SRJC P1

1 [TURN OVER ] SERANGOON JUNIOR COLLEGE 2012 JC2 PRELIMINARY EXAMINATION MATHEMATICS Higher 2 9740/1 Wednesday 15 Aug 2012 Additional materials: Writing paper List of Formulae (MF15) 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 dark 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 al lowed 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. Total marks for this paper is 100 marks. This question paper consists of 6 printed pages (inclusive of this page) and no blank page.

2 Answer all questions [100 marks]. 1 The graph of 2 1y x  for 0x , is shown in the diagram below. Region R is bounded by the x-axis, the y-axis, the line 1x and the curve 2 1y x  . The area of region R may be approximated by the total area, A, of n rectangles, each of width 1 n , as shown in the diagram. (i) Show that 1 0 2n r A nr     . [2] (ii) By considering the exact area of region R, show that 1 0 1 ln 2 n r nr    . [2] Solution (i) Total area of all the n rectangles, 112 1 2 2 2 2 1 0 1 1 1 n n n n A n            1 2 2 2 2 0 1 1 2 1 n n n n n n n n             2 2 2 2 0 1 2 1n n n n n         1 0 2 () n r shownnr     (ii) 1 1 0 0 2 d 2ln 1 2ln 21 xxx     Area of the n rectangles > Area of region R 3 n 2 n 1 1n n  2n n  3n n  x y 2 1y x  0 1 n

3 [TURN OVER ]  1 0 2 2ln 2 n r nr     1 0 1 ln 2 n r nr    (Shown) 2 A sequence of real numbers x1, x2, x3, … satisfies the recurrence relation