TMJC 2026 Prelim P2 QP

Tampines Meridian Junior College 2026 JC2 Preliminary Examination H2 Mathematics CANDIDATE NAME: ___________________________________________________ CIVICS GROUP: _______________________________________________________ __________________________________________________________________________________ H2 MATHEMATICS 9758/02 Paper 2 21 SEPTEMBER 2026 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 9 printed pages and 1 blank page. TAMPINES MERIDIAN JUNIOR COLLEGE JC2 PRELIMINARY EXAMINATION [Turn Over

2 Section A: Pure Mathematics [40 marks] 1 (a) On the same diagram, sketch the graphs of y x a=− and ( ) 21,yx=+ where a is a real constant such that 1.a You should show clearly the axial intercepts of both graphs. [3] (b) Hence, solve the inequality ( ) 213xx+  − , giving your answer in exact form. [3] 2 Referred to the origin O, points A and B have position vectors a and b respectively. Point C lies on OA such that ,OC kOA= where 0 1.k Point D lies on OB such that 52OD OB= (see diagram above). (a) Show that the vector equation of the line AD can be written as 2 5= + −r a b a , where  is a parameter. [1] (b) Find, in a similar form, the vector equation of the line BC in terms of .k Hence, find the position vector of point E, where line s BC and AD intersect, expressing your answer in terms of , and . kab [5] (c) Find the area of triangle OEB. Express your answer in the form ( )f, abk  where ( )f k is a function of k. [2] (d) Hence, or otherwise, give the geometrical interpretation of ( ) 1 lim f k k →    ab . [1] O A B C D E

3 [Turn Over 3 A tentage is designed in the shape of a right pyramid comprising of a regular hexagonal base and six identical triangular surfaces. The pyramid has height h m. Each side of the regular hexagonal base measures x m (see figure above). The interior volume of the tentage, 3 mV , is set to be 375 3 m2 . A layer of solar reflective coating is applied to the six exterior triangular surfaces of the rooftop to reflect sunlight and infrared heat away from the shelter. To reduce the amount of coating used, the total surface area of the six exterior triangular surfaces, denoted by 2 mS , must be minimised. (a) Show that 4 2 27 50625 4 xS x=+ . [4] (b) Using differentiation, calculate the value of x for which S is a minimum and find the corresponding minimum value of S. (You need not prove that the value of S is a minimum.) [4] [The volume of a pyramid is 1 3 base area  height.] h m x m

4 4 The function f is defined by ( ) πsin for 0 1,f 2 2 for 1 2, x axx x a x   +   =  − + +   where a is a positive real number. It is given that ( ) ( )f f 2xx=+ for all real values of .x (a) Sketch the graph of ( )fy x= for 0 6.x  [3] (b) Find the exact value of ( ) 3 0 f dxx in terms of a. [4] 5 The function f is defined by 2f : 4 12 for .x x x x − +   (a) Explain, with the aid of a sketch or otherwise, why f does not have an inverse. [2] The domain of f is now restricted to xk , where k is an integer, such that the inverse function 1f − exists. (b) State the greatest possible value of k. [1] For the remaining parts of this question, take k to be the value found in part (b). (c) Define 1f − in a similar form. [3] The function g is defined by g : ln , for , 0 5.x x x x     (d) Show that the composite function fg exists. [2] (e) Find the range of fg in exact form. [2]

5 [Turn Over Section B: Probability and Statistics [60 marks] 6 Four students, two female teachers and one male teacher stand in a line for a photo-shoot. (a) Find the number of arrangements if there are no restrictions. [1] (b) Find the number of arrangements in which the two female teachers are together. [1] (c) Find the number of arrangements in which the three teachers are all separated. [2] The students and teachers attend a workshop in which they are seated at a round table. (d) Find the probability that the two female teachers are separated by exactly one student given that the two female teachers are separated. [4] 7 The table below gives the median monthly household employment income per household member, x dollars, and the total life expectancy at birth, y years, in Singapore for certain years from 2003. x 1337 1448 1736 2129 2503 2799 3033 3621 y 79.1 80.3 81.4 82.1 82.9 83.4 83.2 83.7 Source: Singapore Department of Statistics (a) Give a sketch of the scatter diagram, as shown on your calculator. State what the scatter diagram suggests about the relationship between x and y. [3] (b) Find, correct to 5 decimal places, the value of the product moment correlation coefficient between (i) ln x and y, [1] (ii) x and y. [1] (c) Using part (b), explain which of the model lny a x b=+ or y c x d=+ gives a better fit to the data and find the equation of the regression line for this model. [2] (d) Use the equation of the regression line found in part (c) to estimate the total life expectancy at birth when the median monthly household employment income per household is $5000. Explain whether your estimate is reliable. [2]

6 8 A food delivery platform TMFoodRide, has two types of delivery riders: motorcycle riders and bicycle riders. Data from peak-hour orders show that • 65% of peak-hour orders are assigned to motorcycle riders. • A peak-hour order assigned to a motorcycle rider has a probability of 0.14 to be delivered late. • A peak-hour order assigned to a bicycle rider has a probability of 0.3 to be delivered late. (a) Find the probability that a randomly chosen peak-hour order is delivered late. [2] (b) Given that a peak -hour order is delivered late, find the probability that it was assigned to a bicycle rider. [2] A regular customer places 14 peak-hour orders over a month on TMFoodRide. Let X be the number of peak-hour orders that are delivered late. (c) State, in the context of the question, two assumptions needed for X to be well- modelled by a binomial distribution. [2] It is now given that X can be modelled by a binomial distribution. (d) Find the probability that at least 3 peak-hour orders are delivered late. [2] Employees of a company place 10 peak -hour orders on TMFoodRide to be delivered to the company at the same time. Explain why it may not be appropriate to use a binomial distribution for the number of peak-hour orders that are delivered late. [1]

7 [Turn Over 9 A dairy company produces cartons of milk which are labelled as containing a volume of 1000 ml. A consumer group suspects that the volume stated on the cartons is overstated and raises a complaint. A food quality control inspector takes a