DHS 9758 2023 Prelim P2

© DHS 2023 This document consists of 19 printed pages and 5 blank pages. [Turn over Name: Centre/Index Number: Class: DUNMAN HIGH SCHOOL Preliminary Examination 2023 Year 6 MATHEMATICS 9758/02 Paper 2 19 September 2023 3 hours Candidates answer on the Question Paper. Additional Materials: List of Formulae (MF26) READ THESE INSTRUCTIONS FIRST Write your Centre number, index number, name and class 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. 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 spe cifically 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. For teachers’ use: Qn Q1 Q2 Q3 Q4 Q5 Q6 Q7 Q8 Q9 Q10 Q11 Total Score Max Score 8 9 10 13 5 8 6 7 10 12 12 100

2 DHS 2023 Year 6 H2 Mathematics Preliminary Examination Section A: Pure Mathematics [40 marks] 1 (a) Given that f is a continuous and increasing function, explain with the aid of a sketch, why 1 0 1 f n r r nn − =  ∑ is less than 1 0 f( ) d .xx∫ [3] (b) Find a similar expression that is greater than 1 0 f( ) d .xx∫ [1] (c) It is given that 2f( ) 1xx= + and 10.n= (i) Using the expression in part (a) and your expression in part (b), find the lower and upper bounds of 1 0 f( ) d .xx∫ [2] (ii) Comparing your answers from part (c)(i) with the calculated value of 1 0 f( ) d ,xx∫ comment which limit is a better estimate. [2] 2 (a) A sequence of real numbers 123, , . . .uu u is such that 1 3 0.5 , for 1.nnu un+ = −≥ It is given that 1uc= where c is a constant. (i) Describe how the sequence behaves when (A) 5,c= [1] (B) 2.c= [1] (ii) Find the value of c for which 322 5.uu=− [3] (b) Another sequence is defined by 12 , 2,v pv= = where p is a constant, and 21 2 1, for 1.n nnv vv n++ =+− ≥ (i) If 3 21 12 ,v uu+= − find a relationship between c and p. [2] (ii) Find the value of p for which 5 77.v = [2]

3 DHS 2023 Year 6 H2 Mathematics Preliminary Examination [Turn over 3 Fig. 1 shows the net of a triangular prism cut from a squar