SRJC_H2_MATH_P2_Solution

1 [TURN OVER] SERANGOON JUNIOR COLLEGE 2018 JC2 PRELIMINARY EXAMINATION MATHEMATICS Higher 2 9758/2 17 Sept 2018 3 hours Additional materials: Writing paper List of Formulae (MF 26) TIME : 3 hours READ THESE INSTRUCTIONS FIRST Write your name and class on the cover page and on all the work you hand in. Write in 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 calcul ator are not allowed in a question, you are required to present the mathematical steps us ing mathematical notatio ns 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. Total marks for this paper is 100 marks This question paper consists of 7 printed pages (inclusive of this page) and 1 blank page.

2 Section A: Pure Mathematics [40 marks]. 1 R is the region enclosed by the line 5y =− and the curves 2 25yx x=− + − and () 22 5 141 6 yx ++= as shown in the diagram below. Find the volume generated by region R when it is rotated 2π radians about the x-axis. Leave your answer correct to 2 decimal places. [3] Solution Required volume ( ) () () ( ) 2 21.09 0 22 21 . 0 9 51 6 4 d 25 d 2 5 2xx x x xππ π − −− =− − − + − + − − OR ( ) () () 2 21.09 0 0 222 21 . 0 9 2 5 1 64 d 2 5d 5dxx xx x xππ π − −− − =− − − + − + − − −  = 146.91 Using Shell method Vol () 25 8.36 521 4 4 d 4 yyy yπ − −  +=− − − − + −   = 146.91 2 A curve has parametric equations tanx θ= , 2secy θ= for 02 θπ≤< . The equation of the tangent to the curve at the point P with parameter p is given by ()2si n 2c osyp x p=+ . (i) The tangent at P meets the x and y axes at the points A and B respectively. Find the Cartesian equation of the locus of the mid-point of AB as p varies. [3] (ii) The tangent at P meets the line 2yx= at the point S and the line 2yx=− at the point T. Show that the area of the triangle OST is independent of p, where O is the origin. [4] Solution (i) Coordinates of A: () cot ,0p− Coordinate