MJC H2 MATH P2 Question

Write your name and civics group 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 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. MERIDIAN JUNIOR COLLEGE JC2 Preliminary Examination Higher 2 ___________________________________________________________________ H2 Mathematics 9740/02 Paper 2 18 September 2015 3 Hours Additional Materials: Writing paper List of Formulae (MF 15) ___________________________________________________________________ READ THESE INSTRUCTIONS FIRST ___________________________________________________________________ This document consists of 9 printed pages. [Turn Over

2 MJC/2015 JC2 Preliminary Examinations/9740/02 Section A: Pure Mathematics [40 marks] 1 The sequence 1 2 3, , ,...u u u is such that 1 3 and u  1 23 , for all 1.n n un u     (i) By writing the first 3 terms of this sequence, s how that a possible conjecture for nu is 1 1 1 n n a a    , where a is a positive integer to be determined. [2] (ii) Using the value of a found in part (i), prove by induction that 1 1 1 n n n au a    for all 1n . [4] (iii) Hence determine if the limit of 1 2 3 ... nu u u u exists as n . [2] 2 Fig. 1 shows a rectangular piece of cardboard ABCD of sides 10 cm and 20 cm. A trapezium shape is cut out from each corner , to give the shape shown in Fig. 2. This shape consists of 2 isosceles triangles and 3 rectangles of different sizes. The remaining cardboard shown in Fig. 2 is folded along the dotted lines, to form a closed triangular prism shown in Fig. 3. (i) Show that the volume 3 cmV of the closed triangular prism is given by   4 3 21 10 25 2505V h h h h    . [4] (ii) Use differentiation to find the maximum value of V, proving that it is a maximum. [5] 10 20 Fig. 1 Fig. 2 Fig. 3 h y A B D C h h x

3 MJC/2015 JC2 Preliminary Examinations/9740/02 3 (a) It is given that 1 2i is a root of 32 50x ax bx    , where a and b are real. Find the values of a and b and the other roots. [4] (b) (i) Find the possible values of z such that 4 2z  and 4z is a negative real number. [3] Let 1 2 3 4, , and z z z z be the values of z found in part (i) such that        1 2 3 4arg arg arg argz z z z      . (ii) Write down a complex number w such that 21z wz . [1] (iii) Hence, or otherwise, find the exact value of 2 33 *wz w z , where *w is the conjugate of w. [2] [Turn Over

4 MJC/2015 JC2 Preliminary Examinations/9740/02 4 The planes p and q have equations 1 11 1   r and 1 02 1   r respectively, and meet in the line l. (i) Find the acute angle between p and q. [2] (ii) Find a vector equation for l. [2] Let P be the set of planes 1 2 3, , ,p p p , such that the c artesian equation of np is given by ,nnx u y z S   where nu is the nth term of a geometric progression with first term 1 and common ratio 1 2 , and nS is the sum of the first n terms of the geometric progression. (iii) Write down a c artesian equation of kp in terms of k, where k  . Hence state the limit of the acute angle between 1p and kp as k . [3] (iv) Show that l lies in all planes in P. [2] Let  be a plane with equation a bd c   r . (v) If ac , describe the relationship between  and two randomly chosen planes in P. [1] (vi) If 1a c d   and 1 2b , find the perpendicular distance from the point  2, 1,0 to  . [3]

5 MJC/2015 JC2 Preliminary Examinations/9740/02 Section B: Statistics [60 marks] 5 A group of nine people consists of five men and four women. (a) The nine people are to form a queue. Find the number of different arrangements if there are no two people of the same gender standing next to each other. [2] (b) The group of nine people finds a round table with eight seats. Assuming only eight people are to be seated, find the number of different arrangements if (i) there are no restrictions, [2] (ii) two particular woman are seated at the round table but are not next to each other. [2] 6 It is known that p% of college students in Singapore score distinction for their Mathematics final year examination. (a) Given the probability that at least 2 students out of 10 randomly chosen stude nts score distinction for their Mathematics final year examination is 0.95, form an equation in terms of p and hence find the value p. [3] (b) 50 samples of 10 students each are randomly chosen. Given that 40p , find the probability that the mean number of students per sample who score distinction for their Mathematics final year examination is less than 3.5. [3] [Turn Over

6 MJC/2015 JC2 Preliminary Examinations/9740/02 7 The number of people arriving at a particular bus stop in a period of 5 minutes is a random variable with the distribution  Po 2 . (i) Given that the probability that more than 9 people arrive at that particular bus stop in a period of t minutes is less than 0.759 , show that the greatest possible integer value of t is 30. [2] (ii) The number of people leaving that particular bus stop in a period of 5 minutes is a random variable with the distribution  Po 2.5 . At 0800 on a particular day, there are 5 people at the bus stop. Using a suitable approximation, estimate the probability that there are more than 15 people at the bus stop at 0900. (You may assume that there are always people at that particular bus stop during this period. You should assume also that people arriving and leaving that particular bus stop are independent of each other.) [5] (iii) Explain why a Poisson model would probably not be valid if applied to a time period of several hours. [1] 8 The time required by a student to complete the homework given in a week is a normally distributed continuous random variable X. Over a long period, it is known that the mean time required is 16 hours. The students are encouraged to switch off their mobile phones while doing their homework, and afterwards the time required, x hours, is measured for a random sample of 14 students. The results are summarised as follows.     2 16 23.8 16 149.127xx    (i) Find unbiased estimates of the population mean and variance. Test, at the 5% significance level, whether there has been a change in the mean time required by a student to complete the homework given in a week. [6] (ii) It is given that the standard deviation of X is 3 hours and the mean time taken for another 14 randomly chosen students to complete the homework is x . Find the set of values of x for which the result of the tes