NYJC TJC VJC FM 2024 Prelim P1

1 TEMASEK JUNIOR COLLEGE JC2 Preliminary Examination Higher 2 FURTHER MATHEMATICS 9649/01 Paper 1 09 September 2024 3 hours Additional Materials: Answer Booklet List of Formulae (MF26) READ THESE INSTRUCTIONS FIRST An answer booklet will be provided with this question paper. You should follow the instructions on the front cover of the answer booklet. If you need additional answer paper ask the invigilator for a continuation booklet. 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. This document consists of 7 printed pages and 1 blank page. [Turn over

2 1 Polar equations of the form cos ,r a a n =+ , , 2a n n are commonly called petal curves because of the shapes of their graphs. The number and sizes of the petals are dependent on the values of a and n. (a) State the relationship between the number of petals and n. [1] (b) Show that the area bounded by the curve is independent of n. [3] 2 Show that ( ) ( )1 i tan sec cos isin k k kk   + = + . [1] Hence or otherwise, show that 1 0 cos sec cot sin sec , n kn k kn     − = = provided  is not an integer multiple of .2  [5] 3 Let 2 0 cos dn nI x x x  = . (a) Prove that for 2,n ( ) 212 n nnI n n I − = − − . [3] (b) R is the region enclosed by the graph of 2 1sin 2y x x= , the line 2x = and the x-axis. By using the result in (a), find the exact volume of the solid generated when R is rotated 2 radians about the x-axis. [5] 4 Let nP be the vector space of real polynomials of degree n or less. The transformation 43:T P P→ is defined by, ( )( ) ( ) ( ) ( ) 23f f 0 f(1) f 1 f 2 .T x x x x = + + − + Show that T is a linear transformation. [2] Find a basis for the null space of T. [7]

3 5 The half-line L with equation 8y mx=+ , 0x , 0m , is tangential to the locus 4z = at point Q represented by complex number 4z . (a) Sketch the locus 4z = and L on a single Argand diagram. [