CJC P2

9758/02/PRELIMS/2020 [Turn over CATHOLIC JUNIOR COLLEGE General Certificate of Education Advanced Level Higher 2 JC2 Preliminary Examination CANDIDATE NAME CLASS INDEX NUMBER MATHEMATICS 9758/02 Paper 2 16 September 2020 3 hours Candidates answer on the Question Paper. Additional Materials: List of Formulae (MF26) 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 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 graphi ng 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. The number of marks is given in brackets [ ] at the end of each question or part question. Question 1 2 3 4 5 6 7 8 9 10 11 Total Marks Total 5 6 7 12 10 8 8 8 9 13 14 100 This document consists of 28 printed pages.

2 9758/02/PRELIMS/2020 Section A: Pure Mathematics [40 marks] 1. The graph of a c urve C is a cubic polynomial and it passes through the origin, ( )2, 0 and ( )2.55, 0.0631− . Given that it has a turning point when 0.785x= , find the equation of C, giving its coefficients correct to 1 decimal place. [5] 2. (i) By means of the substitution cos ecx θ= , show that 2 3 2 322 63 1 d sin ( )d 1 x xx π π θθ= −∫∫ . You are to show all workings clearly. [3] (ii) Hence evaluate exactly 2 322 3 1 d 1 x xx −∫ . [3] 3. (i) By writing ( )( ) 3 3 13 4rr++ in partial fractions, find ( )( )1 3 3 13 4 n r rr= ++∑ in terms of n. [4] (ii) Hence find the exact value of 3 13 13 1 22 25 25 28 28 31×+×+×+ . [3] 4. Botanists are studying how excessive logging is affecting the population of forest trees in a country in South-East Asia. The population of trees in a forest after t years is denoted by x, in thousands. One simple model proposed is that the population increases at a rate proportional to its population size. It is known that t rees are chopped off at a constant rate of 3 thousand trees per year and the population stays constant when xp= . (i) Find a differential equation involving x and t. [3] (ii) The initial population of trees in a forest is 5 thousand trees. Solve the differential equation in part (i) and show that ( ) 3 5e tpxpp= −+ . [5] (iii) Given that the population of tree decreases in a finite tim e, state an inequality satisfied by p. Sketch the graph of x against t for this case, showing clearly the coordinates of the graph’s point of intersection with the horizontal axis. Hence explain the meaning of this axial intercept in the context of the question. [4]

3 9758/02/PRELIMS/2020 [Turn over Section B: Probability and Statistics [60 marks] 5. (a) In the triangle shown below, one vertex is origin O, and the two other vertices are A and B where OA= a    , OB= b   . A median of a triangle is a line segment joining a vertex ( O, A or B) to the midpoint of the opposite side (e.g. AD is a median of triangle OAB from vertex A). It is given that F is the point of intersection between the medians of triangle OAB from vertices A and B. (i) By finding AD   and BE   , show that ( )1 3OF = + ab   . [3] (ii) Prove that F also lies on OC, the median of triangle OAB from vertex O. [2] (b) With reference to the origin O, the points N, P and Q have position vectors n, p and q respectively, and Π has an equation 0⋅=rn . It is known that 0≠⋅=⋅pn qn . (i) Show that PQ   is perpendicular to n . Hence describe the geometrical relationship between the line PQ and the plane Π. [3] (ii) Find a vector equation of the plane that contains points P and Q, and is perpendicular to plane Π, leaving your answer in terms of n, p and q. [2] 6. A bag contains 4 red counters and 6 blue counters. 4 counters are drawn from the bag at random, without replacement. (a) Calculate the probability that (i) all the counters drawn are blue, [1] (ii) at least 3 blue counters are drawn, [2] (iii) at least 1 counter of each colour is drawn, [2] (iv) at least 3 blue counters are drawn, given that at least 1 of each colour is drawn. [2] O A B D E F

4 9758/02/PRELIMS/2020 (b) State with a reason whether or not the events “at least 3 blue counters are drawn” and “at least 1 counter of each colour is drawn” are independent. [1] 7. Flyers of a tuition centre for General Paper (GP) tuition are given to JC2 students in a particular college with a large population. On average, 8% of the JC2 students will sign up for the GP tuition. (i) State, in context, two assumptions needed for the number of JC2 students signing up for the GP tuition to be well modelled by a binomial distribution. [2] Assume now that the number of JC2 students signing up for the GP tuition follows a binomial distribution. (ii) It is given that 120 JC2 students receive a copy of the flyer. (a) Find the probability that more than 15 students sign up for the GP tuition. [3] (b) Find the variance of the number of students signing up for the GP tuition. [1] Flyers of another tuition centre for Mathematics tuition are given to JC2 students in the same college. The number of students who sign up for the Mathematics tuition is denoted by M. Assume that M has a distribution ( )B 320, p . (iii) It is known that on average the proportion of students who sign up for Mathematics tuition is higher than the proportion of students who sign up for GP tuition. Given that ( )30 or 31 0.037986PM = = correct to 5 significant figures, find the value of p. [2] 8. A box contains 1 red, 1 blue and n yellow discs, where 2n≥ . Two discs are randomly drawn from the box, without replacement. The random variable Y is the number of yellow discs drawn from the box. (i) Show that ( ) ( )( ) 4P1 12 nY nn= = ++ . Hence find the probability distribution of Y. [3] (ii) Show that ( )E 2 AnY n= + , where A is a constant to be determined. [2] (iii) Hence, find ( )Var Y , giving your answer in the form ( ) ( ) ( ) 2 f 21 n nn++ where ( )f n is a quadratic polynomial to be determined. [3] 9. Legox bricks come in different colours and sizes. Bricks of the same size and colour are identical . Jamie has 9 Legox bricks. The following table shows the various colours and sizes of Legox bricks that Jamie has. Small Medium Large Green 0 0 2 Blue 1 1 1 Red 2 2 0 (a) Jamie arranges these 9 bricks in a row. Find the number of different possible arrangements if (i) each of the green bricks are at the ends of the row, [2]

5 9758/02/PRELIMS/2020 [Turn over (ii) all the red bricks are together and the 2 green bricks are separated. [3] (b) Jamie decides to use only 4 bricks to arrange in a row . Find the number of different possible arrangements if all 3 colours are used.