MJC JC2 H2 Maths 2012 Paper 1

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 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 notatio ns 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 9740/01 Paper 1 11 September 2012 3 Hours Additional Materials: Writing paper List of Formulae (MF 15) ___________________________________________________________________ READ THESE INSTRUCTIONS FIRST ___________________________________________________________________ This document consists of 7 printed pages and 1 blank page. [Turn Over]

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3 MJC/2012 JC2 Preliminary Examination/9740/01 1 By using an algebraic method, solve the inequality 2 464 1 xx x   . [4] 2 The graph of a cubic polynomial passes through the origin and only has one stationary point at  1 ,3 . Find the cubic polynomial. [4] 3 (a) Given 2 ln 2x xy y    , find d d y x . [2] (b) (i) Find   2d 2d x x . [1] (ii) Hence find 22 ln 2 dxx x . [3] 4 The complex number z satisfies 2 4i 4z   and  0 arg 2 4z    . (i) On an Argand diagram, sketch the region in which the point repre senting z can lie. [3] (ii) Find the smallest value of  arg 2 4i .z [1] (iii) It is further given that the value of 2z is minimum at point P. Find the complex number w representing P in the form iab , giving the exact values of a and b. [2] 5 Relative to the origin O, t he point s A and B have position vectors 2i j k and 32  i j k respectively. (i) The point M lies on AB extended such that AB:AM = 4:5. Find the position vector of M. [2] (ii) Give a geometrical interpretation of OA OB OB  . [1] (iii) Find the shortest distance from the point (1,3,8)C to the plane containing O , A and B . [3] [Turn Over]

4 MJC/2012 JC2 Preliminary Examination/9740/01 6 Given that  f ( ) ln e 2 ,xx find (3)f (0), f (0), f (0) and f (0).   [2] (i) Hence write down the first four non -zero terms in the Maclaurin series for f ( ).x [1] (ii) Using the seri es found in part (i), find the Maclaurin series for 2 e2x , up to and including the term in x2. [1] (iii) By considering the standard series for (1 ) , nx verify that the series obtained in part (ii) is correct. [2] 7 A sequence 1 2 3, , , ...a a a is such that 1 1 2a  and   1 1 , for 121 nna a n nn      (i) By considering the values of 23,aa and 4a , write down a conjecture for na in the form of 11 cn , where c is a constant to be determined. [2] (ii) Use the method of mathematical induction to prove the conjecture. [4] (iii) Hence find 2 1 1 1 1 ...2 6 12 NN     in terms of N. [4]

5 MJC/2012 JC2 Preliminary Examination/9740/01 8 In the d iagram below, triangle PQR is inscribed inside a circle of centre O and constant radius r. PR is a line that passes through the centre of the circle O. As point Q moves along the circle, the area of triangle PQR changes. (i) Using differentiation, find the length of the sides PQ and QR such that the maximum area of triangle PQR is obtained. Leave your answers in terms of r. [7] (ii) Given that QR increases at a rate of 0.2 units per second, find the rate of change of QPR when 3QPR  and 2.r [3] 9 Runners A and B are undergoing two types of a training programme in preparation for a marathon. Runner A: Runs 2.4 km on day 1, and on each successive day, the distance covered is increased by 1 10 of the previous day. Runner B: Runs 4 km on day 1, and on each successive day, the distance covered is increased by 800 m. (i) Find the total distance, to the nearest metre, runner A would have covered in the first 15 days. [2] (ii) On day n, runner B meets her target of covering a distance of 42 km for the first time. Find n. [2] (iii) In order for runner A to cover the same total distance as runner B by the end of day 3, the distance covered by runner A has to be inc reased by %x of the previous day on each successive day. Find x. [6] P Q R O [Turn Over]

6 MJC/2012 JC2 Preliminary Examination/9740/01 10 (a) By means of the substitution y xz , show that the differential equation     2de 1 e 1 d xx y y x x x y     can be reduced to the form d e 1 d e 1 x x zz x   . Hence find the general solution for 2y in terms of x. [5] (b) There was an island where initially there was no one living on it. The total capacity of the islan d is 9 000. The population increases at a rate which is inversely proportional to the remaining capacity of the island. At the same time, the rate at which the population decreases is 1 20 of the population size. When the population re aches 4 000, it remains at this value. The population size (in thousands) is x at time t months, show that      45d .d 20 9 xxx tx   Find t in terms of x. Hence find the time when the population reaches 2 000. [7]

7 MJC/2012 JC2 Preliminary Examination/9740/01 11 (i) Find the fifth roots of 32, giving the roots in the form of ier  , where 0r and      . Show these roots on a sketch of an Argand diagram. [5] (ii) The set of points in the Argand diagram representing the roots is denoted by S. State the number of points in S which are also in the locus of points representing the complex number v such that 2vv . [1] (iii) Two of the roots found in (i) are denoted by 1z and 2z , where    120 arg argzz    . The complex number z is represented by the point of intersection of the loci of 12z z z z   and 2z  . Find z, in the form of ixy , leaving the values of x and y in 2 decimal places. [2] (iv) Using (i), show that all the roots of the equation in         432 2 2 2 4 2 8 2 16 0w w w w         can be expressed in the form of i 104cos e10 p p    , and state the values of p that give all the roots of the equation. [4] [Turn Over]

8 MJC/2012 JC2 Preliminary Examination/9740/01 12 (a) A curve C has equation 2 21xy xk   where k . (i) State a sequence of transformations which transform the graph of C to the graph of   2 412 21 xy xk   . [3] (ii) Given 4k  , find the exact value of 0.5 21 21 dx xxk     . [6] (b) The diagram below shows the graph of  gyx . Sketch the following graphs on separate diagrams, (i)  2gyx , [3] (ii)   2 gyx  , [2] showing clearly in each case the inter