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NSW HSC 2006 : MATHEMATICS EXTENSION-1

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2006 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) dx Find . 49 + x 2 2 (b) Using the substitution u = x4 + 8 , or otherwise, find x 3 x 4 + 8 dx . 3 (c) Evaluate lim 2 (d) Using the sum of two cubes, simplify: 2 sin 5 x . x 0 3 x sin3 + cos3 1, sin + cos for 0 < < (e) . 2 For what values of b is the line y = 12x + b tangent to y = x3 ? 3 3 Marks Question 2 (12 marks) Use a SEPARATE writing booklet. (a) Let (x) = sin 1 ( x + 5 ) . (i) 2 (ii) Find the gradient of the graph of y = (x) at the point where x = 5 . 2 (iii) (b) State the domain and range of the function (x) . Sketch the graph of y = (x) . 2 (i) By applying the binomial theorem to show that n (1 + x ) (ii) n 1 n n = + 2 x + 1 2 (1 + x )n and differentiating, n + r x r 1 + r n + n x n 1 . n 1 Hence deduce that n n3n 1 = + 1 1 n + r 2r 1 + r n + n 2n 1 . n Question 2 continues on page 5 4 Marks Question 2 (continued) (c) y Q R T U P x The points P (2ap, ap2 ), Q (2aq, aq2 ) and R (2ar, ar2 ) lie on the parabola x2 = 4ay. The chord QR is perpendicular to the axis of the parabola. The chord PR meets the axis of the parabola at U. The equation of the chord PR is y = 1 ( p + r ) x apr . 2 The equation of the tangent at P is y = px ap2 . (Do NOT prove this.) (Do NOT prove this.) (i) Find the coordinates of U. 1 (ii) The tangents at P and Q meet at the point T. Show that the coordinates of T are (a ( p + q), apq) . 2 (iii) Show that TU is perpendicular to the axis of the parabola. 1 End of Question 2 5 Marks Question 3 (12 marks) Use a SEPARATE writing booklet. (a) 4 Find sin 2 x dx . 0 2 (c) By considering ( x ) = 3loge x x , show that the curve y = 3loge x and the line y = x meet at a point P whose x-coordinate is between 1.5 and 2. 1 (ii) Use one application of Newton s method, starting at x = 1.5, to find an approximation to the x-coordinate of P. Give your answer correct to two decimal places. (b) 2 (i) Sophie has five coloured blocks: one red, one blue, one green, one yellow and one white. She stacks two, three, four or five blocks on top of one another to form a vertical tower. (i) How many different towers are there that she could form that are three blocks high? 1 (ii) How many different towers can she form in total? 2 Question 3 continues on page 7 6 Marks Question 3 (continued) (d) P K Q N T M The points P, Q and T lie on a circle. The line MN is tangent to the circle at T with M chosen so that QM is perpendicular to MN. The point K on PQ is chosen so that TK is perpendicular to PQ as shown in the diagram. (i) Show that QKTM is a cyclic quadrilateral. 1 (ii) Show that KMT = KQT . 1 (iii) Hence, or otherwise, show that MK is parallel to TP. 2 End of Question 3 7 BLANK PAGE 8 Marks Question 4 (12 marks) Use a SEPARATE writing booklet. (a) The cubic polynomial P (x) = x3 + rx2 + sx + t , where r, s and t are real numbers, has three real zeros, 1, and . (i) Find the value of r. 1 (ii) Find the value of s + t . 2 (b) A particle is undergoing simple harmonic motion on the x-axis about the origin. It is initially at its extreme positive position. The amplitude of the motion is 18 and the particle returns to its initial position every 5 seconds. (i) Write down an equation for the position of the particle at time t seconds. (ii) How long does the particle take to move from a rest position to the point halfway between that rest position and the equilibrium position? (c) 1 2 3 2 A particle is moving so that x = 18 x + 27 x + 9 x . Initially x = 2 and the velocity, v, is 6. (i) Show that v2 = 9x2 (1 + x)2 . 2 (ii) Hence, or otherwise, show that 2 1 dx = 3t . x (1 + x ) 2 (iii) It can be shown that for some constant c, 1 loge 1 + = 3t + c . x (Do NOT prove this.) Using this equation and the initial conditions, find x as a function of t. 9 Marks Question 5 (12 marks) Use a SEPARATE writing booklet. (a) Show that y = 10 e 0.7t + 3 is a solution of (b) Let ( x ) = loge 1 + e x ( ) dy = 0.7 ( y 3) . dt for all x. Show that (x) has an inverse. 