ASRJC Prelims 2025 H2 Math P2 Questions

ANDERSON SERANGOON JUNIOR COLLEGE H2 MATHEMATICS Preliminary Examination Paper 2 (100 marks) 9758/02 3 hours Additional Material(s): Printed Answer Booklet 01 September 2025 MF27 Formula Booklet CANDIDATE NAME CLASS / READ THESE INSTRUCTIONS FIRST Answer all questions. Write your answers on the Printed Answer Booklet. Follow the instructions on the front cover of the answer booklet. 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 must present the mathematical steps using mathematical notations and not calculator commands. You must show all necessary working clearly. The number of marks is given in brackets [ ] at the end of each question or part question. This booklet contains 8 printed pages.

2 Section A: Pure Mathematics [40 marks] 1 Relative to the origin O , the position vectors of two points A and B are a and b respectively. The length of a is k units and b is a unit vector. The angle between a and b is 6 π radians. It is g iven that the point M lies on the line segment AB such that AM :AB is 3:4, find exact area of OAM∆ , giving your answer in terms of k. [4] 2 (a) Without the use of a graphing calculator, solve the inequality 42 12 xx x ++ ≥+ . [3] (b) Hence solve 3 e21 e1 2 x x ++≥ + . [2] 3 (a) Given that y = 1 1tan 5 x− , show that ( ) 2 2 2 dd25 2 0 dd yyxx xx+ += . [2] (b) By further differentiation of this result, find the Maclaurin’s series of y up to and including the term in x3. [3] 4 The curve C is defined by the parametric equations x = cos t + 1 2 cos 5t, y = sin t + 1 2 sin 5t for 0 ≤ t ≤ π 2 . (a) Find the coordinates of the point on the curve C corresponding to t = 0. [1] Another curve D has the following parametric equations. x = 2 + cosk θ , y = 3 sin2 θ for 0 ≤ θ ≤ 2π and k > 0. (b) Find the cartesian equation for curve D. [2] (c) Sketch the curves C and D on the same diagram. [2] (d) Hence determine the range of values of k such that the equation ( ) ( ) 22 cos 0.5cos5 2 4 sin 0.5sin 5 19 t t tt k +− + += has no real solutions. [1]

3 [Turn Over 5 (a) Solve the following integral. (i) 2 12 d 32 x x xx − +− ⌠ ⌡ . [2] (ii) ( ) 2ln 2 dx xx −∫ , where 22 x− << . [3] (b) A function f is defined by f( ) e 2 xx = − . (i) Sketch the graph of f( )yx= , indicating clearly the equation(s) of asymptote(s), if any. [2] (ii) Hence find ln 3 0 f( ) dxx∫ , giving your answer in exact form. [3] 6 The points P and Q are represented by the complex numbers p and q respectively, where 2 2ip = + , 2arg( ) 3q π= and 2q = . (a) Find p and arg( )p in exact form. [2] (b) Sketch the points P and Q on an Argand diagram. [2] (c) Use the Argand diagram to deduce Re( q) and Im(q), giving your answers in exact form. [2] (d) The point R is represented by the complex number pq+ . What can you deduce about the shape of quadrilateral OPRQ, where O is the origin? [1] (e) By considering arg( )pq+ or otherwise, show that 11t a n 632 224 π =+++ . [3]

4 Section B: Probability and Statistics [60 marks] 7 Anthony plays a game using a four-sided fair die with the numbers 1, 2, 3 and 4 printed on its sides. Anthony rolls the die and should the outcome be a prime number, he wins an amount equivalent to that number in dollars. Conversely, if the outcome is a non- prime number, Anthony will lose an amount equivalent to that number in dollars. T he random variable X represents the amount of money Anthony wins in a roll of the die. (a) Tabulate the probability distribution of X and find the value of E(X). [2] (b) Find the value of the variance of X. [1] (c) The random variables X1 and X2 are two independent observations of X. By drawing a table of outcomes for 12XX+ or otherwise, find E( )12XX+ . [2] Assume now that a biased six-sided die with numbers 2, 3, 4, 5, 6 and 7 printed on its sides, is used instead for the same game. It is known that for this biased die, the probability of obtaining a “3” is p, the probability of obtaining a “5” is 2p and the remaining four numbers each have an equal probability of occurring. (d) Find the exact value of p for the game to be fair using the biased die. [3] 8 A study was conducted on a group of pre-diabetic seniors to examine whether the number of hours of exercise per week ( t) has an y effect on their fasting blood sugar levels ( S, measured in mg/dL). The results were as follows. t 1.5 1.8 2.1 2.4 2.5 3.1 3.5 4.2 S 119 108 100 95 93 88 86 85 (a) Draw the scatter diagram for these values, labelling the axes. [1] (b) Comment on whether a linear model would be appropriate, referring both to the scatter diagram and the context of the question. [2] (c) By considering the shape of the following graphs, explain which model is the most appropriate for the data collected, where a and b are positive constants. A: 2S a bt= + , B: bSa t= + , C: lnS ab t= + . [1] Using the appropriate model selected above, find (d) the value of the product moment correlation coefficient, [1] (e) the equation of the regression line of S on t, and hence find an estimate of the exercise time required for the fasting blood sugar to be 99 mg/dL. Comment on the reliability of your answer. [3]

5 [Turn Over 9 In this question, you should state clearly all the distributions that you use, together with the values of the appropriate parameters. A toy designer creates weighted wooden blocks used in balance -based educational toys. There are two types of blocks: cylindrical blocks and cuboidal blocks. The mass, in grams, of each cylindrical block and cuboidal block follow a normal distribution with the following parameters: Mean Standard Deviation cylindrical block 280 6 cuboidal block 220 5 (a) Find the probability that the mass of a randomly selected cylindrical block exceeds the mass of a randomly selected cuboidal block by more than 65 grams. [2] (b) The probability that the total of twice the mass of a cylindrical block and thrice the mass of a cuboidal block being less than M g is 0.355. Find the value of M, giving your answer to the nearest integer. [2] To build a deluxe seesaw model, the designer uses: • 2 cylindrical blocks, • 3 cuboidal blocks, • 7 screws, • 1 wooden plank. The blocks are drilled before assembly to fit screws: • Each cylindrical block loses 5% of its mass after drilling. • Each cuboidal block loses 3% of its mass after drilling. The masses of the remaining components follow a normal distribution with the following parameters: Mean Standard Deviation screw 8 1.2 wooden plank 540 8 When assembling the deluxe seesaw model, all of its components are chosen at random. (c) Find the probability that the total mass of all the modified components in a deluxe seesaw model is more than 1785 g. [4]

6 10 Green Earth Factory produces solar panels . On average, 2% of the solar panels produced are found to contain defects. The solar panels are packed in batches of 10. (a) State, in the context of solar panel production, two assumptions needed for the number of defective solar panels in a batch to be well modelled by a binomial distribution. [2] It is now given that the number of defective solar panels per batch follows a binomial distribution. (b) Find the probability that a batch contains at least 2 defective solar panels. [2] As part of Green Plan initiative, the Education Department plans to install solar panels on the rooftops of 100 primary