SRJC H2 MATH P2

1 [TURN OVER] SERANGOON JUNIOR COLLEGE 2018 JC2 PRELIMINARY EXAMINATION MATHEMATICS Higher 2 9758/2 17 Sept 2018 3 hours Additional materials: Writing paper List of Formulae (MF 26) TIME : 3 hours READ THESE INSTRUCTIONS FIRST Write your name and class on the cover page and on all the work you hand in. Write in 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 calcul ator are not allowed in a question, you are required to present the mathematical steps us ing mathematical notatio ns 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. At the end of the examination, fasten all your work securely together. Total marks for this paper is 100 marks This question paper consists of 7 printed pages (inclusive of this page) and 1 blank page.

2 Section A: Pure Mathematics [40 marks]. 1 R is the region enclosed by the line 5y =− and the curves 2 25yx x=− + − and () 22 5 141 6 yx ++= as shown in the diagram below. Find the volume generated by region R when it is rotated 2π radians about the x-axis. Leave your answer correct to 2 decimal places. [3] 2 A curve has parametric equations tanx θ= , 2secy θ= for 02 θπ≤< . The equation of the tangent to the curve at the point P with parameter p is given by ()2si n 2c osyp x p=+ . (i) The tangent at P meets the x and y axes at the points A and B respectively. Find the Cartesian equation of the locus of the mid-point of AB as p varies. [3] (ii) The tangent at P meets the line 2yx= at the point S and the line 2yx=− at the point T. Show that the area of the triangle OST is independent of p, where O is the origin. [4] 3 Given that 1tanln e xy − = , show that () () 2 2 2 dd1l n 1 2 dd yyxy x xx += + − . [2] (i) Find the Maclaurin’s series for y up to and including the term in 3x . [4] (ii) Deduce the Maclaurin’s series for y, where 1tanln e 1 xyx − =+ − , up to and including the term in 3x . [3] x y R

3 [TURN OVER] 4 Given that the equation () ( ) ( ) 32f2 2 4 2 8 0zz a z az a=++−+− = has no real solution, explain clearly why a is not a real number. [2] (i) It is known that f has a factor ()2iz + , find a. Hence find all the roots of ()f0 z = , showing your workings clearly. [4] (ii) Deduce the roots of the equation () 23 24 2820 aaa ww w − −−+ − = , where a takes the value obtained in (i). [2] 5 A hollow metallic ramp, in the shape of a prism, is constructed for the marching contingent to march onto to reach an elevated platform from the ground during the national day parade. The diagram below shows the prism with O as the origin of position vectors and the unit vectors i, j and k are parallel to OA, OC and OE respectively. It is given that OE = CD = 1 m, OA = CB = 2 m and OC = AB = ED = 4 m. A laser beam in the form of a line l has Cartesian equation , 12 ax zy+ == , where a ∈¡ , is emitted onto the plane ABDE. (i) Find, in terms of a, the coordinates of the point of intersection, M, of the laser beam and the plane ABDE. [4] For the following parts of the question assume 0a = . (ii) The laser beam is reflected about the plane ABDE. By finding the foot of perpendicular from ()0,1, 0Q to the plane ABDE, find the equation of the reflected beam. [5] (iii) The path traced out by an ant crawling on the floor OABC is given by 21 02 , 00 ββ −     =+ ∈       r ¡ . Let P be the point on the path, located under the ramp, whereby the ant is equidistant between the planes ABDE and OCDE. Find the position vector of point P exactly. [4] j O E A C B D k i 4 m 1 m 2 m

4 Section B: Statistics [60 marks] 6 A jackfruit farm produces 2 types of j ackfruit pulps namely Grade A and Grade B. The pulps are randomly packed into boxes of 8. The probability that a box contains at most 3 Grade A jackfruit pulps is 0.00245. A fruit trading wholesaler places a monthly order of 1000 boxes of jackfruit pulps for 5 years. Find the approximate probability that the mean number of boxes that contain at most 3 Grade A jackfruit pulps in a month is more than 3. [You may assume that there are 12 months in a year.] [4] 7 A player throws an unbiased six-sided die and the number shown on the top face is noted. If it is not a six, the score is the number shown on the top face. If it is a six, he throws the die a second time and the score is the total score obtained from his two throws. The player has at most two throws. (i) Construct the probability distribution table on the score for the player. [2] Given that the expected player’s score is 49 12 , (ii) find the exact value of the variance of the player’s score, showing your workings clearly. [3] 8 The length of time that Somesong phones last in between charges has a normal distribution with mean 20 hour s and standard deviation 2 hours. The length of time that Apfel phones last in between charges has a normal distribution with mean μ hours and standard deviation σ hours. (a) The average length of time two randomly chosen Somesong phones and one randomly chosen Apfel phone will last in between charges is equally likely to be less than 19 hours or more than 23 hours, with the probability known to be 0.02275. (i) Calculate the values of μ and σ . [3] (ii) Find the probability that twic e the length of time a Somesong phone lasts in between charges differs from 40 hours by at least 2 hours. [2] (b) Six randomly chosen Somesong phone s are examined. Find the probability that the sixth Somesong phone is the fourth Somesong phone that lasts more than 22 hours in between charges. [2]

5 [TURN OVER] 9 A customer wishes to investigate the shelf life of the durian puffs produced by a baker before they turn bad when placed at room temperature and pressure. It is assumed that the shelf life of the durian puffs are independent of one another. Based on past records, the baker claims that the mean shelf life of the durian puffs is at least 8 hours. To test this claim, the customer recorded the shelf life of 55 randomly chosen durian puffs and found th at its mean is 7.4 hours and standard deviation is 2.6 hours. (i) Carry out a test, to determine whethe r there is any evidence to doubt the baker’s claim at 5% significance level. [5] (ii) Explain, in the context of the questio n, the meaning of ‘at 5% significance level’. [1] (iii) Suppose the population standard deviat ion is 3.2 hours now and the baker claims that the mean shelf life the duri an puffs is 8 hours. A new sample of 10 durian puffs is taken. Using this sample, the customer conducts another test and found that the baker’s claim is not rejected at the 5% significance level. Stating a necessary assumption for the test, find the set of values that the sample mean shelf life, y , can take. [4]