HCI 2023 JC1 MYE 9649

HWA CHONG INSTITUTION 2023 JC1 BLOCK TEST FURTHER MATHEMATICS Higher 2 Paper 1 Wednesday 28 June 2023 3 hours Additional materials: 12-page Answer Booklet List of Formula (MF26) 4-page Additional Answ er Booklet (upon request) READ THESE INSTRUCTIONS FIRST Write your name and class on the 12-page Answer Booklet and any other additional 4- page Answer Booklets 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 paper clips, highlighters, glue or correction fluid. Do not write anything on the List of Formula (MF26). 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 differ ent level of accuracy is specified in the question. You are expected to use a graphing calculator. Unsupported answers from a graphing ca lculator are allowed unless a question specifically states otherwise. Where unsupported answers from a graphing calc ulator are not allo wed 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, slot any additional 4-page Answ er Booklets used in your 12-page Answer Booklet and indicate on the 12-page Answer Book let the number of additional 4-page Answer Booklets used (if any) This question paper consists of 9 printed pages. 9649/01

2 1. Prove by mathematical induction that 22 n nnab a b  for n  , where a and b are positive. [5] 2. A point P lies on the curve C with polar equation 2c o s, 22ra     , where a is a positive constant. The point N is the foot of perpendicular from the pole to the tangent of C at P . (a) Sketch ,CP and N on the same diagram. [2] (b) By considering the polar coordinates of N , show that, as P varies, the locus of N is given by the polar equation (1 cos ), .ra      [4] 3. The sequence  nu is given by the recurrence relation 21 56nn nuu u , n ,      together with terms 12 and ua ub . (a) Find the expression of nu in terms of a and b. [4] (b) Find algebraically th e possible limits of 1 n n u u     . [3]

3 4. (a) A straight line l passes through the focus F of a conic with polar equation 1c o s kr e   and meets the conic at 2 points A and B . Show that 11 2 AF BF k . [4] (b) The diagram above shows an ellipse with foci 1F and 2F . The points ,PQ and R lies on the ellipse with PQ and PR passing through 1F and 2F respectively. By using the result in part (a), or otherwise, show that 12 12 PF PF FQ F R