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NSW HSC 2008 : MATHEMATICS EXTENSION-1

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2008 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) The polynomial x 3 is divided by x + 3. Calculate the remainder. 2 (b) Differentiate cos 1 (3x) with respect to x . 2 (c) Evaluate 1 1 (d) 1 4 x 2 dx . 2 Find an expression for the coefficient of x 8y4 in the expansion of (2x + 3y)12. 4 2 (e) Evaluate cos sin 2 d . 0 2 (f) Let ( x ) = loge ( x 3) ( 5 x ) . 2 What is the domain of ( x ) ? 3 Marks Question 2 (12 marks) Use a SEPARATE writing booklet. e2 (a) 1 Use the substitution u = log e x to evaluate x loge x e ( ) 2 dx . (b) A particle moves on the x-axis with velocity v . The particle is initially at rest at x = 1. Its acceleration is given by x = x + 4. Using the fact that x = (c) 3 3 d 1 2 v , find the speed of the particle at x = 2. dx 2 The polynomial p(x) is given by p(x) = ax 3 + 16x 2 + cx 120, where a and c are constants. 3 The three zeros of p(x) are 2, 3 and . Find the value of . (d) The function ( x ) = tan x loge x has a zero near x = 4. Use one application of Newton s method to obtain another approximation to this zero. Give your answer correct to two decimal places. 4 3 Marks Question 3 (12 marks) Use a SEPARATE writing booklet. (i) Sketch the graph of y = 2 x 1 . 1 (ii) (a) Hence, or otherwise, solve 2 x 1 x 3 . 3 (b) Use mathematical induction to prove that, for integers n 1, 1 3 + 2 4 + 3 5 + + n ( n + 2) = (c) 3 n ( n + 1)( 2n + 7) . 6 A P O C x Q A race car is travelling on the x-axis from P to Q at a constant velocity, v. A spectator is at A which is directly opposite O, and OA = metres. When the car is at C, its displacement from O is x metres and OAC = , with . < 2 2 (i) Show that d dt (ii) = v 2 + x 2 . 2 Let m be the maximum value of d . dt 1 Find the value of m in terms of v and . (iii) There are two values of for which Find these two values of . 5 d m = . dt 4 2 Marks Question 4 (12 marks) Use a SEPARATE writing booklet. (a) A turkey is taken from the refrigerator. Its temperature is 5 C when it is placed in an oven preheated to 190 C. Its temperature, T C, after t hours in the oven satisfies the equation dT = k ( T 190 ) . dt (i) Show that T = 190 185e kt satisfies both this equation and the initial condition. 2 (ii) The turkey is placed into the oven at 9 am. At 10 am the turkey reaches a temperature of 29 C. The turkey will be cooked when it reaches a temperature of 80 C. 3 At what time (to the nearest minute) will it be cooked? (b) Barbara and John and six other people go through a doorway one at a time. (i) In how many ways can the eight people go through the doorway if John goes through the doorway after Barbara with no-one in between? 1 (ii) Find the number of ways in which the eight people can go through the doorway if John goes through the doorway after Barbara. 1 Question 4 continues on page 7 6 Marks Question 4 (continued) (c) y x 2 = 4 ay Q (2aq, aq 2 ) M P (2ap, ap2 ) x O K L T The points P (2ap, ap2 ), Q (2aq, aq2 ) lie on the parabola x 2 = 4ay. The tangents to the parabola at P and Q intersect at T. The chord QO produced meets PT at K, and PKQ is a right angle. (i) Find the gradient of QO, and hence show that pq = 2. 2 (ii) The chord PO produced meets QT at L. Show that PLQ is a right angle. 1 (iii) Let M be the midpoint of the chord PQ. By considering the quadrilateral PQLK, or otherwise, show that MK = ML . 2 End of Question 4 7 Marks Question 5 (12 marks) Use a SEPARATE writing booklet. (a) Let ( x ) = x 1 2 x for x 1. This function has an inverse, 1 ( x ) . 2 (i) Sketch the graphs of y = ( x ) and y = 1 ( x ) on the same set of axes. (Use the same scale on both axes.) 2 (ii) Find an expression for 1 ( x ). 3 3 (iii) Evaluate 1 . 8 1 (b) A particle is moving in simple harmonic motion in a straight line. Its maximum speed is 2 m s 1 and its maximum acceleration is 6 m s 2. 3 Find the amplitude and the period of the motion. (c) T 3 C2 P L M Q C1 K Two circles C1 and C2 intersect at P and Q as shown in the diagram. The tangent TP to C2 at P meets C1 at K. The line KQ meets C2 at M. The line MP meets C1 at L . Copy or trace the diagram into your writing booklet. Prove that PKL is isosceles. 8 Marks Question 6 (12 marks) Use a SEPARATE writing booklet. (a) From a point A due south of a tower, the angle of elevation of the top of the tower T, is 23 . From another point B, on a bearing of 120 from the tower, the angle of elevation of T is 32 . The distance AB is 200 metres. T North O 120 B A (i) Copy or trace the diagram into your writing booklet, adding the given information to your diagram. 1 (ii) Hence find the height of the tower. 3 (b) It can be shown that sin 3 = 3 sin 4 sin3 for all values of . (Do NOT prove this.) 3 Use this result to solve sin 3 + sin 2 = sin for 0 2 . (c) Let p and q be positive integers with p q . (i) Use the binomial theorem to expand (1 + x ) p + q , and hence write down the term of (ii) Given that (1 + x ) xq (1 + x ) p+ q x q 2 p+ q which is independent of x . = (1 + x ) p q 1 1 + , apply the binomial theorem x and the result of part (i) to find a simpler expression for p q p q 1 + + + + 1 1 2 2 9 p q . p p 3 Marks Question 7 (12 marks) Use a SEPARATE writing booklet. y L M w V h O P Q N x r A projectile is fired from O with velocity V at an angle of inclination across level ground. The projectile passes through the points L and M, which are both h metres above the ground, at times t1 and t2 respectively. The projectile returns to the ground at N. The equations of motion of the projectile are x = Vt cos and y = Vt sin (a) Show that t1 + t2 = 2V sin g AND 12 g t . (Do NOT prove this.) 2 t1t2 = 2h . g Question 7 continues on page 11 10 2 Marks Question 7 (continued) Let LON = and LNO = . It can be shown that tan = h h and tan = . (Do NOT prove this.) Vt1 cos Vt2 cos (b) Show that tan + tan = tan . (c) Show that tan tan = 2 gh 2V 2 cos2 . 1 Let ON = r and L M = w . (d) Show that r = h ( cot + cot ) and w = h ( cot cot ) . 2 Let the gradient of the parabola at L be tan . (e) Show that tan = tan tan . (f) Show that 3 w r . = tan tan 2 End of paper 11 STANDARD INTEGRALS x n dx = 1 dx x = ln x , x > 0 e ax d x = 1 ax e , a 0 a cos ax dx = 1 sin ax , a 0 a sin ax dx 1 = cos ax , a 0 a sec 2 ax dx = 1 tan ax , a 0 a sec ax tan ax dx = 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 2008

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Additional Info : New South Wales Higher School Certificate Mathematics Extension-1 2008
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