TJC H2 MATHS P1 Questions

TEMASEK JUNIOR COLLEGE, SINGAPORE Preliminary Examination 2015 Higher 2 MATHEMATICS 9740/01 Paper 1 31 August 2015 Additional Materials: Answer paper 3 hours List of Formulae (MF15) READ THESE INSTRUCTIONS FIRST Write your Civics group and name on all the work that 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 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. This document consists of 5 printed pages. © TJC 2015 [Turn over

TJC/MA 9740/Preliminary Exam 2015 2 1 The equation of a circle M is given by 22 0x y Ax By C     where A, B and C are real constants. The line y = 2(x + 1) passes through the centre of M and the graph of y = | x | intersects M at the points where x = 2 and x = 8. Find the equation of M. [4] 2 The diagram below shows the graph of y = g( x). The graph has a minimum point at (0, 2) and a maximum point at 13, 2   . The equations of the asymptotes are x = 1, y = 0 and y = 2x. On separate diagrams, sketch the graphs of (i) y = g(x), [2] (ii) 1 g( )y x , [2] showing clearly in each case, the equations of the asymptotes and the coordinates of the turning points and axial intercepts, where applicable. 3 Without using a calculator, solve the inequality 23 112 x x  . Hence solve 23 112 x x  . [5] 4 The sequence of real numbers 1 2 3, , , . . .u u u is defined by 11 2 and , where 1 and .4 nn nu u u a n an      (i) Prove by mathematical induction that for 1.12 ( 2)( 3)n nau nn   [4] (ii) Determine the limit of 1 ( 2) nunn u as .n [2] 5 The complex number z satisfies the equation 3 3 1 3 i1 z z   . Without the use of a graphing calculator, express z3 in the form rei where r  0 and  <   . Hence find the roots of the equation. [6] y = g(x) y x y = 2x x = 1 13, 2   2 (0, 2) 0

TJC/MA 9740/Preliminary Exam 2015 3 6 The figure below shows a rectangle OACB where 2OA OB . Point D is on AC produced such that : :1AD AC  where  is a constant. The lines OD and AB intersect at point E. It is given that OA a , OB b and OEA  . Find OD in terms of a and b, and show that   2 4OD AB  b . [4] In the case when E is the foot of perpendicular from A to OD, deduce the value of . [2] Using this value of  and given that 4 4 2 a   and 2 1 2 b   , find OE . [2] 7 The function f is defined by 14f: 1 xx x   , x ¡ , x  k. (i) With the aid of a graph, find the least value of k such that f has an inverse. [2] (ii) Using the least value of k found in (i), (a) find f 1(x) and state its domain, [3] (b) find the exact solution(s) of the equation f(x) = f1(x). [2] Describe a sequence of two transformations which would tr ansform the graph of y = f(x) onto the graph of 24 2 xy x   . [2] 8 (i) Use the substitution 2sinx  , where π0 2 and 01 x , to show that 1d sin (1 ) where is an arbitrary cons tant.1 x x x x x c cx     [5] (ii) The region R is bounded by the curve 1 4 1 xy x   and the lines y = 4x  1 and 1 4x . Find the volume of revolution formed when R is rotated completely about the x-axis, giving your answer in exact form. [5] [Turn over D O A C B E θ

TJC/MA 9740/Preliminary Exam 2015 4 9 The planes 1p and 2p have equations 22xz and 0 20 1   r respectively. (i) Obtain a vector equation of the line of intersection, l , between 1p and 2p . [2] (ii) A third plane 3p contains l and is perpendicular to 1p . Find a vector equation of 3p , in scalar product form. [3] (iii) The point S lies on 1p and the point T lies on 3p such that the line ST is perpendicular to 2p . If the coordinates of S are  2, 3, 2 , find the coordinates of T. [4] (iv) Find the acute angle between ST and 1p . [2] 10 P and Q are two points lying 20 m apart on a horizontal straight line . Two particles A and B are initially located at P and Q respectively. A begins to move towards Q and B begins to move away from Q. At time t s, the distance travelled by A and B are a m and b m respectively where 0 20a . The fixed point R is located 20 m vertically above point Q such that angle ARB =  . By considering  as the sum of two acute angles, show that  20 20tan 400 20 ab b ab   . [3] (a) On day 1 , A and B move in such a way that the distance of B from Q is always twice the distance of A from P, that is, b = 2 a. Find , using differentiation, the value of a when  is maximum. [4] [You do not need to show that  is maximum.] (b) On day 2, A and B resume their starting positions at P and Q, and move such that  remains a constant. (i) Show that 20 40 ab a  . [2] (ii) If A moves at a constant speed of 0.5 ms1, find the speed of B at t = 30. [3] R P Q A B 20 m 20 m  a b

TJC/MA 9740/Preliminary Exam 2015 5 11 (a) By using small angle approximations, where x is small enough for x3 and higher powers of x to be neglected, show that 2 πsin 2 14 22 sin 4 x px qxx         , where p and q are constants to be determined. [5] (b) A curve has equation y2  xy = 4  sin x. (i) Show that there is no tangent to the curve that is parallel to the y-axis. [4] (ii) Given that y = 2 when x = 0, find the Macla urin’s series for y up to and including the term in x2. [3] 12 A curve C has parametric equations 27 4sinxt , 34 3sinyt where ππ 22 t   . (i) Show that the equation of the tangent to the curve at the point with parameter t is 38 9 sin 63sin 12sin 32 0y x t t t     . This tangent passes through a fixed point ( X, Y). Give a brief argument to explain why there cannot be more than 3 tangents passing through (X, Y). [5] (ii) Sketch the curve C. [2] (iii) Show that the coordinates of the points of intersection between C and the line 8y + 9x  83 = 0 are (3, 7) and 296, 8   . [3] (iv) Find the area of the region bounded by C and the line 8y + 9x  83 = 0. [3]    End of Paper   