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NSW HSC 2006 : MATHEMATICS

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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 General Instructions Reading time 5 minutes Working time 3 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 212 Total marks 120 Attempt Questions 1 10 All questions are of equal value Total marks 120 Attempt Questions 1 10 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) 0.5 Evaluate e correct to three decimal places. 2 (b) Factorise 2x2 + 5x 3 . 2 (c) Sketch the graph of y = x + 4 . 2 (d) 5 NOT TO SCALE 9 2 33 Find the value of in the diagram. Give your answer to the nearest degree. (e) Solve 3 5x 2 . (f) Find the limiting sum of the geometric series 2 2 13 13 13 + + + 5 25 125 . 2 Marks Question 2 (12 marks) Use a SEPARATE writing booklet. (a) Differentiate with respect to x : (i) 2 (ii) (b) x tan x sin x . x + 1 2 (i) Find 1 + e 7 x d x . 2 3 8 x dx . (ii) Evaluate 0 1 + x 2 (c) 3 Find the equation of the tangent to the curve y = cos 2 x at the point whose x-coordinate is . 6 3 3 Marks Question 3 (12 marks) Use a SEPARATE writing booklet. (a) y NOT TO SCALE D A(1, 4) O x C( 3, 1) B(5, 4) In the diagram, A, B and C are the points (1, 4), (5, 4) and ( 3, 1) respectively. The line AB meets the y-axis at D. (i) Show that the equation of the line AB is 2x + y 6 = 0. 2 (ii) Find the coordinates of the point D. 1 (iii) Find the perpendicular distance of the point C from the line AB. 1 (iv) Hence, or otherwise, find the area of the triangle ADC. 2 Question 3 continues on page 5 4 Marks Question 3 (continued) 4 (b) Evaluate 1 r . 1 r =2 (c) On the first day of the harvest, an orchard produces 560 kg of fruit. On the next day, the orchard produces 543 kg, and the amount produced continues to decrease by the same amount each day. (i) How much fruit is produced on the fourteenth day of the harvest? 2 (ii) What is the total amount of fruit that is produced in the first 14 days of the harvest? 1 (iii) On what day does the daily production first fall below 60 kg? 2 End of Question 3 5 Marks Question 4 (12 marks) Use a SEPARATE writing booklet. (a) NOT TO SCALE D 5 6 B A C In the diagram, ABCD represents a garden. The sector BCD has centre B and 5 DBC = . 6 The points A, B and C lie on a straight line and AB = AD = 3 metres. Copy or trace the diagram into your writing booklet. 2 . 3 (i) Show that DAB = (ii) Find the length of BD. 2 (iii) Find the area of the garden ABCD. 2 1 Question 4 continues on page 7 6 Marks Question 4 (continued) (b) 3 y y = x2 + 1 y=5 O x In the diagram, the shaded region is bounded by the parabola y = x2 + 1, the y-axis and the line y = 5 . Find the volume of the solid formed when the shaded region is rotated about the y-axis. (c) A chessboard has 32 black squares and 32 white squares. Tanya chooses three different squares at random. (i) What is the probability that Tanya chooses three white squares? 2 (ii) What is the probability that the three squares Tanya chooses are the same colour? 1 (iii) What is the probability that the three squares Tanya chooses are not the same colour? 1 End of Question 4 7 Marks Question 5 (12 marks) Use a SEPARATE writing booklet. (a) A function (x) is defined by (x) = 2 x 2 (3 x). (i) Find the coordinates of the turning points of y = (x) and determine their nature. (ii) Find the coordinates of the point of inflexion. 1 (iii) Hence sketch the graph of y = (x), showing the turning points, the point of inflexion and the points where the curve meets the x-axis. 3 (iv) What is the minimum value of (x) for 1 x 4? (b) 3 1 (i) (ii) Show that d loge ( cos x ) = tan x . dx 1 3 y y=x y = tan x x= O 4 x The shaded region in the diagram is bounded by the curve y = tan x and the lines y = x and x = . 4 Using the result of part (i), or otherwise, find the area of the shaded region. 8 Marks Question 6 (12 marks) Use a SEPARATE writing booklet. (a) B C P A D In the diagram, AD is parallel to BC, AC bisects BAD and BD bisects ABC. The lines AC and BD intersect at P. Copy or trace the diagram into your writing booklet. (i) 1 (ii) Prove that ABP CBP. 2 (iii) (b) Prove that BAC = BCA. Prove that ABCD is a rhombus. 3 A rare species of bird lives only on a remote island. A mathematical model predicts that the bird population, P , is given by P = 150 + 300e 0.05t where t is the number of years after observations began. (i) According to the model, how many birds were there when observations began? 