TMJC 9758 2025 Prelim P2

Tampines Meridian Junior College 2025 JC2 Preliminary Examination H2 Mathematics CANDIDATE NAME: ___________________________________________________ CIVICS GROUP: _______________________________________________________ __________________________________________________________________________________ H2 MATHEMATICS 9758/02 Paper 2 22 SEPTEMBER 2025 3 hours Additional materials: Printed Answer Booklet List of Formulae and Results (MF27) ________________________________________________________________________________ READ THESE INSTRUCTIONS FIRST Answer all the questions. Write your answers in the spaces provided in the Printed Answer Booklet. Follow the instructions on the front cover of the Printed 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 are reminded of the need for clear presentation in your answers. The number of marks is given in brackets [ ] at the end of each question or part question. _________________________________________________________________________________ This document consists of 7 printed pages and 1 blank page. TAMPINES MERIDIAN JUNIOR COLLEGE JC2 PRELIMINARY EXAMINATION [Turn Over

2 Section A: Pure Mathematics [40 marks] 1 (a) Without using a calculator, solve the inequality 2 5 12 5 3 x xx + − + + . [4] (b) Hence, solve exactly the inequality 2 cos 5 12cos 5cos 3 x xx + − + + for 0 x  . [2] 2 It is given that ( ) 2f a b c  = + + , where a, b and c are real constants. The curve with equation ( )fy = passes through the point with coordinates ( )0,1 and has a turning point at 31,.32 − (a) Find the exact values of a, b and c. [3] (b) In a triangle ABC, 1AB= , 3AC= and angle 6BAC =+ radians. Given that  is sufficiently small for 3 and higher powers of  to be neglected, show that ( ) 2 f.BC  [3] (c) Hence, show that 2331. 28BC  + + [3] 3 The function f is defined by ( ) 2 for 4 1,f for 1 . 4 2 2 x x xx x −  −= −   + + It is given that ( ) ( )f f 6xx=+ for all real values of .x (a) Sketch the graph of ( )fy x= for .6 6x−  [3] (b) Hence sketch the graph of ( )f12y x= − for .6 6x−  [3]

3 4 A viral trend on social media gains popularity rapidly but eventually slows down due to oversaturation. Let P represent the popularity score of the trend at time t (measured in days), where 0 100.t The change in the trend’s momentum is modelled by 22 2 dd 0.05 .dd PP tt =−  (a) By substituting d d Pv t= , show that 20v tC= + where C is a real constant. [3] (b) Given that 0 and 10Pv== when 0,t= find the particular solution of P in terms of t. [4] (c) Find the popularity score of the trend when 30.t = [1] 5 Referred to the origin O, the points A, B and C have position vectors a, b and c respectively where a is a non-zero vector and bc . (a) Given that  = a b a c , show that ,−=b c a where  is a non-zero constant. [2] It is given that the area of the trapezium OABC is 10 units2, 3,=a 5=c and the angle between a and c is 30 . (b) Show that 5 units.BC = [3] (c) State the value of k such that OA kCB= , where k is a positive constant. [1] (d) The line OC meets the line AB at point D. Find OD in terms of c. [5] Section B: Probability and Statistics [60 marks] 6 For events A, B and C, it is given that ( ) ( ) ( )P 0.3, P 0.4, P 0.42,A B C= = = ( )P' A B C x  = and ( )P 0.02.A B C  = It is also given that events A and B are independent, and that events A and C are independent. (a) Find ( ) ( )P and P .A B A C [2] (b) Draw a Venn diagram to represent this situation, showing the probability in each of the eight regions, in terms of x where necessary. [3] (c) Find the greatest and least possible values of ( )P ' ' .A B C [2] [Turn Over

4 7 A coffee shop owner claims that its signature coffee has a mean caffeine content of 120 mg per cup. To investigate this claim, a random sample of 60 cups of coffee is selected and the caffeine content, x mg per cup, is summarised as follows. 6960x= 2 822465x = (a) Calculate unbiased estimates of the population mean and variance for the caffeine content per cup. [2] (b) Test, at the 5% level of significance, whether the coffee shop owner’s claim is supported by the data. You should state your hypotheses and define any symbols that you use. [4] The coffee shop also sells premium coffee, where its caffeine content is normally distributed with population variance 200 mg2. The coffee shop owner now claims that its premium coffee has a mean caffeine content less than 120 mg per cup. Another large random sample of n cups of premium coffee is selected and the sample mean caffeine content per cup is found to be 116.6 mg. A test is carried out at the 12 2 % level of significance and the result supports the owner’s claim. (c) Find the set of values that n can take. [4] 8 A manufacturer produces mystery boxes, each containing either a regular or a seasonal toy. On average, %p of the mystery boxes contain a seasonal toy. Amber orders n mystery boxes from the manufacturer. The number of seasonal toys that Amber gets is the random variable S. (a) State, in the context of the question, two assumptions needed for S to be well modelled by a binomial distribution. [2] You are now given that S can be modelled by a binomial distribution. (b) Given that ( ) ( )P 2 P 3SS= = = and ( )E 2.96S = , find the value of p. [4] Assume now that 5p= . The manufacturer now packs the mystery boxes into cartons of 12 each for sale. Each carton is checked for quality control. If there is at most 1 mystery box containing a seasonal toy, the carton is accepted. Otherwise, the carton is rejected. (c) Given that a randomly chosen carton is rejected, find the probability that no more than 30% of the boxes in the carton each contains a seasonal toy. [3]

5 9 A car dealer is investigating how the value of a car depreciates over time. A random sample of eight cars of the same model is selected and the current resale value, y thousand dollars of each car is recorded along with its age in x years. The results are shown in the table. Age of car (x years) 0 1 2 3 4 5 6 7 Resale Value (y thousand dollars) 30.0 25.8 22.3 19.2 16.5 14.4 12.5 10.9 (a) Draw the scatter diagram for these values, labelling the axes clearly. [1] It is thought that the resale value of a car, y , can be modelled by one of the formulae or ln ,y a bx y c dx= + = + where a, b, c and d are real constants. (b) Find, correct to 5 decimal places, the value of the product moment correlation coefficient between (i) y and x, (ii) ln y and x. [2] (c) Use your answers to parts (a) and (b) to explain which of y a bx=+ and ln y c dx=+ is the better model. [2] It is required to estimate the age of a car with a resale value of $18000. (d) Find the equation of a suitable regression line and use it to find the required estimate. [2] (e) Without the use of a graphing calculator, r e-write your equation from part (d) so that it can be used to estimate the resale value when the age is given in months. [2] [Turn Over

6 10 Wardrobe A contains 4 distinct blouses, labelled 1 2 3 4, , and T T T T . Wardrobe B contains 3 distinct pairs of