TJC 9758 2023 Prelim P2

1 TEMASEK JUNIOR COLLEGE 2023 PRELIMINARY EXAMINATION Higher 2 CANDIDATE NAME CENTRE NUMBER S INDEX NUMBER MATHEMATICS 9758/02 Paper 2 13 September 2023 3 hours Candidates answer on the Question Paper. Additional Materials: List of Formulae (MF26) READ THESE INSTRUCTIONS FIRST Write your name, centre number and index number on the work you hand in. Write in dark blue or black pen. You may use an HB pencil for any diagrams or graphs. Do not use staples, paper clips, glue or correction fluid. Answer all the questions. Write your answers in the spaces provided in the Question Paper. 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 are required to present the mathematical st eps 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. The total number of marks for this paper is 100. This document consists of 23 printed pages and 1 blank page. [Turn over For Examiner’s Use Question Number Marks Obtained 1 2 3 4 5 6 7 8 9 10 Total Marks

2 DO NOT WRITE IN THIS MARGIN DO NOT WRITE IN THIS MARGIN DO NOT WRITE IN THIS MARGIN DO NOT WRITE IN THIS MARGIN DO NOT WRITE IN THIS MARGIN @TJC 2023 9758 Section A: Pure Mathematics [40 marks] 1 (a) Prove the identity 44 311 422n n nn  + −− ≡ +     . [1] (b) The sum NS is defined by 3 1 N N n Sn = =∑ . Using the identity in part (a) find NS as a polynomial in terms of N. [5] (c) Explain why 4 NNS− converges and state the value that it converges to. [2] 2 The function f is defined by 2 2 for 0 4,f: 12 28 for 4 , xxx xx xa  + << − − ≤≤  where a is a constant. (a) State the largest value of a∈  for which the function 1f − exists. [1] For the rest of the question, let the domain of f be restricted to (0,5] . (b) Sketch the graph of f, indicating clearly the coordinates of the end point(s). [2] (c) Find 1f− in similar form. [4] The function g is defined by g( ) 1 3 for xx x += +− ∈  . (d) Show that gf exists and find its corresponding range. [2] 3 (a) By using suitable small angle approximations and standard series from the List of Formulae (MF26), find the Maclaurin’s series for 2cos 2 1 sin 2 x x+ up to the 2x term. [3] (b) It is given that ( )ln 1 sin 2yx= + , for 44 xππ−<≤ . (i) Show that 2 2 dde sin 2dd h y yy kxxx  +=   , where h and k are constants to be determined. [3] (ii) Find the first three non- zero terms of the Maclaurin expansion of ( )ln 1 sin 2 x+ . [3] (c) Using the result in part (b)(ii), verify the correctness of your answer in part (a). [1]

3 @TJC 2023 9758 [Turn over DO NOT WRITE IN THIS MARGIN DO NOT WRITE IN THIS MARGIN DO NOT WRITE IN THIS MARGIN DO NOT WRITE IN THIS MARGIN DO NOT WRITE IN THIS MARGIN 4 (a) With reference to the origin O, the points A, B and C are such that aOA=    , bOB=   and cOC =   . The length of OA and OB are 1 and 2 respectively. (i) Give the geometrical meaning of ab× . [1] (ii) Given that c is perpendicular to ×ab , explain why c can be expressed as c abµλ= + for some constants andµλ . [1] (iii) Given that angle 2AOC π= radians and angle 3AOB π= radians and the length of OC is 3 , show that µλ=− and hence find all possible value(s) of λ.[4] (b) Planes p1 and p2 have vector equations 11 24 1 r  ⋅= − and 11 2 12 1 r  ⋅= − respectively. The point A has coordinates( )2 11, 4, 10 . (i) Find the shortest distance from A to p1. [2] (ii) By finding the distance between p1 and p2, or otherwise, determine with clear justification whether A is in between the planes p1 and p2. [2] (iii) Given that the point B lies on p 1 and has coordinates ( )0, 4, 12−− , find the position vector of the point C on p2 such that the line BC is perpendicular to both p1 and p2. [3]

4 DO NOT WRITE IN THIS MARGIN DO NOT WRITE IN THIS MARGIN DO NOT WRITE IN THIS MARGIN DO NOT WRITE IN THIS MARGIN DO NOT WRITE IN THIS MARGIN @TJC 2023 9758 Section B: Statistics [60 marks] 5 A group of 12 artistes, comprises 7 men and 5 women are invited to participate in an episode of TV variety show. In one of the challenges, 3 pairs of artistes, each consisting of a man and a woman, are to be selected to play a game in a night market. (a) Find the number of ways the 3 pa irs can be selected and grouped into pair A , B and C. [2] In the game, the woman is required to generate a number, 1 to 9, by using a number generator with the probability of obtaining a number x is 1 45 x. If the number is an even number, the male artiste must draw a ball from a red box with 3 black and 5 yellow balls. If the number is odd, he must draw a ball from a blue box with 8 black and 4 yellow balls. If a yellow ball is picked, the pair will win. Otherwise, the pair will lose. (b) Pair A plays the game once. Show that the probability that the pair loses the game is 29 54 . [2] Pair B plays a match with Pair C . Each pair takes turns to play the game. The ball is replaced before the next pair plays the game. The match will stop only when either one of them wins a game. Pair C starts first. (c) Find the probability that Pair B is the winner. [3] 6 A multiple choice test consists of 30 questions, with each question having 4 choices of which only 1 is correct. Each correct answer gains 3 marks but each incorrect answer loses 1 mark. Charlie’s strategy for this test is to answer all 30 questions. For each question, he will choose his answer at random with each choice having equal chance of being chosen. Let C denote the number of correct answers that Charlie achieves. (a) State, in context, what must be assumed for C to be well modelled by a binomial distribution. [1] Assume now that C has the binomial distribution B(30,0.25) . (b) Find the probability that Charlie answers at most 10 questions correctly. [1] (c) Show that the expected number of marks that Charlie can achieve is zero and find the variance of Charlie’s marks. [4] The passing mark for this test is 32. (d) Find the probability that Charlie will pass the test. [2]

5 @TJC 2023 9758 [Turn over DO NOT WRITE IN THIS MARGIN DO NOT WRITE IN THIS MARGIN DO NOT WRITE IN THIS MARGIN DO NOT WRITE IN THIS MARGIN DO NOT WRITE IN THIS MARGIN 7 A vending machine is set to dispense cups of drink of β ml. It is known that the volume of drink dispensed per cup is distributed normally with standard deviation of 6.3 ml. The company manager does a check by taking a random sample of 8 cups of drink to check if the volume of drink dispensed for each cup is less than β ml. He finds the volume of drink, in ml, are as follows. 190 190 192 194 195 200 203 207 (a) Find the mean of the sample of these 8 cups of drink. [2] (