VJC H2 MATH P2

1 VICTORIA JUNIOR COLLEGE Preliminary Examination MATHEMATICS (Higher 2) 9740/02 Paper 2 September 2015 3 hours READ THESE INSTRUCTIONS FIRST Write your name and CT group 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, glue or correction fluid. Answer all the questions. Give non-exact numerical answers correct t o 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 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. This document consists of 5 printed pages © VJC 2015 VICTORIA JUNIOR COLLEGE [Turn over Additional Materials: Answer Paper Graph Paper List of Formulae (MF15)

2 Section A: Pure Mathematics [40 marks] 1 It is given that 1e sin 22 y x . (i) Show that 22 2 dd 4 2edd yyy xx    . [2] (ii) By repeated differentiation of the result in part (i), find the first four non–zero terms of the Maclaurin series for y, giving the coefficients in exact form. [4] (iii) Show that the same result in part (ii) can be obtained using the standard results given in the List of Formulae (MF15). [3] 2 Relative to the origi n O , the points A , B and C have position vectors a , ac and c respectively. The point X is on AC produced such that :AC AX is 1: 4 and the point Y is such that AXYB is a parallelogram. (i) The lines AY and BX intersect at the point N. Find, in terms of a and c , the position vector of N . [2] (ii) Given that the area of triangle OAB is 2 square units, find the area of triangle AXB. [3] (iii) Give the geometrical interpretation of BXAX BX        . Using the results from part (ii), show that BXAX BX        k mn ca , where k, m and n are constants to be determined. [4] 3 The complex number z satisfies the equation 5 7i 6z   . (i) Show the locus of z on an Argand diagram. [2] (ii) If the locus in part (i) intersects the locus  arg z – – 2i 2a  at two distinct points, where a is a real number, find the set of values that a can take. [2] (iii) Given that the locus in part (i) intersects the locus 2 4izk   at exactly one point, find two possible exact values of k. [3] Using one value of k found, find exactly the value of z represented by the point of intersection, giving your answer in the form x + iy. [3]

3 4 The diagram below shows the curve with equation 2 ln xy x , 0x . The curve cuts the x-axis at  1,0 and has a maximum point at A. (i) Find the exact coordinates of A. [3] (ii) Without using a calculator, find the exact area of the finite region bounded by the curve, the x-axis and the line 2x . [4] (iii) Find the volume of the solid generated when the region bounded by the curve, the tangent at A and the line 1x is rotated completely about the x-axis, giving your answer correct to 3 significant figures. [3] (iv) The geometric series S is defined by     2323 2 4 6 4e ln 8e ln2eln1 xxx x x x    . A student claims that, if the value of x is larger than 1, then the sum to infinity of S exists. State, with a reason, whether you agree with him. [2] Section B: Statistics [60 marks] 5 A college has 540 students in Year One and 660 students in Year Two. The college intends to carry out a survey to investigate students’ opinions about the gymnasium facilities available at the college. (i) Describe how to obtain a stratified random sample of 60 students to take part in the survey. [2] (ii) State how a better stratified random sample of size 60 could have been achieved. [1] 6 A group of 12 people consists of 6 married couples. (i) The 12 people are to be seated randomly at a round table. Find the num ber of ways in which the 12 people can be arranged if each married couple is seated together. [2] (ii) The group is going on a flight and is assigned to sit in three distinct rows of four seats each. Find the number of ways in which the 12 people can be arranged if each row has at least 1 woman. [5] [Turn over A O  1,0 y x

4 7 A game is to be played between two players, A and B. A bag contains 5 balls each with A’s name and 8 balls each with B’s name. Starting from A, the players will take turn to pick 3 balls randomly in a single draw from the bag, note down the names and return all 3 balls to the bag. Assuming that the balls are identical in size, t he winner is the first player to get 3 balls of the player’s name in a single draw. (i) Find the probability that A wins the game. [4] (ii) Find the probability that A wins on A’s first draw given that A wins the game. [3] Suppose A eventually wins the game on A’s 4th turn. Let ( a1, a2, a3, a4) denote A’s draw sequence where ai denote the number of balls with A’s name picked by A on A’s ith turn. Find the total number of possible draw sequence for A. [2] 8 Each night in the month of August, Amy observes the number of meteors from the telescope set up in her laboratory. Amy observes meteors at an average rate of 2 per minute. (i) State two conditions needed for the number of meteors observed in a randomly chosen period of 1 minute to be well modelled by a Poisson distribution. [2] Assume that the conditions in (i) are satisfied. (ii) Find the probability that Amy observes exactly 6 meteors in a randomly chosen period of 4 minutes. [1] (iii) Find the probability that Amy observes exactly 3 meteors in each of the two successive 2- minute intervals. [2] (iv) Explain why the answer to part (ii) is greater than the answer to part (iii). [1] (v) In a randomly selected period of n minutes (where 10n ), the probability that Amy sees more than 3n meteors is less than 0.005 . Using a suitable approximation, determine an inequality satisfied by n and, hence, find algebraically, the smallest possible integer value of n. [5] 9 Past records shows that female students in tertiary institutions have a mean height of 162 cm. A random sample of 150 female students is taken from a particular institution and the height, x cm, of each female student is measured. The results are summarised by     2 160 480, 160 8837.xx    Test, at the 5% significance level, whether the mean height o f 162 cm is an understated value for this institution. [5] Explain the meaning of 5% significance level in the context of this question. [1] Two tests, each with a sample size of 10 taken from this particular institution, are carried out with the sa me hypotheses as above. Assume that the un biased estimate s of the population variance are the same for both the samples. (i) What change would there be in carrying out the two tests as compared to the previous test? State whether any assumption is needed for the two tests to be valid. [2] (ii) The first test has a sample mean height of m cm. Based on this test, the null hypothesis is not rejected at the 5% sig