SRJC H2 MATHS P2 Qn

1   [Turn Over   SERANGOON JUNIOR COLLEGE 2014 JC2 PRELIMINARY EXAMINATION MATHEMATICS Higher 2 9740/2 Tuesday 26 Aug 2014 Additional materials: Writing paper List of Formulae (MF15) 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 dark 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 usin g 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. 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 and 1 blank page.

2 Section A: Pure Mathematics [40 marks] 1 The line ℓ with Cartesian equation 4 43 x z  , y = 1 contains the point B with position vector j + 3k. A point A, not lying on ℓ, has position vector 2i + (15 )j ‒ k. (i) Given that c denotes a unit vector parallel to ℓ, find AB  c and give a geometrical interpretation of this quantity. [3] (ii) Hence find the shortest distance from A to ℓ. [2] The foot of perpendicular from A to ℓ is denoted by F and the foot of perpendicular from F to AB is denoted by G. (iii) Write down the ratio between the area of ∆AGF and area of ∆BGF. [1] (iv) Hence, deduce the ratio AG:GB and find the position vector of G. [2] 2 (a) (i) By using a graphic calculator, find the x-coordinates of the points of intersection of the curves xy e and 21yx  . Hence solve the inequality 21xex  . [2] (ii) Hence, find the exact value of 1 2 21 dxex x   . [3] (b) Find 2 21 d. 47 x xxx   [3] (c) Find   sin ln cos d .x xx [2]

3 A r 60 with (i) An fou leak cyli If a che (ii) ight circula  stands on h h a type of Find the v volume. engineer d und in (i). T kages occur indrical tank a crack is fou emical is lea find the ex inverted c mh mh ar cone-shap horizontal g liquid chem value of r in decides to b u To prevent r, an inverte k as shown und at the c aking into th xact rate of cone half an ped structure ground. A cy mical, is insc n terms of h uild the ab o the liquid ed cone of s in the diagr centre of the he inverted change of t n hour after t 3 e of fixed h ylindrical ta cribed insid when the c ove structur chemical f semi-vertic ram below. e base of the cone at a ra the surface the leaking height h m a ank of radiu de the cone. cylindrical ta re with cyl i from conta m al angle of e cylindrica ate of 0.3 m3 area of the l starts. mr mr and semi-ver us r m, whic ank has a m indrical tank minating t h 45 will be al tank and t 3/min, liquid chem [Tur rtical angle ch is fully fi maximum nk of radius he ground w e attached t o the liquid mical in the rn Over illed [5] r as when o the [5]

4 (a) The complex number z is such that 2 3z  and arg( i ) 4z   . Find w in the form iab , where ,ab  , if 23wz  and 2 5arg 6 z w    . [4] (b) Solve the equation z4 = – 81 , giving the roots in the form reiθ, where r > 0 and   . [2] (i) Hence, express z4 + 81 as the product of two quadratic factors with real coefficients, giving each factor in exact non-trigonometrical form. [3] The roots of the equation z4 = – 81 are represented by z1, z2, z3, z4 such that     1234arg arg arg argzz zz . (ii) Explain why the locus of all points z such that | z – z3 | = | z – z2 | passes through the origin. [1] (iii) The points A, B, C and D represent the complex numbers z1, z2, v and z4 respectively, with 3 1 2vz . Find the area enclosed by the points A, B, C and D. [2]   Section B: Statistics [60 marks]  5 (a) Five couples are seated in a row. Find the number of ways in which three particular men must not be seated next to each other. [2] (b) These five couples are now seated at a round table. (i) Find the number of ways in which the wives must be seated next to her husbands. [1] (ii) Find the number of ways in which the five women are not all seated together. [2] (c) A funfair game consists of a segment whic h requires the player to draw coloured balls from a box. The box contains five bl ue balls and two red balls. In order for the player to win the game, he has to pick two red balls consecutively. Whenever a blue ball is drawn, it will be repl aced back into the box and the drawing continues. However, when a red ball is dr awn, it will not be replaced back into the box. Calculate the probability that the player wins the game. [3] 4

5   [Turn Over   6 In a particular high school, 94% of its students own a Fr iend Book account. A random sample of 80 students is taken fr om the high school. The random variable X denotes the number of students in the sample who own a Friend Book account. (i) State, in the context of this question, an assumption needed to model this situation by a binomial distribution. [1] (ii) If the sample has at least 70 students who own a Friend Book account, find the probability that there are at most 74 students who own a Friend Book account in the sample. [2] (iii) Estimate the probability that there are exactly 75 students who own a Friend Book account in the sample. [3] (iv) Sixty samples each of size 80 are taken from the high school. Find the probability that the average number of students who own a Friend Book account in each sample exceeds 76. [2] 7 (a) A famous zoologist Elsa claims that the mean tail length of Proboscis Monkeys is at most 65 cm on a particular remote island. The tails of a random sample of 20 Proboscis Monkeys are measured and found to have mean 65.5 cm and standard deviation 0.9 cm. Test at the 1% significance level whether Elsa’s claim is valid. [5] (b) Another famous zoologist Anna claims that the mean tail length of Proboscis Monkeys on another island is 63 cm. The tails of a new random sample of 25 Proboscis Monkeys on the island have been measured and the mean length of these monkeys is found to be t cm. (i) Assuming that the tail lengths of Proboscis Monkeys on the island are normally distributed with standard deviation 5.8 cm, find the set of values of t for Anna’s claim to be rejected at the 5% level of significance. [4] (ii) Explain 5% level of significance in the context of the question. [1]

6 8 (a) It is given that the regression line y on x for the following bi variate data is 60 . 5yx . x 20 22 24 26 28 30 32 34 y 14 19 16 a 20 22 25 18 Find a. [2] (b) The data below show the average height s of Japanese maple trees planted in a botanical garden which were observed over a period of 10 years. It is assumed that the height of a tree is dependent on its age. Age of trees in years (x) 1 2 3 4 5 6 7 8 9 10 Average Height in feet (y) 4 6.1 7.3 8.5 9.3 9.7 9.8 10.1 10.5 10.6 (i) Give a sketch of the sc atter diagram for the above data for the model ya b x . [2] (ii) For the model used in (i), give a