SAJC 9758 2025 Prelim P2

[Turn Over ST ANDREW’S JUNIOR COLLEGE General Certificate of Education Advanced Level Higher 2 MATHEMATICS 9758/02 Paper 2 16 September 2025 (Tuesday) Preliminary Examination 3 hours Additional Materials: Printed Answer Booklet List of Formulae and Results (MF27) ______________________________________________________________________________ READ THESE INSTRUCTIONS FIRST Answer all questions. Total marks: 100 Write your answers on the Printed Answer Booklet. Follow the instructions on the front cover of the answer booklet. Give non-exact numerical answers correct to 3 significant figures, or 1 decimal place in the case of angles in degrees, unless a different level of accuracy is specified in the question. You are expected to use an approved graphing calculator. Unsupported answers from a graphing calculator are allowed unless a question specifically states otherwise. Where unsupported answers from a graphing calculator are not allowed in a question, you must present the mathematical steps using mathematical notations and not calculator commands. You must show all necessary working clearly. The number of marks is given in brackets [ ] at the end of each question or part question. This document consists of 8 printed pages.

2 [Turn Over Section A: Pure Mathematics [40 marks] 1 (i) Solve the inequality 23 4 2xx−  + . [4] (ii) Hence, solve 2 2 3 1 214 xxxx +− . [2] 2 (a) It is given that 64f ( )x rpx qx++= , where p, q, and r are real constants. (i) Show that if x = is a root of f ( ) 0x = , then x =− is also a root. [1] (ii) Given now that 5x= and x = are roots of f ( ) 0x = , where Re( ) 0  and Im( ) 0  , write down all the remaining roots. [3] (b) The complex number z satisfies the equation 32 (3 ) (2 6i) 6 0z a z z+ − − + − = , where a is a complex number. It is given that one of the roots is 1 + i. Find a and the other roots of the equation. [5] 3 The diagram below shows  ABC and  AED.  ABC is an equilateral triangle with each side measuring 2 units and  AED is a right-angled triangle with 4ADE x  = + . Points C and E both lie on the line BD. (a) Show that 31 tan 4 BD x=+  + . [2] (b) Given that x is sufficiently small for 3x and higher powers of x to be neglected, show that 2BD a bx cx + + where a, b and c are exact constants to be determined. [4] A B C D E 4 x + 2

3 [Turn Over 4 A sequence 1 2 3, , , ...u u u is such that 123 19 16, for 0rru u r+ = +  and 1 2u = . (i) Write down the value of 10u , giving your answer correct to 4 decimal places. [1] (ii) It is given that as r→ , rul → . Show that 4l = . [1] (iii) Hence, find the smallest integer r for which ru exceeds 99.9% of l . [3] It is known that the kth term of this sequence is given by 1 1942 23 k ku − =−  . (iv) Given that a new series nS is defined by ( ) 1 n nk k S u c = =− , where c is a real constant. Express nS in the form of 191 23 n An B  +−   , where A is a constant in terms of c and B is a real number. [3] (v) Determine the possible values of c for nS to diverge as n→ . [1] 5 A curve C has parametric equations πtan cot , tan cot , for 0 . 4xy     = − = +   (i) Show that d cos 2d y x =− . [3] The line L is the tangent to C at point P with parameter p. (ii) Show that the equation of L is (cos2 ) 2sin 2y p x p+= . [3] Let Q and R be the points where L cuts the x-axis and y-axis respectively. (iii) Given that p increases at a constant rate of 0.2 radians per second, find the exact rate of change of the area of  OQR when π 6p = , where O is the origin. [4]

4 [Turn Over Section B: Statistics [60 marks] 6 A group of 8 tourists – 2 married couples and 4 singles – travels to the airport in two taxis, X and Y. Each taxi can take 4 passengers. The 8 tourists divide themselves into two groups of 4, one group for taxi X and one group for taxi Y. (i) Find the number of different ways in which this can be done if each married couple must travel together in the same taxi. [2] Each taxi can take 1 passenger in the front and 3 passengers in the back (see diagram). (ii) Find the number of seating arrangements so that each married couple sit next to each other at the back of the taxi. [3] 7 (a) With reference to the Venn diagram below, S is the universal set and A and B are non-empty proper subsets of S. Explain whether A and B are independent, and write down the value of P( | ).AB [2] Front Back Taxi X Front Back Taxi Y A B S

5 [Turn Over (b) A fair die has three sides numbered 2, two sides numbered 1 and one side numbered 0. A game is played by throwing this fair die three times and the score, X, is the sum of the numbers obtained. (i) Show that 7P( 2) 72X == . [2] (ii) The table shows an incomplete probability distribution of X. Find the missing probabilities, showing your workings clearly. [3] x 0 1 2 3 4 5 6 P( )Xx= 1 216 1 36 7 72 7 24 The game is played 5 times. (iii) Find the probability that exactly one of the games obtained a score of 0 and exactly 2 games obtained a score of more than 2. [3] 8 A company sells toys in blind boxes. Each blind box contains exactly one toy. On average, 2% of such blind boxes contain a faulty toy. A collector buys 15 such blind boxes at random. (i) State, in context, two assumptions needed for the number of faulty toys he gets to be well modelled by a binomial distribution. [2] Assume now that the number of faulty toys the collector gets has a binomial distribution. (ii) Find the probability that the collector gets no less than 3 faulty toys. [1] (iii) Find the probability that the fourteenth box is the second box with a faulty toy. [2] The company also sells keychains. The probability of a keychain being faulty is p. The number of faulty keychains follows a binomial distribution. Faults on keychains are independent of faults on toys in blind boxes. (iv) Write down an expression, in terms of p, for the probability that in a random sample of 3 keychains, exactly one is faulty. [1] A surprise pack contains 2 randomly chosen blind boxes and 3 randomly chosen keychains. Given that a randomly selected surprise pack contains at most 1 faulty item, the probability that one of the toys is the only faulty item in the pack is 0.035. (v) Write down an equation satisfied by p. Hence, find the value of p. [4]

6 [Turn Over 9 (i) Sketch a scatter diagram that might be expected when x and y are related approximately as given in each of the cases (A) and (B) below. In each case your diagram should include 6 points, approximately equally spaced with respect to x, and with all x- and y- values positive. The letters a, b, c and d represent constants. (A) 2y a bx=+ , where a is positive and b is negative, (B) dyc x=+ , where c is positive and d is positive. [2] Alvin is investigating the relationship between the amount of screen time per day and the sleep quality of youths. He recorded the sleep quality score, y, on a scale of 1 to 10 with 1 being the worst and 10 being the best, of particular youths with x hours of screen time per day. x 1.6 1.7 1.9 2.0 2.5 2.8 3.9 y 9 8 7 6 4 3 1 (ii) Draw a scatter diagram for these values, labelling the axes. [1] (iii) Explain which of the two cases in part (i) is the most appropriate for modelling these values and calculate the product moment correlation coefficient for this case. [2] (iv) Alvin wants to estimate the sleep quality score when the screen time per day is 3 hours. Use the case that you have identified in part (iii) to find the equation of a suitable regression line and use your equation to find the required estimate. Explain w