EJC 9758 2023 Prelim P1

2023 JC2 H2 Mathematics Preliminary Examination Paper 1 [Turn over EUNOIA JUNIOR COLLEGE JC2 Preliminary Examination 2023 General Certificate of Education Advanced Level Higher 2 CANDIDATE NAME CIVICS GROUP INDEX NO. MATHEMATICS Paper 1 [100 marks] 9758/01 12 September 2023 3 hours Candidates answer on the Question Paper. Additional Materials: List of Formulae (MF26) READ THESE INSTRUCTIONS FIRST Write your name, civics group and index number 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. DO NOT WRITE ON ANY BARCODES. Answer all 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. This document consists of 23 printed pages and 1 blank page. Q1 Q2 Q3 Q4 Q5 Q6 Q7 Q8 Q9 Q10 Q11 Q12 Total

2 2023 JC2 H2 Mathematics Preliminary Examination Paper 1 1 The general solution of a differential equation is given by 22ee exx xyA B C −= ++ , where A, B and C are arbitrary constants. Given that a particular solution satisfies 2y= , d 3d y x =− and 2 2 d 11d y x = when 0x= , find the values of A, B and C. [4] 2 A sequence of numbers 1 23, , , ...uuu has sum nS where 1 n nr r Su = =∑ . It is given that 1 2 218 3 n n nS + −= − . (a) By finding an expression for nu , or otherwise, show that the sequence is a geometric progression. [3] An arithmetic progression has first term 4− . The sum of the first 9 terms of the arithmetic progression is equal to 1 r r u ∞ = ∑ . (b) Find the common difference of the arithmetic progression. [3] 3 With reference to the origin O, the points A , B and C are such that 2 1 1 OA  =−    , 0 1 2 OB  =−   and 1 3 0 OC  =   . Point P lies on AB, between A and B, such that :1:AP PB λλ −= where 0 1λ< < . Point Q lies on BC , between B and C, such that 1::BQ QC λλ= − . (a) Express PQ   in terms of λ. [3] The point X has coordinates 22 5, 1, −− . Points P, Q and X are collinear. (b) Find the value of λ. [3] 4 (a) Using the method of differences, find 2 2 2 1 n r r= −∑ . [4] (b) Hence, find ( ) 31 11 2 2 n r rr − = +∑ . [3] 5 (a) Find sin 3 cos dx xx