DHS H2 MATHS P2 QP

This document consists of 8 printed pages (including this cover page). Name: Index Number: Class: DUNMAN HIGH SCHOOL Preliminary Examination Year 6 MATHEMATICS 9740/02 (Higher 2) 22 September 2014 Paper 2 3 hours Additional Materials: Answer Paper List of Formulae (MF15) READ THESE INSTRUCTIONS FIRST Write your Name, Index Number 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 signi ficant 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. The number of marks is given in brackets [ ] at the end of each question or part question. At the end of the examination, attach the question paper to the front of your answer script. The total number of marks for this paper is 100. For teachers’ use: Qn Q1 Q2 Q3 Q4 Q5 Q6 Q7 Q8 Q9 Q10 Q11 Total Score Max Score 9 10 10 11 4 6 8 9 10 11 12 100

2 © DHS 2014 Year 6 H2 Mathematics Preliminary Examination [Turn over Section A: Pure Mathematics [40 marks] 1 (i) Prove by method of mathematical induction that           2231 3 3 7 1 1 1 2 ... 112 23 34 1 2 1 nn n n nn n         for all positive integral values of n. [4] (ii) Hence, by considering    2 11 1 rr rr    ,1 A Br rr   where A and B are constants to be determined, find an expression for 1 n r r   in terms of n. [5] 2 An event company builds a tent (as shown in the diagram) which has a uniform cross-sectional area consisting of an equilateral triangle of sides x metres and a rectangle of width x metres and height h metres. The length of the tent is 3x metres. (i) It is given that the tent has a fixed volume of k cubic metres and is fully covered (except the base) with canvas assumed to be of negligible thickness. Show that the area of the canvas used, A, in square metres, is given by 22 33 86. 23 kAx x x  Use differentiation to find, in terms of k, the value of x which gives a stationary value of A. Determine if A is minimum or maximum for this value of x. [7] (ii) It is given instead that th e tent has a volume of 360 cubi c metres and the area of canvas used is 300 square metres. Find the value of x and the value of h. [3] x x x 3x h

3 © DHS 2014 Year 6 H2 Mathematics Preliminary Examination [Turn over 3 (i) Given 42 22 0 ,ww  find the roots of the equation, giving your answers in the exact form ie,r  where 0r and ππ .  [4] (ii) Show the roots on an Argand diagram. [2] (iii) The roots in (i) represented by 123 4, , and www w are such that 1234π arg( ) < arg( ) < arg( ) < arg( ) < π.wwww Explain why the locus of all points z such that 1 πarg( ) = 8zw passes through the point that represents 3.w Draw this locus on your Argand diagram and find its exact cartesian equation. [4] 4 The line l has equation 2 ,1 ,11 xz a y   where a is a real constant and the plane p1 has equation 32 5 .xy z  The point A has the position vector 22i j with respect to the origin O. (i) Find the acute angle between l and p1. [2] (ii) Find the perpendicular distance from the point A to p1. [3] (iii) Given that l is the line of intersection of the planes p2 and p3 with equations 46xy z  and ,x yb zc  where b and c are real constants, find b and c. [3] (iv) The point B varies such that the midpoint of AB is always in p1. Find a cartesian equation for the locus of B. [3] Section B: Statistics [60 marks] 5 The organiser of an overseas education fair would like to survey 1% of the people attending the one-day event on their opinions about the fair. (a) The organiser has identified that the people at the event comprise students and working adults and has decided to targ et these two groups for his survey. Give one advantage of using a stratifie d sample in comparison to a sample obtained by quota sampling. Suggest why it would be difficult for the organiser to obtain a stratified sample. [2] (b) Explain how the event organiser can conduct his survey us ing a systematic sample. [2]

4 © DHS 2014 Year 6 H2 Mathematics Preliminary Examination [Turn over 6 For a particular activity, three boxes A, B a nd C are placed in a circular arrangement. Box A contains 5 red and 5 white balls. Box B contains 4 red and 6 white balls. Box C contains 3 red and 7 white balls. For each round, a participant of this activity will draw a ball randomly from a box at any one time without replacement. The partic ipant begins his first draw from box A, and the subsequent draws are based on the following rules:  If a red ball is drawn, the next draw is from the box in the clockwise direction.  If a white ball is drawn, the next draw is from the box in the anti-clockwise direction. (i) Show that the probability that a participant has not drawn from box C from the first to the sixth draw is 1 36 . [2] (ii) Given that a participant drew from box B on the fourth draw, find the probability that he has not drawn from box C from the first to his sixth draw. [4] 7 A group of eight girls and four boys take part in a Mathematics competition. Among this group, there are six Chinese, four Malay, and two Indian students. For the individual segment, all students are arranged ra ndomly in a row. Find the probability that (i) the boys are all separated from one another. [2] (ii) a particular boy is between two girls. [2] For the team segment, all students are arranged in a circle to facilitate discussion. (iii) Find the probability that the boys are all next to one another. [1] The group won the first prize and only seven students are chosen to represent the group to receive the prize. (iv) Find the probability that at least one student from each race are chosen to represent the group. [3] A B C

5 © DHS 2014 Year 6 H2 Mathematics Preliminary Examination [Turn over 8 (i) Sketch a scatter diagram that might be expected for the case when x and y are related approximately by lnyabx  , where a is positive and b is negative. Your diagram should include 6 points, approximately equally spaced with respect to x, and with all x- and y-values positive. [1] About a year ago, Amy decided to go on a healthy eating lifestyle and an exercise regime in order to lose weight. She monitored her weight, y kg, x months after she started and the data is provided below. No. of months, x 1 3 5 7 9 11 Weight, y 61.4 60.1 59.5 58.9 58.5 58.2 (ii) Draw the scatter diagram for these values, labelling the axes. [1] (iii) Explain which model, lnyabx  or ,ycd x  is better for modelling these values. [1] For the better model that you have identified in part (iii), (iv) give a contextual interp retation of the value of a or c and calculate the product moment correlation coefficient. [2] (v) use a suitable regression line to estimate the month in which she will reach a weight of 55 kg. You may assume that the model is suitable for shor