RI H2 Maths P1 Qn

MATHEMATICS PAPER 1 9740/1 Higher 2 16 September 2014 Total Marks: 100 3 hours Additional materials: Answer Paper Graph Paper List of Formulae (MF15)   READ THESE INSTRUCTIONS FIRST Write your name and CT 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 signifi cant 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 ar e 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 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. RAFFLES INSTITUTION Mathematics Department  RI2014 [Turn over RAFFLES INSTITUTION 2014 YEAR 6 PRELIMINARY EXAMINATION

2 H2 MA 9740/2014 RI Year 6 Preliminary Examination/01  1 A function f is given by 2 f( )   x ax bx x for ,0 xx R where a and b are positive real constants. Find f' ( )x and hence sketch the graph of f' ( )yx , stating the equations of any asymptotes and the coordinates of the points where the curve crosses the axes. [4] 2 A sequence 012, , , ...uuu is such that 0 20u  and 1 1 , for all 0.100 n nn uur u n   (i) Given that 2 3r , find the least value of n such that 1nu  . [2] (ii) Find the value of r such that 20nu  for all values of n . [2] (iii) It is given that 4 3r  and nul  as n . Showing your working, find the exact value of .l [2] 3 Let 22 1f( ) . (1 ) xx x   (i) Find the binomial expansion of f( )x in increasing powers of ,x up to and including the term in 4.x [3] (ii) Find the coefficient of 21rx  in the expansion for f( ) .x [3] 4 Prove by the method of mathematical induction that 1 sin(2 )sin sin( )sin( 1) n r rx x nx n x   for all positive integers n. [4] Hence find the exact value of 2 4 sin 3 sin 4 d. sin xx x x   [3]

3 H2 MA 9740/2014 RI Year 6 Preliminary Examination/01  5 The curve with equation 1y x undergoes a translation of 1 unit in the positive x-direction, followed by a stretch with factor 1 2 parallel to the x-axis and then a translation of 3 units in the positive y-direction. Given that the equation of the new curve is f( )yx . Sketch the curve, stating its asymptotes and the coordinates of the points of intersection with the axes. [4] By sketching the graphs of f( )yx and  fyx on separate diagrams, solve the inequality        f( ) f .x x [4] 6 Referred to the origin O, the points A and B have position vectors a and b respectively. It is given that 3,a 5b and 3 10.ab (i) Give the geometrical interpretation of .ba b [1] (ii) Show that 1.ab [2] (iii) Hence find the shortest distance from A to the line OB, and the area of the triangle OAB. [2] (iv) Given that a, 2a+3b and 2, ab where  is a constant, are position vectors of collinear points, find . [4]  RI2014 [Turn over

4 H2 MA 9740/2014 RI Year 6 Preliminary Examination/01  7 Curves 1C and 2C are given by the equations 22 1 22 2 :2 5 :1 0 0 Cx y Ca x b y   where a and b are positive real constants such that ab . (a) Find the condition(s) on the values of a and b such that 1C and 2C intersect at exactly 4 points. [2] (b) The diagram shows the curves 1C and 2C when 1a  and 9b . Find the area of the shaded region. [3] (c) The diagram shows the curves 1C and 2C when 1a  and 4b . Find the exact volume when the shaded region is rotated through  radians about the axisy . [4] x y 0 x y 0

5 H2 MA 9740/2014 RI Year 6 Preliminary Examination/01  8 (a) Without using a calculator, solve the equation   4 83 i 0z    , giving the roots in the form ier  , where 0r and .   [4] Show the roots on an Argand diagram. [2] (b) One root of the equation 2 *0za zb  , where a and b are real, is w. Show that *w is also a root of this equation. [2] Solve the equation 2 6*9 0zz  , giving your answers in the form ix y . [4] 9 The diagram shows a cuboid with horizontal rectangular base OABC, where 5OA units, 3OC  units and 2OP  units. The edges OP, AQ, BR and CS are vertical, and PQRS is the top of the cuboid. Using O as the origin, unit vectors i, j and k are taken along OA, OC and OP respectively.  The plane 1 contains the points A, C and R, and the plane 2 contains the points Q, S and B. (i) Find, in scalar product form, an equation of 1. [3]  (ii) Find the acute angle between 1 and the horizontal base. [2] (iii)   Hence state the acute angle between 1 and 2. [1] The point X lies on PS such that ,SX SP    where 0  is a constant, and Y is the point with coordinates (5, 2,1).  (iv) Find the equations of the lines XY and OR in vector form. [3] (v) Given that the lines XY and OR intersect at a point W, find  and the ratio :.OW OR [4]  RI2014 [Turn over   O A BC P Q RS i j k

6 H2 MA 9740/2014 RI Year 6 Preliminary Examination/01  10 A curve C is defined by the parametric equations tan , sec  xy for 0 2  . (i) Find the cartesian equation of .C Sketch ,C giving the equation(s) of any asymptote(s). [3] The tangent and normal at tan , sec P meets the x-axis at Q and R respectively. (ii) Show that the area, A of the circle passing through ,PQ and R can be expressed as 2 1tan . 2t a n    A [7] (iii) Show that 2 1 202t t   for all ,0 .tt  [2] Deduce the minimum value of A . [1] 11 The variables x and y are related by the differential equation 2 2 d .d y yx  ----------------- (1) It is also given that d 1d yy x when 0.x (i) Write down the value of 2 2 d d y x when x = 0. [1] (ii) By further differentiation, obtain the values of 3 3 d d y x and 4 4 d d y x when x = 0. Hence write down the first five terms of the Maclaurin series for .y [2] (iii) Show that the substitution d d y ux  reduces (1) to d 2.d u yy  Find u in terms of y , given that 1u  when 1.y [4] (iv) Using your answer from (iii), find y in terms of ,x giving your answer in the form sin( ),yP x Q where P and Q are constants to be determined. [4] (v) Use the standard series in MF15 to find th e Maclaurin’s expansion for your answer in (iv) up to and including the term in 4.x [2]  End of Paper 