CJC P1

9758/01/PRELIMS/2020 [Turn over CATHOLIC JUNIOR COLLEGE General Certificate of Education Advanced Level Higher 2 JC2 Preliminary Examination CANDIDATE NAME CLASS INDEX NUMBER MATHEMATICS 9758/01 Paper 1 1 September 2020 3 hours Candidates answer on the Question Paper. Additional Materials: List of Formulae (MF26) READ THESE INSTRUCTIONS FIRST Write your name and class on all 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. Write your answers in the spaces provided in the Question Paper. 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 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 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. Question 1 2 3 4 5 6 7 8 9 10 11 Total Marks Total 5 6 7 8 8 8 9 10 12 13 14 100 This document consists of 27 printed pages and 1 blank page.

2 9758/01/PRELIMS/2020 1. A function f is defined by ( ) 23f 35 xx x += + . (i) Show that ( )f x can be written in the form 35 ba x+ + , where a and b are constants to be determined. [1] (ii) Hence describe a sequence of transformations which transforms the graph of 1y x= on to the graph of ( )fyx= . [4] 2. The complex number z has modulus 2 and argument 8 π . It is also given that 1iw= + . (i) Given that n is an integer, find * nz w in terms of n, giving your answer in the form ier θ . [3] (ii) Hence, find the smallest two positive integers n such that * nz w is real and negative. [3] 3. Do not use a calculator in answering this question. The equation 3225 0z z pz q+ + += has a root 1iz= −+ , where and pq are real constants. Find and pq and all the other roots of the equation. [7] 4. The function f is given by ( ) 5 , , 22f 1, , 2 xx x x xxx  −∈<=  ∈≥   (i) Sketch the graph of f( )yx= , stating the equation of any asymptotes and the coordinates of any points where the curve crosses the x- and y-axes. [2] (ii) Explain why f has an inverse. Hence find 1f () x− , giving your answer in a similar form. [4] Let ( ) 3g xx= , , 2xx∈≥ . (iii) Determine whether 1gf− exists. [2]