, or otherwise, find e sin x dx . 1 A salad, which is initially at a temperature of 25 C, is placed in a refrigerator that has a constant temperature of 3 C. The cooling rate of the salad is proportional to the difference between the temperature of the refrigerator and the temperature, T, of the salad. That is, T satisfies the equation dT = k(T 3), dt where t is the number of minutes after the salad is placed in the refrigerator. (i) Show that T = 3 + Ae kt satisfies this equation. 1 (ii) The temperature of the salad is 11 C after 10 minutes. Find the temperature of the salad after 15 minutes. 3 4 Marks Question 3 (12 marks) Use a SEPARATE writing booklet. (b) (i) Show that the function g(x) = x2 log e(x + 1) has a zero between 0.7 and 0.9. 1 (ii) (a) Use the method of halving the interval to find an approximation to this zero of g(x) , correct to one decimal place. 2 (i) By expanding the left-hand side, show that 1 sin (5x + 4x) + sin (5x 4x) = 2sin 5x cos4x. (ii) (c) Hence find sin 5 x cos 4 x dx . 2 Use the definition of the derivative, f ( x ) = lim f ( x + h) f ( x ) h h 0 , to find f ( x ) 2 when f ( x ) = x 2 + 5 x . (d) C O B NOT TO SCALE E 4 x A D In the circle centred at O the chord AB has length 7. The point E lies on AB and AE has length 4. The chord CD passes through E. Let the length of CD be l and the length of DE be x . (i) Show that x2 lx + 12 = 0. 2 (ii) Find the length of the shortest chord that passes through E . 2 5 Marks Question 4 (12 marks) Use a SEPARATE writing booklet. (a) 4 Evaluate cos x sin 2 x dx . 0 2 (b) By making the substitution t = tan , or otherwise, show that 2 2 cosec + cot = cot . 2 (c) The points P (2ap, ap2 ) and Q (2aq, aq2 ) lie on the parabola x2 = 4ay . The equation of the normal to the parabola at P is x + py = 2ap + ap3 and the equation of the normal at Q is similarly given by x + qy = 2aq + aq3. (i) Show that the normals at P and Q intersect at the point R whose coordinates are 2 ( apq [ p + q], a [ p2 + pq + q2 + 2]). (ii) The equation of the chord PQ is y = 1 ( p + q) x apq . (Do NOT show this.) 2 1 If the chord PQ passes through (0, a), show that pq = 1. (iii) Find the equation of the locus of R if the chord PQ passes through (0, a). (d) 2 Use the principle of mathematical induction to show that 4n 1 7n > 0 for all integers n 2. 3 6 Marks Question 5 (12 marks) Use a SEPARATE writing booklet. (a) Find the exact value of the volume of the solid of revolution formed when the region bounded by the curve y = sin2x , the x-axis and the line x = is rotated 8 about the x-axis. (b) Two chords of a circle, AB and CD, intersect at E. The perpendiculars to AB at A and CD at D intersect at P. The line PE meets BC at Q, as shown in the diagram. 3 P D E A C B Q (i) 1 (ii) Prove that APE = ABC. 2 (iii) (c) Explain why DPAE is a cyclic quadrilateral. Deduce that PQ is perpendicular to BC . 1 A particle moves in a straight line and its position at time t is given by x=5+ 3 sin 3t cos 3t . (i) 3 sin 3t cos 3t in the form R sin ( 3t ) , where is Express in radians. 2 (ii) The particle is undergoing simple harmonic motion. Find the amplitude and the centre of the motion. 2 (iii) When does the particle first reach its maximum speed after time t = 0? 1 7 Marks Question 6 (12 marks) Use a SEPARATE writing booklet. (a) There are five matches on each weekend of a football season. Megan takes part in a competition in which she earns one point if she picks more than half of the winning teams for a weekend, and zero points otherwise. The probability that 2 Megan correctly picks the team that wins any given match is . 3 (i) Show that the probability that Megan earns one point for a given weekend is 0.7901, correct to four decimal places. 2 (ii) Hence find the probability that Megan earns one point every week of the eighteen-week season. Give your answer correct to two decimal places. 1 (iii) Find the probability that Megan earns at most 16 points during the eighteen-week season. Give your answer correct to two decimal places. 2 Question 6 continues on page 9 8 Marks Question 6 (continued) (b) An experimental rocket is at a height of 5000 m, ascending with a velocity of 200 2 m s 1 at an angle of 45 to the horizontal, when its engine stops. y 5000 m x O After this time, the equations of motion of the rocket are: x = 200 t y = 4.9 t2 + 200 t + 5000, where t is measured in seconds after the engine stops. (Do NOT show this.) (i) What is the maximum height the rocket will reach, and when will it reach this height? 2 (ii) The pilot can only operate the ejection seat when the rocket is descending at an angle between 45 and 60 to the horizontal. What are the earliest and latest times that the pilot can operate the ejection seat? 3 (iii) For the parachute to open safely, the pilot must eject when the speed of the rocket is no more than 350 m s 1. What is the latest time at which the pilot can eject safely? 2 End of Question 6 9 Marks Question 7 (12 marks) Use a SEPARATE writing booklet. (a) An oil tanker at T is leaking oil which forms a circular oil slick. An observer is measuring the oil slick from a position P, 450 metres above sea level and 2 kilometres horizontally from the centre of the oil slick. P 450 m r 2 km (i) (ii) (b) T r At a certain time the observer measures the angle, , subtended by the diameter of the oil slick, to be 0.1 radians. What is the radius, r , at this time? d = 0.02 radians per hour. Find the rate at which the radius dt of the oil slick is growing. At this time, 2 2 Let (x) = Ax3 Ax + 1, where A > 0. 3 . 3 (i) Show that (x) has stationary points at x = (ii) Show that (x) has exactly one zero when A < (iii) By observing that ( 1) = 1, deduce that (x) does not have a zero in the interval 1 x 1 when 0 < A < (iv) Let g( ) = 2cos + tan , where 33 . 2 1 2 1 33 . 2 < < . 2 2 3 By calculating g ( ) and applying the result in part (iii), or otherwise, show that g( ) does not have any stationary points. (v) Hence, or otherwise, deduce that g( ) has an inverse function. End of paper 10 1 BLANK PAGE 11 STANDARD INTEGRALS n x dx = 1 x dx = ln x, x > 0 ax e dx = 1 ax e , a 0 a cos ax dx = 1 sin ax, a 0 a sin ax dx 1 = cos ax, a 0 a 2 sec ax dx = 1 tan ax, a 0 a sec ax tan ax dx = 1 sec ax, a 0 a 1 dx 2 a + x2 = 1 x tan 1 , a 0 a a 1 dx 2 a x2 x = sin 1 , a > 0, a < x < a a 1 dx 2 x a2 = ln x + x 2 a 2 , x > a > 0 1 dx 2 x + a2 = ln x + x 2 + a 2 1 n +1 x , n 1; x 0, if n < 0 n +1 ( ) ( ) NOTE : ln x = loge x, x>0 12 Board of Studies NSW 2005

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Additional Info : New South Wales Higher School Certificate Mathematics Extension-1 2005
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