Student Number 2024 Preliminary Higher School Certificate Examination Mathematics Advanced General Instructions • Reading time – 10 minutes • Working time – 2 hours • Write using black or blue pen • Calculators approved by NESA may be used • A reference sheet is provided at the back of this paper • For questions in Section II, show relevant mathematical reasoning and/ or calculations • Write your Student Number at the top of this page Total marks: 70 Section I – 10 marks (pages 2–6) • Attempt Questions 1–10 • Allow about 15 minutes for this section Section II – 60 marks (pages 7–19) • Attempt Questions 11–26 • Allow about 1 hour and 45 minutes for this section Disclaimer: Every effort has been made to prepare these examinations in accordance with the NESA Documents. All care has been taken to ensure that this examination paper is error free and that it follows the style, format and material content of the Higher School Certificate Examination. No guarantee or warranty is made or implied that the ‘Trial’ Examination papers mirror in every respect the actual HSC Examination question paper in any or all courses to be examined. Real Education Resources Pty Ltd has no liability for any reliance, use or purpose related to these examination papers. Advice on HSC examination issues is only to be obtained from NESA. To ensure integrity and security, examination papers must NOT be removed from the examination room and may not be returned to students until 20th September 2024. These examination papers are supplied Copyright Free, as such the purchaser may photocopy and/or make changes for education purposes within the confines of their School or College. 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. Question 1 Maths Bank ? Consider the functions f pxq “ ´x2 and g pxq “ x. Which of the following represents the graph of y “ f pg pxqq? B.Maths Bank A.Maths Bank y y x O O x D.Maths Bank C.Maths Bank y y O x O –2– x Question 2 Maths Bank $ 2 ’ &x ´ 1 Consider the function f pxq “ 3 ´ 5x ’ % 2 xď0 0ăxď3 xą3 What is the solution to the equation f pxq “ ´3? A. x “ ´2 1 B. x “ 5 6 C. x “ 5 D. x “ 2 Question 3 Maths Bank Consider the curve shown below: y ´1 1 O 4 ´8 Which of the following best represents the equation of the curve? A. y “ 2x3 ´ 8x ´ 2x ` 8 B. y “ ´2x3 ` 8x2 ` 2x ´ 8 C. y “ ´2x3 ´ 8x2 ` 2x ` 8 D. y “ 2x3 ` 8x2 ´ 2x ´ 8 Question 4 Maths Bank 1 ´ 3x The angle between the line y “ and the positive x-axis is θ . 2 What is the value of θ , correct to the nearest degree? A. 34˝ B. 56˝ C. 124˝ D. 146˝ –3– x Question 5 Maths Bank The diagram below shows the circle x2 ` py ´ 2q2 “ 1 and the parabola y “ ax2 , where a is a positive constant. The circle and the parabola intersect at two points A and B. y A 2 B O x How many solutions exist to the equation y ` py ´ 2q2 “ 1? a A. 1 B. 2 C. 3 D. 4 Question 6 Maths Bank π 2 It is known that cos θ “ , where 0 ă θ ă . 3 2 ˆ ˙ 3π What is the exact value of sin ´θ ? 2 2 A. ´ ? 5 2 B. ´ 3 2 C. 3 2 D. ? 5 –4– Question 7 Maths Bank The sector of radius 13 cm shown below has an arc length of 18 cm. 18 cm 13 cm What is the area of the sector? A. 84.5 cm2 B. 117 cm2 C. 162 cm2 D. 234 cm2 Question 8 Maths Bank The graph of the function y “ e2x is shown. y A 2 O Which of the following is true at the point A? A. f pxq ă f 1 pxq ă 0 B. f 1 pxq ă f pxq ă 0 C. 0 ă f pxq ă f 1 pxq D. 0 ă f 1 pxq ă f pxq –5– x Question 9 Maths Bank Consider the statements: I. If two events, A and B, are independent, then P pA X Bq “ P pAq ˆ P pBq. II. If two events, A and B, are mutually exclusive, then P pA Y Bq “ 1. Which of the following is correct? A. Neither statement I nor statement II are always true. B. Only statement I is always true. C. Only statement II is always true. D. Both statements I and II are always true. Question 10 Maths Bank A fair four-sided die is rolled. Let X be the discrete random variable representing the number that appears face down on the die after it is rolled. The probability distribution of the discrete random variable X is shown. x 1 2 3 4 P pX “ xq 0.25 0.25 0.25 0.25 What is the value of Var pXq? A. 1.25 B. 2.5 C. 5 D. 7.5 –6– Section II 60 marks Attempt Questions 11–26 Allow about 1 hour and 45 minutes for this section Answer each question in the space provided. Extra writing booklets are available. For questions in Section II, your responses should include relevant mathematical reasoning and/or calculations. Question 11 (2 marks) ? ? ? 6 ` 3 10 ? Find the value of a such that “ 3 2 ` a. 2 2 .............................................................................. .............................................................................. .............................................................................. .............................................................................. .............................................................................. .............................................................................. Question 12 (2 marks) Maths Bank Solve |2x ´ 7| “ 3. 2 .............................................................................. .............................................................................. .............................................................................. .............................................................................. .............................................................................. .............................................................................. –7– Question 13 (4 marks) Maths Bank Lines ℓ1 and ℓ2 have the equations 3x ´ y ` 7 “ 0 and x ` 3y ´ 11 “ 0, respectively. The lines intersect at point P. (a) 2 Find the coordinates of P. ........................................................................ ........................................................................ ........................................................................ ........................................................................ ........................................................................ ........................................................................ (b) Hence, find the equation of the line passing through P that is parallel to the line 2x ` y ` 7 “ 0. 2 ........................................................................ ........................................................................ ........................................................................ ........................................................................ ........................................................................ ........................................................................ Question 14 (2 marks) Maths Bank 1 1 ` “ 2 sec2 x. Show that 1 ´ sin x 1 ` sin x .............................................................................. .............................................................................. .............................................................................. .............................................................................. .............................................................................. .............................................................................. –8– 2 Question 15 (5 marks) (a) Show that 2 ´ 1 2x ` 7 “ . x`4 x`4 1 ........................................................................ ........................................................................ ........................................................................ ........................................................................ 2x ` 7 . x`4 (b) Hence, sketch the graph of y “ (c) State the domain and range of the graph. ........................................................................ ........................................................................ ........................................................................ ........................................................................ –9– 2 2 Question 16 (3 marks) Maths Bank Sketch the graph of x2 ` y2 ´ 6y “ 0. 3 .............................................................................. .............................................................................. .............................................................................. .............................................................................. Question 17 (3 marks) Maths Bank Solve 2 cos2 x “ 1 for ´180˝ ď x ď 540˝ . .............................................................................. .............................................................................. .............................................................................. .............................................................................. .............................................................................. .............................................................................. .............................................................................. .............................................................................. – 10 – 3 Question 18 (3 marks) Maths Bank Let f pxq “ ax2 ` bx ` c. In the diagram below, the graph of y “ f pxq crosses the x-axis at 5, passes through the point p´1, 4.5q and is symmetrical about the line x “ 1.5. y p´1, 4.5q O 5 x x “ 1.5 Find the values of a, b and c. .............................................................................. .............................................................................. .............................................................................. .............................................................................. .............................................................................. .............................................................................. .............................................................................. .............................................................................. – 11 – 3 Question 19 (7 marks) Maths Bank Zara drives due north along a straight road. At checkpoint A, she notices the top of a telecommunications tower on a bearing of 320˝ with an angle of elevation 28˝ . At point B, Zara observes the top of the same telecommunications tower, of height h metres, on a bearing of 214˝ with an angle of elevation 50˝ . Checkpoints A and B are 6.2 km apart. This information is shown in the diagram below where T and O represent the top and bottom of the tower, respectively. T h O A (a) 6.2 km B N Show that OA “ h tan 62˝ . 1 ........................................................................ ........................................................................ (b) 1 Find a similar expression for OB. ........................................................................ ........................................................................ (c) Find the size of ∠AOB. 2 ........................................................................ ........................................................................ ........................................................................ ........................................................................ Question 19 continues on the next page – 12 – Question 19 (continued) (d) Hence, find the height of the tower, correct to one decimal place. ........................................................................ ........................................................................ ........................................................................ ........................................................................ ........................................................................ ........................................................................ ........................................................................ ........................................................................ ........................................................................ End of Question 19 Please turn over – 13 – 3 Question 20 (3 marks) Maths Bank ? Find the equation of the tangent to the curve y “ e2x`1 ´ 9 at the point where x “ 0. 3 Leave your answer in exact form. .............................................................................. .............................................................................. .............................................................................. .............................................................................. .............................................................................. .............................................................................. .............................................................................. .............................................................................. .............................................................................. Question 21 (4 marks) Maths Bank As it is being filled, the volume of water inside a reservoir, V litres, is described by the equation 7200 V “ 100 ´ t ` 80 where t is the time in minutes since the filling process began. (a) Find the average rate of change in the volume of water inside the reservoir over the first 15 minutes, correct to two decimal places. 2 ........................................................................ ........................................................................ ........................................................................ ........................................................................ (b) Find the rate of change in the volume of water inside the reservoir 8 minutes after the filling process begins, correct to two decimal places. ........................................................................ ........................................................................ ........................................................................ ........................................................................ ........................................................................ – 14 – 2 Question 22 (3 marks) Maths Bank Find the value of x such that 4 ˆ 3x`1 “ 52x , correct to two decimal places. .............................................................................. .............................................................................. .............................................................................. .............................................................................. .............................................................................. .............................................................................. .............................................................................. .............................................................................. .............................................................................. Please turn over – 15 – 3 Question 23 (7 marks) Maths Bank Consider the function f pxq “ (a) 2x2 . x`3 Find f 1 pxq. 2 ........................................................................ ........................................................................ ........................................................................ ........................................................................ (b) Hence, find the equation of the normal to the graph of y “ f pxq at the point where x “ 6. 2 ........................................................................ ........................................................................ ........................................................................ ........................................................................ ........................................................................ (c) The normal at x “ 6 meets the graph of y “ f pxq again when x “ p. Find the value of p. ........................................................................ ........................................................................ ........................................................................ ........................................................................ ........................................................................ ........................................................................ ........................................................................ ........................................................................ – 16 – 3 Question 24 (3 marks) Maths Bank In a particular school, it is known that the probability of a student not completing their homework is 0.2, and that whether or not a student completes their homework is independent of other students. (a) If two students are selected at random, what is the probability that they have both completed their homework? 1 ........................................................................ ........................................................................ ........................................................................ (b) A teacher randomly selects students’ work and wants to be 95% certain that they find at least one student who has not completed their homework. What is the minimum number of students that must be selected? ........................................................................ ........................................................................ ........................................................................ ........................................................................ ........................................................................ ........................................................................ ........................................................................ Please turn over – 17 – 2 Question 25 (4 marks) Maths Bank For constants a, b and c such that 0 ă a ă b, it is known that loga b2 “ c and logb a “ c ´ 1 Find expressions for a in terms of b. .............................................................................. .............................................................................. .............................................................................. .............................................................................. .............................................................................. .............................................................................. .............................................................................. .............................................................................. .............................................................................. .............................................................................. .............................................................................. .............................................................................. – 18 – 4 Question 26 (5 marks) Maths Bank Let X be a discrete random variable such that P pX ď xq “ 0.125x ` 0.015x2 for x “ 1, 2, 3, 4, 5. (a) 2 Find P pX “ 2q. ........................................................................ ........................................................................ ........................................................................ ........................................................................ ........................................................................ (b) Hence, find the expected value of X. ........................................................................ ........................................................................ ........................................................................ ........................................................................ ........................................................................ ........................................................................ ........................................................................ ........................................................................ End of paper – 19 – 3
