NYJC H2 Maths P2

This document consists of 6 printed pages. NANYANG JUNIOR COLLEGE Internal Examinations © NYJC 2014 [Turn Over NANYANG JUNIOR COLLEGE JC2 PRELIMINARY EXAMINATION Higher 2 MATHEMATICS 9740/02 Paper 2 17th September 2014 3 Hours Additional Materials: Cover Sheet Answer Paper Graph Paper List of Formulae (MF15) READ THESE INSTRUCTIONS FIRST Write your name and class on all the work you hand in. Write in dark blue or black pen on both sides of the paper. You may use a soft pencil for any diagrams or graphs. Do not use staples, paper clips, highlighters, glue or correction fluid. Answer all the questions. 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 a graphic calculator. Unsupported answers from a graphic calculator are allowed unless a question specifically states otherwise. Where unsupported answers from a graphic calculator are not allowed in a question, you are required to present the mathematical steps using mathematical notations and not calculator commands. You are reminded of the need for clear presentation in your answers. At the end of the examination, fasten all your work securely together. The number of marks is given in brackets [ ] at the end of each question or part question.

2 NYJC 2014 JC2 Preliminary Examination 9740/02 Section A: Pure Mathematics [40 marks] 1 The three vectors l, m and n are such that  nl m . Show that 0 ln . [1] With respect to an origin O, the position vectors of three non-collinear points A, B and C are a, b and c respectively. The scalars  and  are such that  ab is the projection of c onto the plane containing O, A and B. By considering CC'  , where C’ is the foot of perpendicular from C to the plane containing O, A and B, explain why there is a scalar t such that ()t    ab c a b , and deduce that ()() aa ab ac . [4] 2 The functions f, g and h are defined by     f : 2 , , where 2, g: e , , h: l n 2 , , 3 . x xx a x x a xx xx x x           (i) Show that the composite function gh exists and define gh in a similar form. State the range of gh. [4] (ii) Find, in terms of a, the exact range of x for which    fg h .x ax a  [3] (iii) The curve with equation f( )yx has a stationary point with coordinates  2 22 ,24 aa   . Sketch the graph of  1 fy x , stating clearly the equations of any asymptotes and the coordinates of the stationary point. [3] 3 Given that ln 1 1 xy x   , where 11 x  , show that d12(1 ) 2 d1 yxy x x   . [1] (i) By further differentiation, find the Maclaurin series for y up to and including the term in 3x . [5] (ii) Verify that the same result is obtained if the standard series expansions are used. [3] (iii) Deduce the approximate value of 1 4 0 d.yx   Explain why the approximation is not good. [2] (iv) State the equation of the tangent to the curve y at 0x . [ 1 ]

3 NYJC 2014 JC2 Preliminary Examination 9740/02 [Turn Over 4 (a) The sequence of triangular numbers {}nt is given by the following recurrence relation: 11 1, and 1nntt nn t + + -= + Î = . (i) Write down the first three triangular numbers. [1] (ii) Use the method of difference to show that ()1 12 ntn n=+ for all n +Î . [3] (iii) Find, in terms of N, the sum of the first N triangular numbers, giving your answer in a factorised form. [2] [You may assume that () ( ) 2 1 1 12 16 n r rn n n = =+ +å .] (b) In a certain experiment to study the growth of H2 bacteria in a cultured environment, the number of H2 bacteria (in thousands) exactly n days after the start of the experiment is denoted by na , where 31 , n nan n    . Find (i) the number of days after the start of the experiment when the population of the bacteria first exceeds 2014 thousands, [2] (ii) the rate of growth of the H2 bacteria 2 days after the start of the experiment assuming that na is a continuous quantity. [2] In another experiment, H3 bacteria is introduced together with the H2 bacteria. It is given that exactly 1 day after the start of the experiment, the number of H3 bacteria is 900 thousands and the number increases at a constant rate of 800 thousands per day thereafter. The number of H3 bacteria exactly n days after the start of the experiment is denoted by nb . (iii) Write down an expression for nb in terms of n. [1] (iv) By which day after the start of the experiment will the population of H2 bacteria exceed the population of H3 bacteria? [2]

4 NYJC 2014 JC2 Preliminary Examination 9740/02 Section B: Statistics [60 marks] 5 A school has 880 students. The principal wants to ask the student about the cleanliness of toilets in school. A survey is to be conducted with a random sample of 80 students from the school. Describe briefly how a systematic sample can be obtained. [2] Explain why the sample obtained may not be representative of the school student population. Suggest a better sampling method. [2] 6 Two players, A and B, compete in a racquet match consisting of at most 3 sets. Each set is won by either A or B, and the match is won by the first person to win two sets. Player A has a probability of 0.6 of winning the first set. For each set after the first,  the conditional probability that A wins that set, given that A won the preceding set, is p,  the conditional probability that B wins that set, given that B won the preceding set, is 0.7. Find, in terms of p, the probability that Player A wins the match. [1] (i) The probability that Player B won the first set given that Player B had lost the match is 0.15. Find the value of p. [2] (ii) Given instead that 0.65p and the probability that Player A’s third win occurs in the nth match is 0.10866 correct to 5 significant figures, write down an equation for n, and solve it n u m e r i c a l l y . [3] 7 The bus leaves the bus stop near Adam’s home at X minutes past 0745, where X is normally distributed with mean 25 minutes and standard deviation 3 minutes. Adam reaches the bus stop at Y minutes after 0745, where Y is normally distributed with mean 15 minutes and standard deviation 2 minutes. It is given that X and Y are independent. Find the probability that Adam misses the bus. [3] The bus journey to school lasts W minutes, where W is normally distributed with mean 30 minutes and standard deviation 3 minutes. The random variable T denotes the number of minutes before 0830 at which the bus arrives at the school. Express T in terms of W and X, and find the mean and variance of T. [3] Find the probability that the bus would arrive at the school after 0830. [2]

5 NYJC 2014 JC2 Preliminary Examination 9740/02 [Turn Over 8 (i) Sketch a scatter diagram that might be expected when x and y are related approximately as