CJC_9758_2023_Prelim_P1

9758/01/J2PRELIM/2023 [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 29 Aug 2023 3 hours Candidates answer on the Question Paper. Additional Materials: List of Formulae (MF26) READ THESE INSTRUCTIONS FIRST Write your class, index number and name on the work you hand in. Write in dark blue or black pen. You may use a 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 spec ifically 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 Total Marks Total 4 6 8 9 10 11 13 15 12 12 100 This document consists of 3 printed pages, including this cover page.

2 9758/01/J2PRELIM/2023 1 The complex numbers z and w satisfy the following equations. 21zw+= 2 4 24iwz− = + Find z and w , giving your answers in the form iab+ where a and b are real numbers. [4] 2 (a) The diagram below shows a sketch of the curve 21 xy x = + for 0x . Rectangles, each of width 1 n , where n + , are drawn under the curve for 01 x . Show that the total area of all the rectangles, A , can be written as 2 1 1 2 1 n r r n nr − = + . [3] (b) Find the exact value of lim n A → . [3] 3 The points P , Q and R have position vectors p , q and r respectively. The points P and Q are fixed and R varies. (a) Given that p is non-zero and ( )−r q ×p = 0 , find a linear relationship between p , q and r . Describe geometrically the set of all possible positions of the point R . [4] (b) Given that x y z  = r , 2 5 3  =− p , 4 1 2 − = q and ( ) 0−r q p = , find a relationship between x , y and z . Describe geometrically the set of all possible positions of the point R . [4] O x y   1 1 2 3 nnn 3 2 1n n n n n n − − − 21 xy x = +

3 9758/01/J2PRELIM/2023 4 It is given that f ( )x is a cubic polynomial with real coefficients. The diagram shows the curve with equation f ( ).yx= (a) Wh