YIJC 9758 2023 Prelim P2

This document consists of 22 printed pages and 2 blank pages. [Turn Over YISHUN INNOVA JUNIOR COLLEGE JC 2 PRELIMINARY EXAMINATION Higher 2 CANDIDATE NAME CG MATHEMATICS Paper 2 Candidates answer on the Question Paper. Additional Materials: List of Formulae (MF26) 9758/02 13 SEPTEMBER 2023 3 hours READ THESE INSTRUCTIONS FIRST Write your CG, index number and name on 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 mathem atical 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. For Examiners’ Use Question Marks 1 2 3 4 5 6 7 8 9 10 11 Presentation Total / 100

2 ©YIJC 9758/02/JC2PE/23 Section A: Pure Mathematics [40 marks] 1 The diagram shows a sketch of the curve f( )yx= . The shaded region under the curve between 1x = and 4x = , shown in the diagram, is A . This region is split into 4 vertical strips of equal width, h units. (a) State the value of h and show, by drawing on the above diagram , that [ ] 4 1 f (1 ) n nh h = +∑ is less than the area of A. [2] (b) Find a similar expression that is greater than the area of A. [1] You are now given that f ( ) 2 ln( 1) 7xx = − ++ . (c) Use the expression in part (a) and your expression in part (b) to find lower and upper bounds for the area of A. [2] (d) The curve with equation g( )yx= is obtained when the curve with equation f( )yx= is translated by b units in the positive y-direction, where b is a positive constant. Given that [ ] [ ] 44 11 g(1 ) f (1 ) nn nh h nh h c = = += ++∑∑ , where c is a positive constant, express c in terms of b. [1] 2 The complex number z is given by 3i 3 iz k −= − , where k is a real constant. (a) Given that 1k = , find the exact values of z and arg( )z . [4] (b) Given instead that zz ∗= , find the value of k. [3] y x 1 4 O

3 ©YIJC 9758/02/JC2PE/23 [Turn Over 3 (a) Show that 2 1ln 1 r  − can be expressed as ( ) ( )ln 1 2 ln ln 1r rr−− + + , where 1r > . [1] The sum 2 2 1ln 1 n r r=  −∑ is denoted by nS . (b) Find an expression for nS in terms of n. [4] (c) Find the smallest value of n for which nS is within 0.05 of the sum to infinity. [3] 4 (a) Vectors u and v are such that ≠u 0 and .v×u = 0 (i) Find a linear relationship between u and v. [2] (ii) Find a unit vector n such that ( )22 .− − ×=i j k n0 [2] (b) Referred to the origin O, the points A, B and C have position vectors a, b and 24 3−ab respectively. (i) Given that the area of triangle ABC is 14, show that 12=a×b . [3] (ii) Given further that 5, 3= =ab and AOB∠ is obtuse, deduce the exact value of cos AOB∠ . [2] 5 In the triangle ABC, 1AC = , angle BAC x= radians and angle π1 6ACB = radians (see diagram). (a) Show that 3 in 1 cos sx AB x+ = . [3] (b) Given that x is a sufficiently small angle, show that 21AB a x bx+≈+ , for constants a and b to be determined exactly. [3] (c) The first two terms in the Maclaurin series for ( )ln px q+ are equal to the first two terms in the series expansion in part (b). Using standard series from the List of Formulae (MF26), find the exact values of p and q. [4] Section B: Probability and Statistics [60 marks] 6 For events A and B, it is given that ( )P 0.82 BA =′∪′ and ( )P 0.4AB =∣ . (a) Find ( )P. B [2] (b) Hence find an inequality for ( )P. A B′∩ [2] A third event C is such that ( )P 0.1.C = A and C are mutually exclusive. B and C are independent. (c) Find ( )P BA C′ ′∩∩ . [3] A B C 1 x

4 ©YIJC 9758/02/JC2PE/23 7 There are 2 red discs, 6 green discs and 1 blue disc on a table. The discs are identical in all aspects except colour. (a) The 9 discs are arranged in a line. Find the number of different possible arrangements such that the 2 red discs and the blue disc are next to one another. [2] (b) All the 9 discs are put into a bag. Jack takes 3 discs from the bag at random. The random variable R is the number of red discs taken. (i) Tabulate the probability distribution of R. [3] (ii) For each red disc taken, Jack gains nine points. Otherwise, he loses three points for each disc of any other colour taken. Find the expected change in points Jack has after taking the 3 discs. [3] 8 A company produces packets of scallops with each packet stated to weigh 500 grams. In a quality control inspection, the production manager takes a random sample of 35 packets packed on a particular day at the factory. The masses, x grams, are summarised as follows. ( )500 198.5x −=∑ ( )2500 7188x −=∑ (a) Find unbiased estimates of the population mean and variance. [2] (b) Test, at the 2% level of significance, whether the mean mass of packets of scallops is 500 grams. [4] To increase customer satisfaction, the company changes the machine setting and claims that the mean mass of packets of scallops is now at least 0µ grams. To investigate the claim, the production manager takes a random sample of 40 packets of scallops after the change. The mean and standard deviation of this sample are found to be 510 grams and 11.7 grams respectively. (c) Given that there is no reason to reject the company’s claim at the 5% level of significance, find the range of possible values of 0µ , giving your answer correct to 2 decimal places. [4] 9 A dart board is designed with an inner radius of r cm and an outer radius of 15 cm, as shown in the diagram below. A dart throw is considered a ‘hit’ if it lands within the inner circl e. It is assumed that any dart throw lands randomly on the dart board. r cm 15 cm

5 ©YIJC 9758/02/JC2PE/23 [Turn Over (a) Show that the probability of a ‘hit’ is 2 225 r . [1] (b) Wei plays a round of 8 dart throws. (i) The probability that Wei obtains more than six ‘hits’ is at most 0.08. Write down an inequality satisfied by r . Hence find the range of possible values of r . [2] It is now given that 6r = . (ii) Find the probability that Wei’s fifth throw is the second time that he obtains a ‘hit’. [2] (iii) Find the least number of throws that Wei needs to attempt such that the probability of obtaining at least 1 ‘hit’ from the attempts is at least 0.7 . [3] (iv) Chan also plays a round of 8 dart throws. The number of ‘hits’ Chan obtained is more than the expected number of ‘hits’ Wei obtained. Find the probability that the number of ‘hits’ Chan obtained is fewer than twice the expected number of ‘hits’ Wei obtained. [3] 10 In this question you should state clearly all the distributions that you use, together with the values of the appropriate parameters. The masses, in kg, of pumpkins and cabbages sold in a supermarket are modelled as having independent normal distributions with means and standard deviations as shown in the following table. Mean Mass Standard Deviation Pumpkins 3.7 0.4 Cabbages