Mrs Israel Ms Lau Ms Prosser Ms Stott Mrs Kerr Mr Morrison Mrs Semler Name:.... Teacher:. HIGHER SCHOOL CERTIFICATE TRIAL EXAMINATION 016 Mathematics General Instructions Reading time 5 minutes. Working time 3 hours. Use pencil for Questions 1-10. Write using a black or blue pen for Questions 11-16. Black pen is preferred. Board approved calculators may be used. A reference sheet is provided. In Questions 11-16, show relevant mathematical reasoning and/or calculations. Total Marks 100 Section I Pages 1-5 10 marks Attempt Questions 1-10 Allow about 15 mins for this section Section II Pages 7-16 90 marks Attempt Questions 11-16 Allow about hours 45 minutes for this section Mark /100 Highest Mark /100 Rank
Blank Page
Section I 10 marks Attempt Questions 1-10 Allow about 15 minutes for this section. Use the multiple choice answer sheet for Questions 1-10. 1 The line with equation y = 3x+ 5 is perpendicular to the line with equation y = kx. What is the value of k? (A) (B) (C) (D) 3 3 3 3 A function f, defined on a suitable domain, is given by ( ) What restrictions are there on the domain of f? (A) x 8 or x (B) x 4 or x 4 (C) x 0 (D) x 10 or x 16 f x = x 6x. + 6x 16 f x = x + 1 and g x = 3x 4, on the 3 The functions f and g are defined by ( ) ( ) set of real numbers. Which expression is equivalent to g( f ( x ))? (A) (B) (C) (D) 3x 1 9x 15 9x + 17 3 3x 4x + 3x 4 1
4 Given that f ( x) = x f ( ) (A) 0 (B) 1 (C) 1 (D) 36 4sin 3, what is ' 0? 5 What is ( + ) x 3x dx? (A) (B) (C) (D) 3 x + c 3 x + x + c 1 3 + + x x x c 3 x + x+ c 6 If 4t e = 6, which of the following is an expression for t? (A) (B) (C) (D) 3 t = loge t = t = loge 6 4 6 log 4 e loge 6 t = log 4 e
7 The diagram shows part of the graph of y = a cosbx. The shaded area is 1 unit. 3π 4 What is the value of ( a ) 0 cos bx dx? (A) 1 (B) (C) (D) 1 1 1 1 4 3 8 The volume of a sphere is given by the formula V = π r. 3 What is the rate of change of V with respect to r, at r =? (A) (B) (C) (D) 16π 3 3π 3 16π 3π 3
9 The diagram shows part of the graph of y = f ( x). Which of the following diagrams could be the graph of y f ( x) = + 1? (A) (B) (C) (D) 4
10 The graph of y = f ( x) shown has stationary points at ( p) ( qr ) Here are two statements about f ( x ) (i) f ( x) < 0for s< x< t. (ii) f '( x) < 0 for x< q. 0, and,. Which of the following is true? (A) Neither statement is correct. (B) Only statement (i) is correct. (C) Only statement (ii) is correct. (D) Both statements are correct. 5
Blank Page 6
Question 11 (15 marks) Use a separate writing booklet. (a) Express 6 with a rational denominator in its simplest form. (b) Solve x 4. (c) Solve the equation x 3 x = 1. 3 5 (d) Find the gradient of the tangent to the curve ( ) 4 y = 3x+ 1 when 1 x =. 3 (e) Simplify log m b a log a m and express it in terms of base b. (f) Differentiate 3 x. 1 (g) Find x + dx. x + 4x (h) Evaluate π x sec dx. 0 End of Question 11 7
Question 1 (15 marks) Use a separate writing booklet. (a) In the diagram, AB CD, BAE = 130, EFD = 100. (i) Find the value of y, giving reasons. 1 (ii) Find the value of x, giving reasons. (b) A function f is defined on the set of real numbers by f ( x) = ( x )( x + ) (i) Find where the graph of y = f ( x) cuts 1. (1) the x-axis. 1 () the y-axis. 1 (ii) Find the coordinates of the stationary points on the curve with 4 y = f x and determine their nature. equation ( ) (iii) Sketch the graph of y = f ( x), showing all important features. 3 Question 1 continues on page 9. 8
Question 1 (continued). (c) (i) Prove that PQT and PSR are similar. (ii) Hence, find the length of RS. 1 End of Question 1 9
