JPJC 2026 Prelim P2 Qn

Name:____________________________________ Class:_____________ JURONG PIONEER JUNIOR COLLEGE JC2 Preliminary Examination 2026 MATHEMATICS 9758/02 Higher 2 15 September 2026 Paper 2 3 hours Additional materials: Printed Answer Booklet List of Formulae and Results (MF27) READ THESE INSTRUCTIONS FIRST This document consists of 7 printed pages and 1 blank page. [Turn over 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 by [ ] at the end of each question or part question.

2 Section A: Pure Mathematics [40 Marks] 1 Without using a calculator, solve 2 2 3 4 10 22 3 x x x x     . [4] Hence solve 2 2 3 4 10 22 3 x x x x     . [2] 2 The planes 1p and 2p have vector equations 1 1 2 1 0 1 1 1 1 a b                                r , 2 2 2 4 0 5 1 1 6 a b                                r , respectively, where 1 2 1, ,    and 2 are real parameters, and a and b are constants. (a) Given that 1p and 2p intersect along a line L, write down a vector equation of L, in terms of a and b. [1] (b) Deduce an equation of the plane 3p that passes through (0, 4, 4) and is perpendicular to both 1p and 2p , in terms of a. [1] The plane 4p has equation 1 2 3 3         r . (c) Given that L lies in 4p , find the values of a and b. [4] 3 (a) Using standard series from the List of Formulae (MF27), expand ln( 2 )k x as far as the term in 2x , where k is a positive constant. [2] (b) Find the range of values of x for which the expansion in part (a) is valid. [1] (c) It is given that the first three terms found in part (a) are equal to the first three terms in the series expansion of (1 ) nmx , where m is a constant. Find the exact values of k and m. [4]

3 4 The function f is such that 2f : 4 7, , ,x x x x x k     where k is a real constant. (i) State the greatest value of k for which the function 1f  exists. [1] For the rest of the question, take the value of k as the value found in part (i). (ii) Find 1f  in similar form. [3] The function g is such that 2 10g : 2 , . 1x x x   (iii) Sketch t he graph of g( )y x , g iving the equations of any asymptotes and the coordinates of any turning points. [2] (iv) Show that the composite function gf exists. [1] (v) Find the range of gf. [2] 5 (a) A man borrows $5000 from a lender at the start of the first month. Interest is added immediately at a rate of 2.5% of the amount borrowed and thereafter at the start of every month at a rate of 2.5% of the outstanding loan. The man pays $400 to the lender at the end of every month. (i) Find the outstanding loan at the end of the nth month. [4] (ii) Calculate how many complete months he will take to repay his loan. [3] (b) A supplier delivers steel rods to a factory weekly. In the first week, 100 steel rods are delivered and in each subsequent week, 10 more steel rods are delivered. (i) Find the total number of steel rods delivered to the factory in n weeks. [1] After n weeks, where n < 20, the delivery plan is changed so that each subsequent week’s delivery is 10 steel rods fewer than the previous week’s delivery. By the 20th week, the total number of steel rods delivered is 3180. (ii) Find the value of n. [4] [Turn over

4 Section B: Probability and Statistics [60 Marks] 6 At a carnival, Peter plays a game where two balls are drawn at random without replacement from an urn containing 4 red balls, 4 blue balls and 2 green balls. He wins $ m if the 2 balls drawn are of the same colour, and loses $1 if the 2 balls drawn are of different colours. In addition, he gains $0.50 if at least one of the balls drawn is green. The random variable X denotes the amount of money Peter wins in a single game. (i) Show that P(X = – 0.5) = 16 45. [1] (ii) Determine the probability distribution of X, expressing your answers in terms of m. [3] (iii) Find the least value of m such that it is favourable for Peter to play the game. [2] 7 An investigation was carried out to examine the relationship between wind speed and the electrical power generated by a wind turbine. The wind speed, x ms 1 , and the electrical power generated y kW, were recorded on 6 occasions, as shown in the table. x 4 6 8 10 12 14 y 41 83 165 303 523 805 (a) Draw a scatter diagram for these values. Explain which of the following equations, where a and b are positive constants, provides the most appropriate model for the relationship between x and y. (A) y ax b  (B) 3y ax b  (C) y a x b  [2] (b) Using the model you chose in part (a), write down the equation for the relationship between x and y, giving the numerical values of the coefficients. State the product moment correlation coefficient for this model, giving your answer to 6 decimal places. [2] (c) Give two reasons why it would be reasonable to use your model to estimate the value of y when x = 11. [2] A second wind turbine is tested when the wind speed is 11 ms 1 , and it generates 450 kW of power. (d) Justify if the second turbine appears to perform better than the turbine used in the original investigation. [2]

5 8 A group of 12 players is available for a street football match. The group consists of 5 attackers, 4 defenders and 3 goalkeepers. A team of 5 players is selected at random from the group. (a) Find the (i) number of different teams containing exactly 2 attackers, 2 defenders and 1 goalkeeper, [2] (ii) probability that the team contains exactly 1 goalkeeper and at most 2 attackers. [3] The coach has already decided which goalkeeper will play in a match. To help him select from the 5 attackers and 4 defenders, he conducts a passing drill involving these players. For each pass, 1 player is chosen at random to receive the ball. After each p ass, the ball is returned to the coach before the next player is chosen. A player may receive the ball more than once. The coach makes 4 passes in this manner. Find the probability that (b) exactly 3 passes are made to attackers, [2] (c) either the first 3 passes or the last 3 passes are made to attackers (or both). [3] 9 (a) In this question you should state the parameters of any distribution you use. In a grocery store, the masses (in grams) of oranges from Farm A have the distribution N(245, 82) while those from Farm B have the distribution N(212, 72). (i) Find the probability that the mass of a randomly chosen Farm A orange differs from that of Farm B by more than