YI 2026 H2 Math JC2 Prelim Paper 2 (Qns)

This document consists of 6 printed pages. [Turn Over YISHUN INNOVA JUNIOR COLLEGE JC 2 PRELIMINARY EXAMINATION Higher 2 CANDIDATE NAME CG MATHEMATICS Paper 2 Additional Materials: Printed Answer Booklet List of Formulae and Results (MF27) 9758/02 31 August 2026 3 hours READ THESE INSTRUCTIONS FIRST Write your CG, index number and name on the work you hand in. 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 ©YIJC 9758/02/JC2PE/26 Section A: Pure Mathematics [40 marks] 1 Do not use a calculator in answering this question. The complex number z is such that 1 3i z=− . (a) Find z and arg( )z . [2] The points A, B and C represent the complex numbers z , iz and i2z+ respectively. (b) Mark the origin O and the points A, B and C on an Argand diagram, illustrating clearly the geometrical relationship between them. [3] (c) Hence, find the exact value of ( )arg i 2z+ . [2] 2 (a) The non-zero vectors a, b and c are such that ac is parallel to ab . Show that mn=+c a b , where m and n are constants. [3] (b) Given that a is a non-zero vector and b is a unit vector such that 2=ab and 2.5=ab  , show that the acute angle between the vectors a and b is 1 4tan 5−  and find the exact value of a . [4] 3 The function f is defined by ( )( ) 1f : , 12xx xx −−  , 1x , 2x . (a) Sketch the graph of ( )fyx= , giving the equations of any asymptotes and the coordinates of any turning points. [2] (b) Hence state the range of f. [1] (c) Explain why the function 2f does not exist. [1] The domain of f is further restricted to ( ,a− where a is a constant. (d) State the largest value of a such that the function 1f− exists. [1] (e) Hence find ( )1f x− and state its domain. [4]

3 ©YIJC 9758/02/JC2PE/26 [Turn Over 4 (a) Find 22edxxx . [2] (b) Use the substitution 2wx= to find 32cos dx x x . Hence, using double angle formulae, show that ( ) 3 2 2 2 2 2 411cos d 2 sin 2cos28x x x x x x x C = + + + [5] The diagram shows part of the curves 3 2 21cos 2y x x =  and 1 2 2exyx= , where 0 x  . The region R is bounded by the curves and the line x = . (c) Find the exact volume of the solid generated when R is rotated through 2  radians about the x-axis. [3] 5 The diagram shows an open container made in the form of an upright prism with an equilateral triangular base of side x cm, and height h cm. The material used for the base costs 2 cents per 2cm , while the material used for the sides costs 5 cents per 2cm . The total cost of the container is a fixed value of C cents. (a) Show that the capacity, V cm3 of the container is given by 233 60 2x C x − . [3] (b) Use differentiation to show that V is maximum when the cost of the base is 3 C cents. [4] x y O R x cm h cm

4 ©YIJC 9758/02/JC2PE/26 Section B: Probability and Statistics [60 marks] 6 A mini–Lecture Theatre has 11 seats in 2 rows as shown in the diagram below. Find the number of ways that a class of 5 girls and 3 boys can be seated (a) without restriction, [1] (b) with the girls and the boys occupying different rows. [2] The lecturer has prepared 8 different questions and will ask each student in the class to answer one question in turn. (c) Find the number of ways this can be done if no two consecutive questions are answered by boys. [2] The class needs to form two groups of 4 each for a group activity. (d) Find the number of ways that the groups can be formed if the 3 boys cannot be in the same group. [2] 7 A technician has a set of 5 cables, 3 of which are faulty. He tests the cables one by one , without replacement, in a random order until all the faulty cables are tested. The random variable X denotes the number of tests carried out until all the faulty cables are tested. (a) Show that ( ) 1P3 10X == and find the probability distribution of X. [3] (b) The technician noted that the first cable tested was faulty. Find the probability that exactly 4 tests were required. [2] The testing process incurs a cost, $C, which depends on the number of tests carried out and is given by the formula C = 5 + 10X. (c) Find Var(C). [2] (d) The test ing process is carried out on 80 sets of cables independently. Using a suitable approximation, find the probability that the mean testing cost exceeds $51.50. [3] Lecturer Front row Back row

5 ©YIJC 9758/02/JC2PE/26 [Turn Over 8 The number of active users of a new mobile application, y thousand users, after x number of months since its launch, is given in the following table. x 1 2 4 6 8 10 y 85 55 38 31 27 25 (a) Sketch a scatter diagram of the data. [1] (b) In the context of the question, comment whether a linear model would be appropriate. [1] (c) Suppose that the relationship between x and y is modelled by an equation of the form ayb x=+ , where a and b are constants. Using the scatter diagram, explain whether a is positive or negative. [1] (d) Find the product moment correlation coefficient and the equation of the regression line for the model in part (c). [2] (e) Use your equation to estimate the number of active users of the new mobile application 9 months after the launch. Give two reasons why you would expect this estimate to be reliable. [3] 9 A recycling centre collects plastic bottles from households. It is estimated that, on average, 6% of the plastic bottles collected are non -recyclable. Each day, the recycling centre randomly selects 40 plastic bottles for checking and the number of non-recyclable plastic bottles found is the random variable X. (a) State, in the context of the question, two assumptions needed to model X by a binomial distribution. [2] You are now given that X can be modelled by a binomial distribution. (b) Find the probability that, on a randomly chosen day, at least 4 non -recyclable plastic bottles are found. [2] (c) Find the probability that, in a randomly chosen week of 5 days, at least 4 non -recyclable plastic bottles are found on no more than 2 days. [2] The recycling centre also collects cardboard boxes. The number of non-recyclable cardboard boxes also follows a binomial distribution. The probability that a randomly selected cardboard box is non-recyclable is p. The recyclability of plastic bottles is independent of the recyclability of cardboard boxes. A quality control officer randomly selects a sample of 3 plastic bottles and 2 cardboard boxes for inspection. The probability that the sample contains at most 1 non-recyclable item is 0.9. (d) Write down an equation satisfied by p. Hence, find the value of p. [4]

6 ©YIJC 9758/02/JC2PE/26 10 A company produces three electrical components, Type A, Type B and Type C. The masses, in grams, of Type A components follow the distribution N(250, 3²) and the masses, in grams, of Type B components follow the distribution N(240, 4²). The ma