2020 H2 Prelim P2

ANGLO-CHINESE JUNIOR COLLEGE JC2 PRELIMINARY EXAMINATION Higher 2 CANDIDATE NAME TUTORIAL/ INDEX FORM CLASS NUMBER MATHEMATICS 9758/02 Paper 2 28 August 2020 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 signi ficant 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 expected, 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 __ printed pages. [Turn Over Question Marks 1 /9 2 /9 3 /10 4 /12 5 /5 6 /6 7 /7 8 /7 9 /9 10 /12 11 /14 /100

2 ANGLO-CHINESE JUNIOR COLLEGE 2020 H2 MATHEMATICS 9758/02 Section A: Pure Mathematics [40 marks] 1 (a) If 2=a , 3−=ab and 5 2=a.b , use the cosine rule for triangles to find b . [3] (b) It is given that r ijkabc=++ where a, b and c are constants. Give the geometrical interpretations of (i) r.k , [1] (ii) rk× . [1] (iii) Given that kr p×= and rp k×= , in either order, show that 22 1ab+= and find the value of c. [4] 2 A scientist and his student are investigating the growth of bacteria in a petri dish. At time t minutes after the bacteria is first introduced into the dish, the number of bacteria in the dish is denoted by x, in hundreds . Initially, 100 bacteria were introduced into the dish. In 10 minutes, the number of bacteria became 400. (i) The student claims that the growth rate of the bacteria in the dish is inversely proportional to the square root of the number of bacteria in the dish. Write down a differential equation for this situation and solve it to get x as a function of t. [4] (ii) The scientist claims that ( ) 2d 100d x m xxt = − , where m is a constant. Show that 1e Hmt Hx A −= + , where H is a whole number to be determined. (You need not find the values of A and m.) [4] (iii) Whose model do you think is better, the student’s or the scientist’s? Justify your answer. [1]

3 ANGLO-CHINESE JUNIOR COLLEGE 2020 H2 MATHEMATICS 9758/02 3 (a) The diagram below shows a large rectangular field, with a cor ner of the field being fenced off. The dotted line AB, represents the fence. O is the origin and the points A and B have coordinates ( ),0x and ( )0, y respectively. The point P on line AB, is a permanent sprinkler that is used as a support for the fencing. Point P has coordinates ( ) 2 ,2aa , where a is a positive constant. (i) Show that S, the area of the corner AOB, is given by 2 2 axS xa= − unit2. [2] (ii) Use differentiation to find, in terms of a, the minimum value of S, proving that it is a minimum. [4] (b) In a parallel electrical circuit, the equivalent resistance, PR , is given by 1 11 n iPiRR = =∑ , where iR , 1 , 2 , , in=  , is the resistance of the i-th resistor connected to a battery. The diagram below shows a parallel electrical circuit, with three resistors of resistance, 1R , 2R and 3R respectively, connected to a battery. It is known that 1R changes with time at a rate r , 2R changes with time at a rate 2 r, and 3R remains unchanged. Find the rate of change of the equivalent resistance, d d PR t , when 21RR= , and 31 2RR= , leaving your answer in terms of r. [4] O ( ) 2 , 2Pa a• ( )0,By ( ),0Ax Battery 1R 2R 3R

4 ANGLO-CHINESE JUNIOR COLLEGE 2020 H2 MATHEMATICS 9758/02 4 (a) It is given that ( ) 6 42 f x x ax x b=− −− , where a and b are real numbers. (i) Show that ( ) ( )ff xx= − . [1] The diagram shows the curve with equation ( )fyx= for 0x≥ . The curve crosses the positive x-axis at x β= . (ii) Determine the number of non- real roots of the equation ( )f0 x = . Justify your answer. [2] (iii) Given that 1 iezr θ= , where 0r > and π0 2θ<< , is one of the roots of the equation ( )f0 x = , express all the roots of ( )f0 x = in the modulus -argument form. [3] (iv) Hence, show that ( )f x can be expressed as a product of three quadratic factors of the form ( )( )( ) 2 22 22cos cosx A x Brx r x Crx r θθ− −+ −+ , where A, B and C are constants. [3] (b) Find the modulus of the complex number ( ) ( ) 2 * * i 2i 2 4i zz zz + − , where z is a complex number. [3] x y O β

5 ANGLO-CHINESE JUNIOR COLLEGE 2020 H2 MATHEMATICS 9758/02 Section B: Probability and Statistics [60 marks] 5 For events A and B, it is given that ( )P 0.8AB∪= and ( )P | 0.5AB = . It is also given that events A and B are independent. (i) Find ( )P B . [2] A third event C is such that events A and C are independent and events B and C are independent. (ii) Given also that ( )P 0.5C = , find exactly the maximum and minimum possible values of ( )P ABC∩∩ . [3] 6 Drew has a sock drawer which contain s 3 pairs of red socks, 2 pairs of white socks and 1 pair of black socks. In the mornings, Drew randomly pulls out socks from the drawer, one sock at a time, until he has a pair of socks which matched in colour. The random variable S denote the total number of socks Drew pulls out in a morning. (i) Determine the probability distribution of S. [3] (ii) Find ( )E S and ( )Var S . [2] (iii) Find the probability that Drew took more than 2 pulls to obtain a pair of matching socks for 7 consecutive mornings. [1] 7 Peter and Calvin are brothers. They went to a birthday party with 6 other friends. (i) The 8 friends started off the party with dinner, sitting down on 8 chairs around a circular dining table. Given that t here was at least one other person seated between the two brothers, find the number of different arrangements of the 8 friends at the table. [2] (ii) The 8 friends then decided to play a game of musical chairs. 5 chairs were placed in one row. When the music stopped, Peter could not get a seat but Calvin managed to sit on the chair in the middle of the row. Find the number of ways that the 5 chairs could be occupied. [1] (iii) The 8 friends decided to watch a movie to end off their night. There are 4 movi es for selection. Each person will vote for a movie to watch, and the movie with the highest number of votes will be played. Find the total number of ways the 8 friends could have voted such that there were movies tied for the highest number of votes. [4]

6 ANGLO-CHINESE JUNIOR COLLEGE 2020 H2 MATHEMATICS 9758/02 8 Wally is looking for an investment plan that generates a monthly income of more than $200. He shortlisted an investment plan whose prospectus promises investors a variable monthly income with a mean of $200 per month. Wally decides to research furth er by collecting a random sample of 36 of the plan’s past monthly income returns. The monthly income returns, $x, are summarised as follows. ( )200 662.4x−=∑ ( ) 2 200 307141.56x−=∑ (i) Test, at the 10% level of significance, whether Wally should invest in this plan. You should state your hypotheses and define any symbols you use. [6] (ii) Explain what is meant by the phrase ‘10% significance level’ in the conte