IJC H2 MATH P2 (9758)

INNOVA JUNIOR COLLEGE JC 2 PRELIMINARY EXAMINATION in preparation for General Certificate of Education Advanced Level Higher 2 CANDIDATE NAME CIVICS GROUP INDEX NUMBER Mathematics Paper 2 Additional materials: Answer Paper Cover Page List of Formulae (MF 26) 9758/02 12 September 2017 3 hours READ THESE INSTRUCTIONS FIRST Do not open this booklet until you are told to do so. Write your name, class and index number 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 a graphic calculator. Unsupported answers from a graphic calculator are allowed 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, 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. Innova Junior College [Turn over

2 IJC/2017/JC2 9758/02/Sep/17 Section A: Pure Mathematics [40 marks] 1 The complex number z is such that 1z  and arg z  , where 0 4  . (i) Mark a possible point A representing z on an Argand diagram . Hence, mark the points B and C representing 2z and 2zz respectively on the same Argand diagram corresponding to point A. [2] (ii) State the geometrical shape of OACB. [1] (iii) Express 2zz in polar form,    cos cos i sinpq k k     , where p, q and k are constants to be determined. [2] 2 The function f is given by 1f : 3 2x x  for ,2xx  . (i) Find 1f ( ) x and state the domain of 1f  . [3] (ii) Explain why the composite function 2f exists. [1] (iii) Find the value of x for which 2f() x x . Explain why this value of x satisfies the equation 1f ( ) f ( )x x . [3] 3 It is given that a curve C has parametric equations 2 2 1, 1 xt t y t    for 22 t  . (i) Sketch C, indicating clearly the coordinates of the end points and the points where C cuts the y-axis. [4] (ii) Find the equation of the tangent to C that is parallel to the y-axis. [4] (iii) Express the area of the region bounded by C, the tangent found in part (ii) and both axes, in the form f( ) d b a tt , where the function f and the constants a and b are to be determined. Hence find this area, leaving your answer in exact form. [5]

3 IJC/2017/JC2 9758/02/Sep/17 4 A farmer owns a plot of farmland. To prep are for wheat planting, the farmer has to plough the farmland before sowing wheat seeds. At the start of the first week, 2300 m of the farmland is ploughed. The farmer ploughs another 2100 m of the farmland at the beginning of each subsequent week. To sow wheat seeds, the farmer is considering two different options. (a) In the first option, the farmer sows wheat seeds on 60% of the unsown ploughed land at the end of each week. (i) Find the area of unsown ploughed land at the end of the second week. [1] (ii) Show that the area of unsown ploughed land at the end of the thn week is given by    120.4 300 1 0.4 mnn k   , where k is an exact constant to be determined. [3] (iii) Find the number of complete weeks required for the area of unsown ploughed land to first fall below 270 m . [3] (b) In the second option, the farmer sows 280 m of the unsown ploughed land at the end of the first week. At the end of each subsequent week, he sows 220 m of the unsown ploughed land more than in the previous week. This means that the area of sown ploughed land is 2100 m in the second week, 2120 m in the third week, and so on. (i) Find, in terms of n, the area of unsown ploughed land at the end of the thn week. [4] (ii) Find the number of complete weeks required for the farmer to finish sowing all the ploughed farmland in this option. Deduce the area of ploughed land to be sown in the final week. [4] Section B: Statistics [60 marks] 5 A group of twelve people cons ists of six married couples. Each couple consists of a husband and a wife. (i) The twelve people are to stand in a stra ight line. Find the nu mber of different arrangements if each husband must stand next to his wife. [2] (ii) The group of twelve people finds a round table with ten chairs. Assuming only ten people are to be seated, find the probability that five married couples are seated such that each husband sits next to his wife and husbands and wives alternate. [3] [Turn over

4 IJC/2017/JC2 9758/02/Sep/17 6 Seven red counters and two blue counters are placed in a bag. All the counters are indistinguishable except for their colours. Clark and Kara take tu rns to draw a counter from the bag at random with replacement. The first player to draw a blue counter wins the game and the game ends immediately. If Clark draws first, find the probability that (i) Clark wins the game at his third draw, [2] (ii) Kara wins the game. [3] 7 An archer shoots an arrow into a circular ta rget board that has a radius of 60 cm. The target board further consists of three inner concentric circular sections, with radii 40 cm, 20 cm and 10 cm respectively as shown in the diagram. The archer scores  50 points if the arrow lands in the centre circle of radius 10 cm,  20 points if the arrow lands in the ring with outer radius 20 cm,  10 points if the arrow lands in the ring with outer radius 40 cm,  0 point otherwise. Assume that the arrow will definitely hit the ta rget board and is equally likely to hit any portion of the target board. (i) Let X be the number of points scored for one arrow shot. Find the expectation of X, leaving your answer in 4 significant figures. [3] (ii) Interpret, in this context, the value obtained in part (i). [1] (iii) The archer shot at the target board forty times. Find the probability that the average score obtained by the archer is between 10 and 20 points (inclusive). [4] 20 cm 40 cm 10 cm 60 cm

5 IJC/2017/JC2 9758/02/Sep/17 8 At a hospital, records show that 84.5% of pati ents turn up for their appointments. It is known that on any day, the doctor has time to see 20 patients. On one particular day, there are 20 patients who make appointments to see the doctor. (i) State, in this context, on e condition that must be met for the number of patients who turn up for their appointments to be well modelled by a binomial distribution. [1] For the remainder of this question, assume that the condition stated in part (i) is met. (ii) Find the probability that more than 15 pa tients turn up for their appointments. [2] (iii) Given that at least 12 patient s turn up for their appoint ments, find the probability that more than 2 patients fail to turn up for their appointments. [3] (iv) To improve efficiency, the hospital decides to increase the number of appointments that can be made on each day. Given th at there will still be enough time for the doctor to see 20 patients, find the greatest number of appointments that can be made so that there is a probability of at le ast 0.85 of the doctor having time to see all patients who turn up. [2] 9 In order to recruit the best possible employees, a large corporation has designed an entrance test that consists of three components, namely Logica