NYJC 9758 2025 prelim P2

This document consists of 6 printed pages and 2 blank pages. NANYANG JUNIOR COLLEGE Internal Examinations © NYJC 2025 [Turn Over NANYANG JUNIOR COLLEGE JC2 PRELIMINARY EXAMINATION Higher 2 CANDIDATE NAME CT CLASS 2 4 Centre Number/ Index Number / MATHEMATICS 9758/02 Paper 2 17 September 2025 3 hours Additional Materials: Printed Answer Booklet List of Formulae and Results (MF27) READ THESE INSTRUCTIONS FIRST Answer all questions. 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. The total number of marks for this paper is 100.

2 NYJC 2025 JC2 Preliminary Examination 9758/02 Section A: Pure Mathematics [40 marks] 1 (a) Find the set of values of x for which 3 123 xx− − . [2] (b) Without using a calculator, solve 2 18 15 14 x xx + +− − . [4] 2 (a) It is given that the roots to the equation 32i 5i 0x x ax b− + + + = , where a and b are purely imaginary, are 2 i, 2 i and 1.+− Explain why the complex roots occur in conjugate pairs. [2] (b) By using (a) and an appropriate substitution, find the roots of the equation 3 2 *i 5i 0x x a x b+ + + = , where the complex conjugate of a is denoted by *a . [4] 3 (a) Given that f is a continuous function, explain, with the aid of a sketch, why the value of 2 2 4 2lim f 1 f 1 ... f 1 n n n n n n→      + + + + + +           is ( ) 3 1 f dxx . [3] (b) Hence evaluate 1 22lim ln 1 n n k k nn→ = + exactly. [4] 4 The equation of a curve C is 33 2,x xy y k+ + = where k is a constant. (a) Find d d y x in terms of x and y. [2] It is given that C has a tangent which is parallel to the y-axis. (b) Show that the y-coordinate of the points of contact of the tangent with C must satisfy 63216 4 0y y k+ + = . Hence show that 1 54k  . [6] (c) Find the possible values of k in the case where the line 6x=− is a tangent to C. [2]

3 NYJC 2025 JC2 Preliminary Examination 9758/02 [Turn Over 5 Two planes 1p and 2p have respective cartesian equations given by 48x y z+ + = and 4 3 0x y z+ − = . (a) Find the sine of the acute angle between 1p and 2p in the form m n where m and n are positive integers to be determined. [3] (b) Verify that the point A with coordinates (1,0,4) lies on 1p and 2p . Hence, w ithout the use of a calculator, find the vector equation of the line l formed by the intersection of 1p and 2p . [3] (c) It is given that B is a point on 1p with coordinates (1,3,1). Show that AB is perpendicular to l and hence use (a) to deduce exactly the shortest distance from B to 2p . [3] (d) Find the cartesian equation of the plane 3p which contains B and is perpendicular to both 1p and 2p . [2] Section B: Probability and Statistics [60 marks] 6 Two fair six-sided dice are thrown. Events A, B and C are defined as follows. A: sum of the two scores is odd B: at least one of the two scores is greater than 4 C: the two scores are equal (a) Find, giving your reasons clearly in each case, which pair of the events are (i) mutually exclusive, [1] (ii) independent. [3] (b) Find P(C | B). [2] 7 The random variable S is the number of successes in 5 independent trials of an experiment in which the probability of success in any trial is 1 3 . The random variable D is the difference between the number of successes and the number of failures in 5 such trials. (a) State the values that D can take. [1] (b) Show that 1P) 8(1 40D== . [1] (c) By finding the probability distribution of D, find the exact value of E(D2). [3] (d) Find the exact values of E( )S and E(S2). [2] (e) Hence, by showing that 22 4 20 25D S S= − + , verify the correctness of the value of E( D2) found in part (c). [2]

4 NYJC 2025 JC2 Preliminary Examination 9758/02 8 Helen is a conservationist who monitors the health of fish in a river. She is investigating whether a new water-treatment plant has affected the mass of adult trout in the river. Previously, the mean mass of adult trout in the river has been w kg. Helen carries out a test, at the 10% level of significance to find out whether there has been a change in the mean mass of adult trout in the river. Helen catches 45 adult trout and measures the mass, X, kilograms, of each fish. Her summarised data are as follows. n =45 80.1x= 2( 24.29)x x−= (a) Calculate unbiased estimates of the population mean and variance of the mass of adult trout. [2] (b) Use an algebraic method to calculate the set of values of w for which there is insufficient evidence to conclude that there has been a change in the mean mass of adult trout in the river. You should state your hypotheses and define any symbols you use. [6] (c) Explain why there is no need for Helen to know anything about the population distribution of the mass of adult trout. [2] 9 A small ceramic workshop produces 100 plates each working day. Some of the plates turn out to be faulty. (a) State, in the context of the question, two assumptions needed for the number of faulty plates made in a day to be well modelled by a binomial distribution. [2] Assume now that the number of faulty plates produced each working day has the distribution B(100, p). (b) Show that the probability that exactly 3 faulty plates are produced on a randomly chosen working day is 3 97161700 (1 ) .pp − [1] (c) Given that the most likely number of faulty plates produced on a working day is 3, find the possible range of values of p, leaving your answer in exact form. [4] The workshop also produces bowls on each working day. The number of faulty bowls also follows a binomial distribution. The probability that a bowl is faulty is q. Faults on plates are independent of faults on bowls. The plates and bowls are sold in sets of 2 randomly chosen bowls and 2 randomly chosen plates. In the case where 0.01p= , the probability that a set contains at most 1 faulty item is 0.88. (d) Write down an equation satisfied by q. Hence find the value of q. [4]

5 NYJC 2025 JC2 Preliminary Examination 9758/02 [Turn Over 10 The director of an art gallery wishes to find out the relationship between the time spent, t minutes, in the art gallery and the age, a (in years), of the visitors. Ten visitors were randomly selected and the data are summarised as follows: (a) A linear model is proposed for the relationship between t and a. Given that the least squares regression line of t on a is 0.8957914 14.16013ta=+ , show that 40k = . [2] (b) Sketch a scatter diagram of t against a for the data given in the table and draw the line given in (a) on the same diagram. [2] (c) Use the least squares regression line of t on a in (a) to estimate the time spent in the art gallery by a 60-year old visitor. [1] (d) A regression line t ma c=+