PJC H2 MATHS P2

Candidate Name: ________________________ Class: _____________ JC2 PRELIMINARY EXAM Higher 2 MATHEMATICS 9740/02 Paper 2 21 Sept 2015 3 hours Additional Materials: Cover page Answer papers List of Formulae (MF15) READ THESE INSTRUCTIONS FIRST Write your full 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 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. At the end of the examination, fasten all your work securely together. The number of marks is given in brackets [ ] at the end of each question or part question. This document consists of 6 printed pages.

2 @PJC 2015 [Turn Over] Section A : Pure Mathematics [40 marks] 1 The function f is defined by 2f : , , , 2x x x x x     where 2 . (i) Find, in terms of  ,   1f x , stating the domain of 1f  . [4] (ii) On the same diagram, sketch the graphs of  fyx and   1fyx  , showing their graphical relationship clearly. [2] (iii) Find, in terms of  , the solution of the equation     1ff xx  . [3] 2 By sketching the graphs of 3 1y x  and yx , solve the inequality 3 1 xx  . [2] Hence, without using a calculator, evaluate 4 1 34 1 dxx x  . [4] The area bounded by the curves 3 1y x  , yx , the line 2x and the x-axis is rotated completely about the y-axis to form a solid of revolution of volume V. Find the numerical value of V, giving your answer correct to 3 decimal places. [3]

3 @PJC 2015 [Turn Over] 3 Sketch, on a single Argand diagram , the set of points representing all complex numbers satisfying both of the following inequalities: 3 1 3 i2 2 2z    and arg( 2i) 03 z    . [4] Hence find (i) the minimum value of i2 z , [3] (ii) the exact value of the complex number z such that arg( )z is minimum. [3] 4 The parametric equations of a curve are 2xt , 2y t t . (i) The point P on the curve has parameter p. Show that the equation of the tangent at P is   22 2 1py p x p   . [3] (ii) The tangent at P meets the x- and y- axes at the points Q and R respectively. Find the coordinates of Q and R. [2] (iii) Find a cartesian equation of the locus of the midpoint of QR as p varies. [3] (iv) Find the equation of the tangent at the point  4,6 and determine if this tangent meets the curve again. [4] Section B : Statistics [60 marks] 5 There are 8 red cards, and a single letter is printed on each of them. Together they can be arranged to form the words “GOOD LUCK”. The digits 1 to 9 are printed on 9 white cards, with each card having a single digit. A code is formed by laying out 4 red cards and 4 white cards in a row. Find the number of codes that can be formed if the red and white cards must alternate. [5]

4 @PJC 2015 [Turn Over] 6 A popular brand of titbits is holding a “Wi n as You Eat” promotion as its marketing strategy. 2 % of all the standard si ze packets produced during the promotion period contain prize winning coupon s, with each packet containing at most 1 prize winning coupon. The ti tbits are sold as a family pack, each containing 10 randomly chosen standard size packets. (i) Find the probability that a randomly chosen family pack contains no winning coupon. [1] (ii) The family packs are delivered to supermarkets in cartons. Each carton contains 20 family packs. Find the probability that a randomly chosen carton contains more than 16 family packs with no winning coupon. [2] (iii) A particular family buys one family pack every week, for 5 consecutive weeks. Find the probability that the fifth week is the second week that the family does not get any winning coupon. [3] (iv) An event organiser stocks up 1000 standard size packets for an event. The probability of having at least k winning coupons is more than 0.15. Using a suitable approximation, find the largest possible value of k. [4] 7 The duration of a patient’s consultation, in minutes, with a general practitioner (GP) and a specialist are modelled as having independent normal distribution s with mean and standard deviation as given in the table. Mean Standard Deviation Consultation with GP 6.2 1.9 Consultation with specialist 10.7 2.8 (i) Find the probability that the total duration of 3 patients’ consultation with the GP is shorter than twice the duration of a patient’s consultation with the specialist. [3] The consultation fee charged by the GP is made up of 2 components: a fixed component of $10 and a variable component of $1 per minute. Similarly, the consultation fee charged by the specialist has a fixed component of $25 and a variable component of $2 per minute. (ii) Find the probability that the consultation fee of a patient visiting the specialist is at most 3 times that of a patient visiting the GP. [4]

5 @PJC 2015 [Turn Over] 8 A beverage company claims that the vitamin C content of orange juice produced by the company is the same as that in freshly squeezed orange juice. A random sample of 90 packets of orange juice produced by the company is taken and the vitamin C content, x mg per 100 ml, is measured. The results are summarised by 8993x , 2 900240x  . It is known that the vitamin C content of freshly squeezed orange juice has a mean of 101 mg per 100 ml. Test, at the 5% significance level, whether the company’s claim is valid. [5] Another random sample of 10 packets of orange juice produced by the company is taken. Assuming that the standard deviation of vitamin C content of orange juice produced by the company is now known to be 4 mg per 100ml, find the set of values within which the mean mass of this sample must lie for the company’s claim to be valid at the 5% significance level. Give your answer to 2 decimal places, and state any necessary assumption for your calculations to be valid. [3] 9 On average, a travel agency receives 1.2 complaints daily. (i) State, in this context, two conditions that must be met for the number of complaints to be well modelled by a Poisson distribution. Explain why one of your conditions may not be met. [3] For the remainder of this question assume that these conditions are met. (ii) Find the probability that, in a period of 10 days, the total number of complaints received is below the expected value. [2] (iii) Find the probability that there is at least 1 complaint received daily for 2 consecutive days. [2] (iv) Find the least number of consecutive days for which the probability of at least 1 complaint received exceeds 0.999. [4] (v) Find the probability that the average number of complaint received per day over a period of one year (365 days) is less than 1.3. [2]

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