2024 JC2 H2 Math prelim - YIJC

Th is document consists of 23 printed pages and 1 blank page. [Turn Over YISHUN INNOVA JUNIOR COLLEGE JC 2 PRELIMINARY EXAMINATION Higher 2 CANDIDATE NAME CG MATHEMATICS Paper 1 C andidates answer on the Question Paper. Additional Materials: List of Formulae (MF26) 9758/01 28 AUGUST 2024 3 h ours READ THESE INSTRUCTIONS FIRST Write your CG, index number and name on the work you hand in. 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. A nswer 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. T he number of marks is given in brackets [ ] at the end of each question or part question. The total number of marks for this paper is 100. For Examiners’ Use Question Marks 1 2 3 4 5 6 7 8 9 10 11 Presentation Total / 100

2 ©YIJC 9758/01/JC2PE/24 1 (a) Without using a calculator, solve the inequality 6 14 x xx ≥−+ . [4] (b) Hence, solve 6 14 x xx ≥+− . [2] 2 The diagram below shows a sketch of the graph of 2sinyx= for 0 π 2x≤≤ . Rectangles each of width π 2n are drawn under the curve for 0 π 2x≤≤ . (a) Show that A , the total area of all the rectangles, is given by 1 2 1 sin 2 n k ka n π− = ∑ , where a is to be determined. [2] (b) Find the exact value of 1 2 1 lim 2sin n n k ka n π− →∞ = ∑ . [2] (c) Hence, find the value of 0 11 sin dy y− ∫ . [2] 3 Do not use a calculator in answering this question. (a) Given that f( )x is a polynomial of degree 4 with real coefficients, explain whether it is possible for f( ) 0x = to have 3 non-real roots and 1 real root. [1] (b) One of the roots of the equation 4 322 15 63 0x x ax x b− + − += , where a and b are real, is 3 2i− . Find the other roots of the equation and the values of a and b. [6] y x

3 ©YIJC 9758/01/JC2PE/24 [Turn over 4 (a) It is given that ( )d ln lnd yx xy x y yx = +− . Using the substitution w xy= , show that the differential equation can be transformed to ( )d fd w wx = , where the function ( )f w is to be found. [3] (b) Hence, given that 31 e2y = when 2x = , solve the differential equation ( )d ln lnd yx xy x y yx = +− , to find y in terms of x. [5] 5 (a) The graph of f( )yx= is shown below. The graph has a turning point at ( )1, 4A − , and axial in