VJC 2023 JC1 Promos 9649

VICTORIA JUNIOR COLLEGE JC 1 PROMOTIONAL EXAMINATION 2023 H2 Further Mathematics 9649/01 Paper 1 3 hours Additional Materials: Answer Paper Graph Paper List of Formulae (MF26) READ THESE INSTRUCTIONS FIRST Write your name and CT group on the work you hand in. Write in dark blue or black pen on both sides of the paper. You may use an HB pencil for any diagrams or graphs. Do not use staples, paper clips, glue or correction fluid. Answer all the questions. Give non-exact numerical answers correct to 3 significant figur es, 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 an approved graphing calculator. Unsupported answers from a graphing calculator are allowed unless a question specifically states otherwise. Where unsupported answers from a graphing calculator are not allowed in a question, you are required to present the mathematical steps us ing 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. [Turn over

2 1 Do not use a calculator in answering this question. Let w be a non-real cube root of unity. (a) Prove that 2 1.ww  [ 2 ] (b) Hence, or otherwise, show that 22(1 2 3 )(1 2 3 ) 3.ww w w    [ 3 ] 2 The set of points  ,x y which are equidistant from the line x d and the point (0,2 ),d where d is a positive constant, is denoted by P. Using this geometrical definition, show that 2(2 ) (2 ) .yd d dx  [2] Determine the volume of the solid formed when the finite region bounded by P and the y-axis is rotated completely about the x-axis. Leave your answer in exact form in terms of  and d. [5] 3 (a) Show that 22 2 21 23 kk kk  for 0.k [ 2 ] (b) The sequence 123,,,uu u  is such that 1 2 3u  and 1 2 21nn nuu n   . Prove by induction that 2 21 n n nu n    for all positive integers n. [5] 4 Let f( ) , kx xa where k is a positive integer and a > 0. The number 1/ka is a root of the equation f( ) 0.x  (a) Show that for any initial estimate 11 0xx  of 1/ ,ka the Newton-Raphson method gives the iteration formula 1 1 1 (1 ) .nn k n axk x k x      [2] The iteration formula established in part (a) is used to approximate 3 25. (b) Without using a calculator, explain why 1 3x  is reasonable. [1] (c) Determine the value of 3 25, giving your answer correct to 3 decimal places.