ACJC 9758 2023 Prelim P2

ANGLO-CHINESE JUNIOR COLLEGE JC2 PRELIMINARY EXAMINATION Higher 2 CANDIDATE NAME TUTORIAL/ INDEX FORM CLASS NUMBER MATHEMATICS 9758/02 Paper 2 29 August 2023 3 hours Candidates answer on the Question Paper. Additional Materials: List of Formulae (MF26) READ THESE INSTRUCTIONS FIRST Write your index number, class and name on all 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. The use of an approved graphing calculator is ex pected, where appropriate. 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. 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 26 printed pages and 2 blank pages. [Turn over Question Marks 1 /5 2 /8 3 /8 4 /9 5 /10 6 /7 7 /10 8 /9 9 /9 10 /12 11 /13 /100

2 ANGLO-CHINESE JUNIOR COLLEGE 2023 H2 MATHEMATICS 9758/02 Section A: Pure Mathematics [40 marks] 1 Using the substitution cosux= , find 2 sin2 dcos cos 2 x xxx++∫ , giving your answer in terms of x . [5] 2 It is given that ( )coty xa= + for some constant a. (i) Show that ( ) 2d 1d y yx = −+ and hence find 2 2 d d y x and 3 3 d d y x in terms of y. [3] (ii) Explain why the Maclaurin series of y cannot be found when 0a= . [1] In the case where π 2a= , (iii) use your answers in (i) to find the series expansion of y in ascending powers of x, up to and including the term in 3x . [2] (iv) Using your answer in (iii) and the standard series from the List of Formulae (MF26), find the series expansion of 2 y x+ in ascending powers of x , up to and including the term in 3x . [2] 3 The Stoke’s Law states that when a spherical object of radius R is falling at velocity v in a fluid, the resistance force F acting on the sphere is given by 6πF Rv η=− , where η is the coefficient of viscosity of the fluid. The following table shows the coefficients of viscosity of various fluids at 25 C° . Fluid Coefficient of viscosity η, 11kgm s−− Acetone 43.06 10 −× Benzene 51.81 10 −× Castor Oil 0.985 Corn Syrup 1.3806 Ethanol 31.074 10 −× Methanol 45.44 10 −× Olive Oil 28.1 10 −×

3 ANGLO-CHINESE JUNIOR COLLEGE 2023 H2 MATHEMATICS 9758/02 [Turn Over A spherical ball of radius 0.05 m and mass 0.2 kg is released from rest in a huge container filled with corn syrup at 25 C° . It can be shown that as the ball falls through the corn syrup, the velocity v 1ms− at time t seconds after release satisfies the differential equation d6 π d vR vgtm η+= , where 29.780msg −= is the acceleration due to gravity and m kg is the mass of the ball. (i) Find v in terms of t. [5] (ii) State the velocity of the ball after a long time, assuming that it does not reach the bottom of the container. [1] (iii) The distance travelled by the ball from the point it was released is denoted by x m. Using the fact that d d xv t= , find the distance covered by the ball after 30 seconds. [2] 4 (a) (i) Write 2 4 4 12 5rr++ in partial fractions. [1] (ii) Find 2 2 4 4 12 5 n r rr=   ++∑ . [3] A sequence 123, , , ...uuu is such that 1 3u = and 1 2 4 4 12 5 rruu rr −= + ++ for 2r≥ . (iii) By considering, ( )1 2 n rr r uu − = −∑ , express nu in terms of n. [2] (b) The root test is a test for convergence of infinite series of the general form 0 .r r a ∞ = ∑ The test states that the series converges whenlim 1n nn a →∞ < , and diverges when lim 1n nn a →∞ > . When lim 1n nn a →∞ = , the test is inconclusive. Use the test to explain why the series 3 3 0 3 57 r r r rx r ∞ = −+  +∑ converges for all values of x. [3]

4 ANGLO-CHINESE JUNIOR COLLEGE 2023 H2 MATHEMATICS 9758/02 5 A curve C is defined by the parametric equations 11 ππcos , cos ,cos cos 2 2xy θ θθ θθ= − = + −<< . (i) Show that 2 2 d cos 1 d cos 1 y x θ θ −= + . [2] The line L is the normal to C at the point with parameter p. (ii) Show that the equation of L is 22 2 1 cos 1 cos 21 cos cos ppyx pp ++= + −  . [2] (iii) The line L cuts the x- and y- axes at the points P and Q respectively. Find the area of triangle OPQ in terms of p. [3] (iv) Given that p increases at a constant rate of 0.1 units per second, find the rate of increase of the area OPQ when π 3p= . [3] Section B: Probability and Statistics [60 marks] 6 A factory manufactures a large number of chips in a variety of colours and packs them into boxes. Each box contains 36 randomly chosen chips. On average, 30% of the chips are yellow. You may assume that the number of yellow chips in a box follows a binomial distribution. (i) A randomly chosen box is found to have at most 9 yellow chips . Find the probability that it contains more than 4 yellow chips. [2] 75 randomly chosen boxes are packed into a carton for shipment. (ii) Find the probability that the 60th box chosen is the 14th box with at most 9 yellow chips. [2] (iii) 30 cartons are sold to an arcade. Using a suitable approximation, find the probability that the mean number of boxes containing at most 9 yellow chips in each carton is more than 25. [3]

5 ANGLO-CHINESE JUNIOR COLLEGE 2023 H2 MATHEMATICS 9758/02 [Turn Over 7 An electrical appliance supplier is investigating the pote ntial impact of monthly advertisement expenditure, denoted as x (in thousands of dollars ), on the popular social media platform SnapGram, on their monthly sales profits, denoted as y (in thousands of dollars). The supplier has collected the following data over the course of the past year. Month Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov Dec Advertisement Expenditure, x 5 6.5 7 6 8 5.5 10 7.5 8.5 9 4 9.5 Sales Profits, y 16 19 19.5 17 k 16.5 32.5 22 26 28 15.5 30 It is thought that this set of data can be modelled by one of the following equations: Model A: y mx c= + Model B: ln y ax b= + (i) Explain the meaning of the value of m in the context of the data for Model A. [1] (ii) It is given that the least square s regression line for Model A is 0.3604 3.0194yx= + , correct to 4 decimal places. Show that 23.5k = , correct to 1 decimal place. [2] (iii) Draw the scatter diagram for the data, labelling the axes clearly. [1] (iv) By calculating the product moment correlation coefficients and using the scatter diagram from part (iii) , explain whether Model A or Model B is a more appropriate model for the given set of data. [3] Use the more appropriate model selected in part (iv) for the rest of this question. (v) Use a regression line to obtain the estimate for the sales profit if the advertisement expenditure is $11,000 for a particular month. [2] (vi) Comment on the reliability of the estimate obtained in (v). [1]

6 ANGLO-CHINESE JUNIOR COLLEGE 2023 H2 MATHEMATICS 9758/02 8 Kelly has a box of spherical beads labelled with the 26 letters from A to Z, all of which have the same size. For each letter, there are exactly 3 beads of different colours, red, blue and gold. By stringing the labelled beads together in a specific order, Kelly plans to make a keychain. A possibl