SAJC 9758 2023 Prelim P2

Question 1 2 3 4 5 6 7 8 9 10 11 TOTAL Marks 7 9 9 6 9 8 8 10 10 12 12 100 This document consists of 26 printed pages and 2 blank pages including this page. ST ANDREW’S JUNIOR COLLEGE PRELIMINARY EXAMINATION MATHEMATICS HIGHER 2 9758/02 Wednesday 13 September 2023 3 hr Candidates answer on the Question Paper. Additional Materials: List of Formulae (MF26) NAME:_________________________________________( _____ ) C.G.: __________ TUTOR’S NAME: _________________________________________ SCIENTIFIC / GRAPHIC CALCULATOR MODEL: _______________________ NUMBER OF ADDITIONAL PIECES OF WRITING PAPER :________________ READ THESE INSTRUCTIONS FIRST Write your name, civics group, index number and calculator models on the cover page. Write in dark blue or black pen. 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. Total marks : 100 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. The use of an approved graphing calculator is expected, where appropriate. Unsupported answers from a graphing calc ulator 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.

2 [Turn Over Section A: Pure Mathematics [40 marks] 1 Do not use a calculator in answering this question. The complex number z satisfies the equation 23 (5 i) 0,zz α−+ += where α is a real number. It is given that one root is of the form ikk+ , where k is real and non-zero. Find α and k, and the other root of the equation. [7] 2 In triangle PQR, 2 3PRQ π∠= , 6RPQ π θ∠= + , and PR a= units, where a is a positive real constant. (i) Show that 3 cos 3 sin aPQ θθ− = . [4] (ii) Given that θ is sufficiently small, show that ( ) 231 cPQ a b θθ≈+ + , where b and c are real constants. [3] (iii) Given that 0θ > , find the range of values of θ such that the percentage error of the approximation in (ii) is less than 5%. [2] 3 (a) Use the substitution πtan , where 0 , 2xt t= << to find 22 1 d 1 x xx +∫ . [4] (b) (i) Write down ( ) 3d cosd xx . [1] (ii) Hence, find 53sin .x x dx∫