1 (ii) According to the model, what will be the rate of change in the bird population ten years after observations began? 2 (iii) What does the model predict will be the limiting value of the bird population? 1 (iv) The species will become eligible for inclusion in the endangered species list when the population falls below 200. When does the model predict that this will occur? 2 9 Marks Question 7 (12 marks) Use a SEPARATE writing booklet. (a) Let and be the solutions of x2 3x + 1 = 0. (i) Find . (ii) Hence find + 1 1 . 1 (b) A function (x) is defined by (x) = 1 + 2 cos x . (i) Show that the graph of y = (x) cuts the x-axis at x = 2 . 3 (ii) Sketch the graph of y = (x) for x showing where the graph cuts each of the axes. 3 2 and x = . 2 3 3 Write down the discriminant of 2x2 + (k 2) x + 8 , where k is a constant. 1 (ii) Hence, or otherwise, find the values of k for which the parabola y = 2x2 + kx + 9 does not intersect the line y = 2x + 1. 2 (iii) Find the area under the curve y = (x) between x = (c) 1 (i) 10 Marks Question 8 (12 marks) Use a SEPARATE writing booklet. (a) A particle is moving in a straight line. Its displacement, x metres, from the 7 origin, O, at time t seconds, where t 0, is given by x = 1 . t+4 (i) 1 (ii) Find the velocity of the particle as it passes through the origin. 3 (iii) Show that the acceleration of the particle is always negative. 1 (iv) (b) Find the initial displacement of the particle. Sketch the graph of the displacement of the particle as a function of time. 2 Joe borrows $200 000 which is to be repaid in equal monthly instalments. The interest rate is 7.2% per annum reducible, calculated monthly. It can be shown that the amount, $An , owing after the n th repayment is given by the formula: An = 200 000 r n M (1 + r + r 2 + + r n 1) , where r = 1.006 and $M is the monthly repayment. (i) (Do NOT show this.) The minimum monthly repayment is the amount required to repay the loan in 300 instalments. 3 Find the minimum monthly repayment. (ii) Joe decides to make repayments of $2800 each month from the start of the loan. How many months will it take for Joe to repay the loan? 11 2 BLANK PAGE 12 Marks Question 9 (12 marks) Use a SEPARATE writing booklet. (a) Find the coordinates of the focus of the parabola 12y = x2 6x 3 . (b) During a storm, water flows into a 7000-litre tank at a rate of per minute, where 2 dV litres dt dV = 120 + 26t t 2 and t is the time in minutes dt since the storm began. (i) At what times is the tank filling at twice the initial rate? 2 (ii) Find the volume of water that has flowed into the tank since the start of the storm as a function of t. 1 (iii) Initially, the tank contains 1500 litres of water. When the storm finishes, 30 minutes after it began, the tank is overflowing. 2 How many litres of water have been lost? (c) x O a r A cone is inscribed in a sphere of radius a, centred at O. The height of the cone is x and the radius of the base is r, as shown in the diagram. (i) Show that the volume, V, of the cone is given by V = ( ) 1 2ax 2 x 3 . 3 (ii) Find the value of x for which the volume of the cone is a maximum. You must give reasons why your value of x gives the maximum volume. 13 2 3 Marks Question 10 (12 marks) Use a SEPARATE writing booklet. (a) Use Simpson s rule with three function values to find an approximation to the 1.5 3 value of ( loge x ) dx . 0. 5 Give your answer correct to three decimal places. Question 10 continues on page 15 14 2 Marks Question 10 (continued) (b) 6 cm P x cm O K Q NOT TO SCALE 13 cm 13 y cm M L N S R 12 cm A rectangular piece of paper PQRS has sides PQ = 12 cm and PS = 13 cm. The point O is the midpoint of PQ. The points K and M are to be chosen on OQ and PS respectively, so that when the paper is folded along KM, the corner that was at P lands on the edge QR at L. Let OK = x cm and LM = y cm. Copy or trace the diagram into your writing booklet. (i) Show that QL2 = 24x . 1 (ii) Let N be the point on QR for which MN is perpendicular to QR. 3 By showing that QKL ||| NLM, deduce that y = 6 (6 + x ) x 6 (6 + x ) . 2 (iii) Show that the area, A, of KLM is given by A = (iv) Use the fact that 12 y 13 to find the possible values of x. 2 (v) Find the minimum possible area of KLM. 3 End of paper 15 2x . 1 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 2006
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