Question 13 (15 marks) Use a separate writing booklet. (a) In the diagram above, the equations of the lines BE and BC are 4x+ 3y 10 = 0 and y = 10 respectively. P is the point ( 5,5 ). PE BE, and BC PC. (i) Show that the perpendicular distance from P to BE is 5 units. 1 (ii) Hence prove that BCP BEP. 3 (iii) Show that the coordinates of B are ( 5, 10). 1 (iv) Show that the locus of points which are equidistant from the lines BC and BE is given by the equation x+ y 15 = 0. Question 13 continues on page 11 10
Question 13 (continued). (b) (i) Sketch the parabola P, whose focus is (, ) and whose directrix is the line x = 6. Indicate on your diagram the coordinates of the focus, the vertex and the equation of the directrix. (ii) Determine the equation of the parabola, P. 1 (c) For what values of a will the equation ax 5x a + + be positive definite? 3 (d) A ball is dropped from a height of 10 metres and each time it bounces, it reaches 4 5 of it s previous height. What is the total distance travelled by the ball? End of Question 13 11
Question 14 (15 marks) Use a separate writing booklet. (a) The diagram show the curve y = ln x. x The curve crosses the x-axis at A and has a stationary point at B. (i) State the coordinates of A. 1 (ii) Find the coordinates of the stationary point B, of the curve, giving your answer in an exact form (iii) Find the exact value of the equation of the normal to the curve at the point where x= e 3. (b) Sketch the graph y = cos x for π x π. Question 14 continues on page 13 1
Question 14 (continued). (c) The diagram above describes the velocity, v, of an electrical pulse along a wire in metres/second. Refer to the diagram and answer the questions below. (i) When is the pulse travelling in a positive direction? 1 (ii) When is the pulse stopped? 1 (iii) Describe the motion of the particle for 0 < t < 3. (iv) Find the area between the curve and the t axis for 3 t 6. 1 (d) If α and β are the roots of the quadratic equation 3x 4x 1 0 =, find (i) α + β and αβ. 1 (ii) ( α )( β ) + + End of Question 14 13
Question 15 (15 marks) Use a separate writing booklet. (a) The diagram shows a sketch of the curve 4x y =. (i) Use the trapezoidal rule with three function values to find an approximate 1 4 value for x dx. 0 1 4 (ii) Is the approximate for x dx, an under approximation or 1 0 an over approximation? Explain your choice. (b) The concentration of the pesticide, Xpesto, in soil can be modelled by the equation Pt = Pe where P 0 is the initial concentration P t is the concentration at time t t is the time, in days, after the application of the pesticide. 0 kt (i) Once in the soil, the half-life of a pesticide is the time taken for it s concentration to be reduced to one half of its initial value. If the half-life of Xpesto is 5 days, find the value of k correct to significant figures. (ii) Eighty days after the initial application, what is the percentage decrease 3 in concentration of Xpesto? Question 15 continues on page 15 14
Question 15 (continued). (c) The region bounded by the curve y ln ( x ) y = 0 and y = is rotated about the y-axis to form a solid. Find the exact volume of the solid. = and the y-axis between 4 (d) Given that a 10 5sin 3 x dx =, 0 a < π, calculate the value of a. 3 0 3 End of Question 15 15
Question 16 (15 marks) Use a separate writing booklet. (a) Solve tan xsin x = tan x for 0 x π. 3 (b) An open cylindrical can is to have a surface area of 0π cm. The can has no lid. (i) Let r centimetres be the radius of the can and h centimetres be its height. 1 Show that 0 r h = r. (ii) Hence, show that the total volume of the can is given 1 1 3 by V = 10 πr πr. (iii) Show that the maximum volume is obtained when the height of the can 4 is equal to it s radius. (c) (i) Show that the equation 4cosec θ cot θ= k, where k 4, can be written in the form k 1 sec θ= k 4 (ii) Hence, or otherwise, solve the equation 4 ( x ) ( x ) 4cosec + 75 cot + 75 = 5, giving all values of x in the interval 0 < x < 180. End of paper 16