IJC JC2 H2 Maths 2012 Questions Paper 1

INNOVA JUNIOR COLLEGE JC 2 PRELIMINARY EXAMINATION 2 in preparation for General Certificate of Education Advanced Level Higher 2 CANDIDATE NAME CLASS INDEX NUMBER MATHEMATICS Paper 1 Additional Materials: Answer Paper Cover Page List of Formulae (MF15) 9740 /0 1 13 Sep 2012 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 th e 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 signi ficant figures, or 1 decimal place in the case of angles in degrees, unless a different l evel of accuracy is specified in the question. You are expected to use a graphic calculator. Unsupported answers from a graphic calculator are a llowed 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 usin g 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

IJC/2012/JC2 9740/01/S12 2 1 A renovation company wishes to obtain sand, stone a nd brick for construction work. The company sends its requirements for sand, stone and brick, measured in units, to three different suppliers, AngGui, BaBao and CaiTao. Their quotations are as follows: Price per unit ($) AngGui BaBao CaiTao Sand 15.00 11.00 12.00 Stone 10.50 17.30 13.00 Brick 8.10 7.00 10.00 Total price 205.2 229.4 208 (i) Find the number of units of sand, stone and brick required by the company. [3] (ii) Another supplier, DaoBi, charged 10% lower per uni t of sand, stone and brick than the company that charges the lowest for each of the materials. Find the total amount that the renovation company must pay i f all the materials were purchased from DaoBi, leaving your answers to the nearest cent. [2] 2 Find the general solution of the differential equa tion 2 2 dcosec d yx x x = . [3] Find the equation of the solution curve whose tange nt at the origin is parallel to the line 3 5 y x = − . [3] 3 (i) Prove by induction that 3 2 2 1 1 ( 1) . 4 n r r n n = = + ∑ [4] (ii) The th r term of a series is given by ( ) 3 ln 2 ra where a is a positive constant. Show that the sum of the first n terms of the series is given by ( ) 2( 1) ln 16 4 n n n a + . [3] 4 The equation of a curve is given by 2 2 2 4 66 xy y x − + = . (i) Find the exact coordinates of the points on the cu rve where the tangent is parallel to the y-axis. [4] (ii) Show that every line parallel to the x-axis cuts the curve at two distinct points. [3]

IJC/2012/JC2 9740/01/S12 3 5 Sketch, on the same diagram, the graphs of ln(2 9) y x = + and 2 10 y x = − , including the coordinates of the points where the g raphs cross the x-axis and the equations of any asymptotes. [3] Hence solve the inequality 2 ln(2 9) 10 x x + ≥ − and deduce the solution to the inequality 2 ln(2 9) 10 x x + ≥ − . [5] 6 The above diagram shows a circle with radius r units and centre O. The points A and B on the circle are such that OA = a /combarrowextender/combarrowextender/combarrowextender /arrowrightnosp , OB = b /combarrowextender/combarrowextender/combarrowextender /arrowrightnosp and 120 AOB ∠ = ∘. The point N divides AB in the ratio :1 λ λ − and ON is perpendicular to OB . (i) Show that 1 (2 ) 3ON = a b /combarrowextender/combarrowextender/combarrowextender /arrowrightnosp + and hence find the area of triangle OAN in the form of k ×a b , where k is a constant to be determined. [6] (ii) It is given that the point C lies on the circle such that O, N and C are collinear. By considering the length of ON , find OC /combarrowextender/combarrowextender/combarrowextender /arrowrightnosp in terms of a and b . [2] [Turn over A B N O a b C

IJC/2012/JC2 9740/01/S12 4 7 (a) If ( )2 cos isin z θ θ = + , where 0 2 πθ< < , label the points P, Q and S representing the complex numbers z, 4 z− and 4 z respectively on an Argand diagram. [2] It is given that PR is a diagonal of the rectangle PQRS . (i) State, in terms of z, the complex number represented by the point R. [1] (ii) Find, in terms of θ , the area of the rectangle PQRS , leaving your answer as a single trigonometric function. [2] (b) Show that ii 23 3e 6e cos 2 θ θ θ+ = , where 0 . θ π < < [2] Hence find the exact values of a and θ if i3 3e (1 i) 2 aθ+ = + . [3] 8 It is given that ( )e ln 1 2 xy x −= − . (i) Show that ( ) ( )d1 2 2e 1 2 d xyx y x x −− = − − − and hence show that ( ) ( ) 2 2 d d 1 2 2 4e 3 2 d d xy y x x y x x −− = + + − . [3] (ii) Hence find the Maclaurin series for y up to and including the term in 3x . [4] (iii) Verify that the same result is obtained if the sta ndard series expansions for ex and ln(1 ) x+ are used. [2] 9 A curve C has parametric equations 2cos 1, x t = + sin y t = , 0 2 . t π≤ < (i) Sketch C, indicating the coordinates of the intersections with the x-axis. [2] (ii) Find the numerical value of the volume of revolutio n obtained when C is rotated 180 ∘ about the x-axis. [3] (iii) Find a cartesian equation of C. [1] (iv) Describe a sequence of transformations that will tr ansform the curve C into a circle with radius 2 centred at the origin. [2]

IJC/2012/JC2 9740/01/S12 5 10 (a) The functions h and g are defined as follows: g : x x a −/arrowbarright, x a ≥ , 2h : x x a +/arrowbarright, 0x < . (i) Show that the composite function gh exists. [2 ] (ii) Define gh in a similar form. [3] (b) The function f is defined by 1f : , . 1 2sin 2 x x x π π< < +/arrowbarright (i) By using differentiation, show that ( )f x increases as x increases. [2] (ii) Find ( ) 1f x− stating the domain of 1f − . [3] 11 (a) The positive multiples of 5 are grouped into sets M1 = {5}, M2 = {10, 15}, M3 = {20, 25, 30}, …, where the set Mn has n elements. (i) Find the total number of elements in the first n sets and show that the last element of Mn is 5 ( 1) 2 n n + . [3] (ii) Hence find the sum of all the elements in Mn+1 in terms of n. [2] (b) David, a JC student is training for the 2.4 km run of his NAPFA fitness test on the jogging track, where he has to run 6 rounds. Th e time taken for every round is 23 20 of the time taken of his previous round as he gets more tired. (i) If he takes 2 min 30 seconds to run the fourth rou nd, find the time taken for him to run 2.4km, leaving your answer to the nearest second. [3] (ii) Tommy starts the 2.4 km run one minute after Davi d started to run. For Tommy, the time taken for every round is 11 10 of the time taken of his previous round. If Tommy runs his first round in 1 min