CJC 2012 JC2 H2 Prelim Paper 2 Questions

9740/02/PRELIMS/2012 [Turn over CATHOLIC JUNIOR COLLEGE General Certificate of Education Advanced Level Higher 2 JC2 Preliminary Examination MATHEMATICS 9740/02 Paper 2 29 AUGUST 2012 3 hours Additional Materials: List of Formulae (MF15) READ THESE INSTRUCTIONS FIRST Write your name and class on all the work you hand in. Write in dark blue or black pen on both sides of the paper. You may use a soft pencil for any diagrams or graphs. Do not use staples, paper clips, highlighters, glue or correction fluid. Answer all the questions. 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. You are expected to use a graphic calculator. Unsupported answers from a graphic calculator are a llowed unless a question specifically states otherwise. Where unsupported answers from a graphic 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. At the end of the examination, arrange your answers in NUMERICAL ORDER. Place this cover sheet in front and fasten all your work securely together. The number of marks is given in brackets [ ] at the end of each question or part question. Name: ___________________________ Class: ______ __________ Question 1 2 3 4 5 6 7 8 9 10 11 12 13 Total Marks Total 4 10 10 8 8 8 8 8 6 4 6 11 9 100 This document consists of 6 printed pages. Catholic Junior College

2 9740/02/PRELIMS/2012 [Turn over Section A: Pure Mathematics [40 marks] 1 Let f( x) be a cubic polynomial where 6) 2f( = and 15 ) 1f( =− . Also, f( x) takes on stationary values at 2−=x and 1=x . Find f( x). [4] 2 A line, l 1, and a plane p1, have equations ℜ∈           −+           − = λλ , 2 1 3 2 1 1 r and 4 2 0 1 =          − ⋅r respectively. (i) Find the intersection point of l1 and p1. [3] (ii) Find an equation, in scalar product form, for the plane p2 that contains l1 and the point B(0, -1, -2). [3] (iii) Find an equation for the line of intersection, l2, between p1 and p2. [2] (iv) A plane, p3, has an equation 3 1 =           ⋅ b ar . What is the relationship between a and b such that p3 is parallel to l2? [2] 3 The function f is defined as follows. 1 1: f 2 + −→ xx for ℜ∈x . (i) Sketch the graph of y = f( x). [1] (ii) If the domain of f is restricted to kx ≤ , state with a reason the greatest value of k for which the function )(f 1 x− exists. [2] (iii) With the restricted domain in (ii), find an express ion for )(f 1 x− and the real value of a such that )f( )(f 1 aa =− . [4] The function g is defined as follows. 3: g +→ xx for ℜ∈x . (iv) Find a restriction on the domain of g such that gf 1− exists, using the function )(f 1 x− found in part (iii). [3] 4 (i) Find xxx d sin 2 ∫ . [4] (ii) The region R is bounded by the curve xxy sin = , the lines 0=x and π=x , and the x–axis. Find the volume of the solid of revolution f ormed when R is rotated through 4 right angles about the x–axis. [2]

3 9740/02/PRELIMS/2012 [Turn over (iii) Hence calculate the volume of the solid of re volution formed when S is rotated through 4 right angles about the x–axis, where S is the region bounded by the curve xxy sin = , the lines π=x and π=y , and the y–axis. [2] 5 A family of curves is defined by the differential e quation 22 d d yxx yxy += . By substituting ux y = , where u is a function of x, find an expression for the family of curves, expressing it in the form )( f2 xy = . [5] Given that one of the members from the family of cu rves passes through (1, 2), find an expression for the particular solution curve and st ate the values of x when y = 3. [3] Section B: Statistics [60 marks] 6 In the Student Council of Charisma Junior College, there are a total of 31 members, comprising 22 members in the Student Services (SS) Wing and 9 members in the Charismatic Influence (CI) Wing. At the college’s m orning assembly, a total of 6 members are on duty, carried out in pairs. The three different duties every morning consist of the Daily Inspirations, the Flag Raising and the Recital of the national anthem and pledge. All duties can be carried out by any student councillor except the Daily Inspirations co mponent, which must be carried out by members of the CI Wing. Find (i) the total number of possible duty groupings, [2] (ii) the total number of duty groupings where Jill, a member of the CI Wing, is on duty, [2] (iii) the total number of duty groupings where Jack, a me mber of the SS Wing, is on duty. [2] (iv) Hence comment on whether it is fair to the Student Council members if the duty groupings in part (i) were picked at random, justif ying your answer. [2] 7 In Sunny Island Republic, all male citizens are re quired to undergo a National Physical Fitness Assessment (NPFA) Test before enlisting for National Service in the army. Based on historical data, 75% of males who attempt the NPFA Test will pass the test and they are awarded either a Bronze, Silver or Gold standard. T he performance at the NPFA Test is known to influence the chances of these National Se rvicemen being promoted to the rank of an Officer. For male citizens who fail the NPFA Test, they are required to undergo a Physical Assessment Test (PAT) at the end of their basic tra ining. It is known that those who pass the PAT will have a chance to be promoted to the rank o f an Officer. The information is summarized in the tree diagram below.

4 9740/02/PRELIMS/2012 [Turn over (i) It is known that the probability that a randomly ch osen serviceman is an Officer is 0.60. Find the value of a. [2] (ii) Find the probability that a randomly chosen se rviceman failed his NPFA Test given that he is an Officer. [2] (iii) Hence find the probability that a randomly ch osen Officer passed his NPFA Test. [1] (iv) Find the probability that a randomly chosen se rviceman who is a Non-officer had failed his NPFA Test. [3] 8 On average, the probability that John receives at least one Short Message (SMS) on his mobile phone in any given 30-minutes period is 0.95 . A school day consists of 14 such periods of 30 minutes each. The number of periods i n which John receives at least one SMS is the random variable X. (i) State, in the context of this question, the ass umptions needed to model X by a binomial distribution. [2] (ii) Explain why one of the assumptions stated in p art (i) may not hold in this context. [1] Assume now that these assumptions in part (i) hold. (iii) Find the probability that in a school day, th ere are at most 10 periods in which John receives at least one SMS. [2] (iv) Using an appropriate approximation, find the m ost likely number of periods in a week of 5 school days in which John does not receiv e any SMS. State the approximate distribution clearly. [3] NPFA Pass Fail Standard Rank PAT Fail Pass Gold Silver Bronze Officer Non -officer Officer Non -officer Officer Non -officer Officer Non -officer Non -officer 0.75 0.25 0.63 0.17 a 1− a 1.00 0.91 0.09 0.72 0.28 0.54 0.46 0.36 0.64 0.20

5 9740/02/PRELIMS/2012 [Turn over 9 In a school canteen, the number of people joining the noodle stall queue in a period of 30 seconds is a random variable with the distribution Po(1.5). (i) Find the probability that, in a period of 5 min utes, at least 10 people join the queue. [1] (ii) The number of people leaving the same queue in a period of 30 seconds is a random variable with distribution Po(1.9). In a period of 5 minutes, find the probability that at