2023 Prelim - TMJC P2

Tampines Meridian Junior College 2023 JC2 Preliminary Examination H2 Mathematics CANDIDATE NAME: ___________________________________________________ CIVICS GROUP: _______________________________________________________ __________________________________________________________________________________ H2 MATHEMATICS 9758/02 Paper 2 19 SEPTEMBER 2023 3 hours Candidates answer on the question paper. Additional material: List of Formulae (MF26) ________________________________________________________________________________ READ THESE INSTRUCTIONS FIRST Write your name and Civics Group 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. 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. 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 6 printed pages and 0 blank pages. TAMPINES MERIDIAN JUNIOR COLLEGE JC2 PRELIMINARY EXAMINATION For Examiners’ Use 1 2 3 4 5 6 7 8 9 10 Total [Turn Over

2 Section A: Pure Mathematics [40 marks] 1 The position vectors of points A, B and P with respect to an origin O are a, b and p respectively, where 3 2.= −pab (a) Show that A, B and P are collinear and hence find the ratio :.BP AP [4] (b) It is given that b is a unit vector, 4 3=a and the angle between a and b is π .3 The point N is a point on li ne BP which is closest to the origin . Find the value of BN BA . [5] 2 (a) Logarithmic functions are useful as measurement units for phenomena with large scale of values, such as decibels (for sound) or Richter scale (for earthquakes). Logarithmic functions make the measurement units smaller and easier to work with. An example of a logarithmic function is given as ( ) 2ln 1 2 3 .y xx= ++ (i) By using differentiation, find the Maclaurin series for y , up to and including the term in 2.x [4] (ii) By using the standard series in MF26, verify that the answer obtained in part (i) is correct. [2] (b) Find the series expansion of , 4 x x+ up to and including the term in 3.x [4]

3 [Turn Over 3 (a) Given that ( )11 1 1 cos isinzr θθ= + and ( )22 2 2 cos isin ,zr θθ= + prove, using a trigonometric method, that ( ) ( )11 12 12 22 cos isin .zr zr θθ θθ= −+ − Hence state 1 2 arg z z    in terms of 1θ and 2.θ [3] Let i,rwr= + where r is a real constant, 0r≥ . (b) Use the method of differences to express ( ) ( )( )1 1 arg arg n rr r ww− = −∑ in the form ( )arg i ,kn−+ where k is an exact real constant to be determined. [3] (c) Explain why ( ) ( )( )1 1 arg argrr r ww ∞ − = −∑ converges and state its limit. [2] (d) Hence, or otherwise, find the exact value of 333 1i 2i 3iarg arg arg ...2i 3i 4i +++  +++  +++   . [4]

4 4 Projectile motion is the motion of a launched object, subject to only the constant acceleration of gravity, g ms-2. The object is called a projectile, and its path is called a trajectory. A projectile is launched from the origin with an initial velocity of u ms-1 at an angle θ radian above the horizontal, where 0 2 πθ<< , as shown in the diagram above. The trajectory of a projectile depends on its motion in two dimensions – horizontal (x) and vertical ( y) components. The position of the projectile at time t seconds after it is launched can be defined by 21cos and sin , 2x ut y ut gtθθ= = − where and ug are positive real constants. (a) When 0,y= find the value of t in terms of θ, g and u, where 0t ≠ . State what this value of t means in the context of the question. [3] The range of the projectile is defined as the horizontal distance the projectile travels from the time it is launched to the time it returns to the same height at which it is launched. (b) Show that the range of the projectile can be expressed as ( ) 2 sin 2 .ux g θ= [1] (c) Use differentiation to find the value of θ that gives the maximum range of the projectile. Hence find the maximum range of the projectile in terms of g and u. [5] x y O Projectile trajectory Initial launch direction

5 [Turn Over Section B: Probability and Statistics [60 marks] 5 For events A and B, it is given that ( )P ' 0.51B = and ( )P ' ' 0.15AB∩= . (a) Find ( )P' AB∩ . [2] It is also given that ( ) 11P| 29BA = . (b) Find ( )P A . [3] For a third event C, it is given that A and C are independent of each other, and B and C are mutually exclusive. (c) Find the maximum value of ( )P C . [3]

6 6 At a particular game stall, players throw darts at a target board to earn points for redeeming prizes. The target board can be modelled by three concentric circles with radii one unit, three units and five units respectively. A player will get a score of 50, 25 or 0 points if the dart lands on the respective regions labelled 50, 25 or 0 as shown in the diagram above. The probability that a dart lands on any region labelled 50, 25 or 0 is proportional to the area of the region. It can be assumed that the borders between the regions have negligible thickness and the dart will only land on the target board, which has a total area of 225π units . In a game, a player throws two darts at the target board, one after the other. The first dart will be removed before the next dart is thrown. Let S be the combined score that the player achieves from the two throws. (a) Show that ( ) 16P 75 . 625S = = [3] (b) Find the probability distribution of S. [4] (c) Find the expectation and variance of S. [2] 50 25 0 5 3 1

7 [Turn Over 7 A researcher is investigating the relationship between the students’ test marks and the ir average daily duration of sleep one week leading up to their test. The following table gives the average daily duration of sleep, t hours, of 9 students and their test marks, x, out of 100. t 5.5 5.8 6.1 6.5 7.2 7.4 8.0 8.2 9.1 x 39 43 52 60 66 68 77 78 80 (a) Explain whether the product moment correlation coefficient between t and x would differ if the researcher converted the average daily duration of sleep to minutes. [1] The researcher decides to fit a model of the form ( )ln 100 x a bt−= + to the data. (b) Find the product moment correlation coefficient between t and ( )ln 100 . x− What conclusion can the researcher reach about the relationship between t and ( )ln 100 x− ? Justify your answer. [2] (c) Draw a scatter diagram for ln (100 ) x− against .t Draw, by eye, a line of best fit on your scatter diagram. [2] (d) Use your calculator to find the equation of the least squares regression line of ( )ln 100 x− on .t [1] (e) Use the equation in part (d) to estimate the test marks for a student who has an average daily duration of sleep of 10 hours. Explain whether you would expect this estimate to be reliable. [3]

8 8 A grocery store sells two types of apples, Brand A and Brand S. The masses (in grams) of the apples for each brand can be modelled by independent normal