2 2 (c) x cm A hemispherical bowl of radius r cm is initially empty. Water is poured into it at a constant rate of k cm3 per minute. When the depth of water in the bowl is x cm, the volume, V cm3 , of water in the bowl is given by V= 2 x (3r x ) . 3 (Do NOT prove this.) dx k . = dt x ( 2r x ) (i) Show that (ii) Hence, or otherwise, show that it takes 3.5 times as long to fill the bowl 2 2 to the point where x = r as it does to fill the bowl to the point where 3 1 x= r. 3 Question 5 continues on page 11 10 2 Marks Question 5 (continued) (d) (i) Use the fact that tan ( ) = tan tan to show that 1 + tan tan 1 1 + tan n tan ( n + 1) = cot ( tan ( n + 1) tan n ) . (ii) Use mathematical induction to prove that, for all integers n 1, tan tan 2 + tan 2 tan 3 + + tan n tan ( n + 1) = ( n + 1) + cot tan ( n + 1) . End of Question 5 11 3 Question 6 (12 marks) Use a SEPARATE writing booklet. (a) Two particles are fired simultaneously from the ground at time t = 0. Particle 1 is projected from the origin at an angle , 0 < < , with an initial 2 velocity V. Particle 2 is projected vertically upward from the point A, at a distance a to the right of the origin, also with an initial velocity of V. y V V O A x It can be shown that while both particles are in flight, Particle 1 has equations of motion: x = Vt cos y = Vt sin 12 gt , 2 and Particle 2 has equations of motion: x=a y = Vt 12 gt . 2 Do NOT prove these equations of motion. Let L be the distance between the particles at time t. Question 6 continues on page 13 12 Marks Question 6 (continued) (i) Show that, while both particles are in flight, 2 L2 = 2V 2 t 2 (1 sin ) 2aVt cos + a 2 . (ii) An observer notices that the distance between the particles in flight first decreases, then increases. 3 Show that the distance between the particles in flight is smallest when t= a cos 1 sin and that this smallest distance is a . 2V (1 sin ) 2 (iii) Show that the smallest distance between the two particles in flight occurs while Particle 1 is ascending if V > 1 ag cos . 2sin (1 sin ) (b) In an endurance event, the probability that a competitor will complete the course is p and the probability that a competitor will not complete the course is q = 1 p . Teams consist of either two or four competitors. A team scores points if at least half its members complete the course. (i) Show that the probability that a four-member team will have at least three of its members not complete the course is 4pq3 + q4. 1 (ii) Hence, or otherwise, find an expression in terms of q only for the probability that a four-member team will score points. 2 (iii) Find an expression in terms of q only for the probability that a two-member team will score points. 1 (iv) Hence, or otherwise, find the range of values of q for which a two-member team is more likely than a four-member team to score points. 2 End of Question 6 13 Marks Question 7 (12 marks) Use a SEPARATE writing booklet. A gutter is to be formed by bending a long rectangular metal strip of width w so that the cross-section is an arc of a circle. Let r be the radius of the arc and 2 the angle at the centre, O, so that the cross-sectional area, A, of the gutter is the area of the shaded region in the diagram on the right. O B CROSS-SECTION 2 C r r Gutter B C w (a) Show that, when 0 < , the cross-sectional area is 2 A = r 2 ( si cos (b) 2 ). The formula in part (a) for A is true for 0 < < . (Do NOT prove this.) By first expressing r in terms of w and , and then differentiating, show that dA d = w 2 cos (sin cos ) 2 3 for 0 < < . Question 7 continues on page 15 14 3 Marks Question 7 (continued) (c) Let g ( ) = sin cos . 3 By considering g ( ) , show that g ( ) > 0 for 0 < < . (d) Show that there is exactly one value of in the interval 0 < < for which 2 dA = 0. d (e) Show that the value of for which dA = 0 gives the maximum cross-sectional d area. Find this area in terms of w. End of paper 15 2 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 16 Board of Studies NSW 2006

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Additional Info : New South Wales Higher School Certificate Mathematics Extension-1 2006
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