ACJC 2023 JC1 H1 Promo QP

ANGLO-CHINESE JUNIOR COLLEGE JC1 PROMOTIONAL EXAMINATION Higher 1 CANDIDATE NAME TUTORIAL/ INDEX FORM CLASS NUMBER MATHEMATICS 8865/01 Paper 1 29 September 2023 1 hr 30 min Candidates answer on the Question Paper. Additional Materials: List of Formulae (MF26) READ THESE INSTRUCTIONS FIRST Write your index number, class and name on all 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 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 14 printed pages. [Turn over Question Marks 1 /4 2 /4 3 /7 4 /7 5 /7 6 /8 7 /13 /50

2 ANGLO-CHINESE JUNIOR COLLEGE 2023 H1 MATHEMATICS 8865/01 [Turn over 1 Use the substitution lnux= to solve the inequality ( ) 22(ln ) ln e 7xx − , leaving your answer in terms of e. [4] 2 Show that the expression 25 5 2xx−+ is positive for all real values of x. Hence or otherwise, find the set of values of k for which the equation 24 (12 8) ( 2) 0+ − − + =x k x k k has two distinct roots. [4] 3 (a) Differentiate 2eln 25 x x −  + , leaving your answer in the form 4 BA Cx+ + . [3] (b) Given that 2 3 6 d 4 23 k x x = + where 2k  . Find the value of k. [4] 4 The curve C has equation ln( 1) 2yx= − + . (i) Sketch the graph of C, stating the exact coordinates of any point(s) of intersection with the axes and the equation of the asymptote. [3] (ii) By using an appropriate line and the graph of C, solve the equation ln( 1) 7 2xx− + = . [3] (iii) Find the numerical value of the area of the region bounded by C, the x-axis, and the lines 1.5x= and 4x= . [1]

3 ANGLO-CHINESE JUNIOR COLLEGE 2022 H1 MATHEMATICS 8865/01 [Turn over 5 The curve C has the equation 22y k x=− and the line L has equation 2y kx k=− + , where k is a positive constant. Curve C and line L intersect at the points A and B as shown. (i) Find, in terms of k, the coordinates of A and B. [3] (ii) Find the area of the shaded region