TJC 8865 Prelim 2022 Questions

1 TEMASEK JUNIOR COLLEGE 2022 JC2 Preliminary Examination Higher 1 CANDIDATE NAME CENTRE NUMBER S INDEX NUMBER MATHEMATICS 8865/01 Paper 1 26 August 2022 3 hours Candidates answer on the Question Paper. Additional Materials: List of Formulae (MF26) READ THESE INSTRUCTIONS FIRST Write your Centre number, index number 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. The total number of marks for this paper is 100. This document consists of 21 printed pages and 1 blank page. [Turn over For Examiner’s Use Question Number Marks Obtained 1 2 3 4 5 6 7 8 9 10 11 12 Total Marks

2 DO NOT WRITE IN THIS MARGIN DO NOT WRITE IN THIS MARGIN DO NOT WRITE IN THIS MARGIN DO NOT WRITE IN THIS MARGIN DO NOT WRITE IN THIS MARGIN @TJC 2022 8865 Section A: Pure Mathematics [40 marks] 1 Find algebraically the set of values of k for which ( ) 22 4 1 0kx x k− + +  for all real values of x. [4] 2 (a) Differentiate 31 23ln 9 2xx  − with respect to x. [3] (b) Find ( ) 2 23 dx x x −  . [3] 3 (i) Find the exact root of the equation 222e 1 3exx −−= . [3] (ii) Sketch on the same diagram, the graphs of 22e 1xy=− and 23e xy −= , showing clearly the equations of any asymptotes and the coordinates of the y-intercepts. [2] (iii) Use (i) and (ii) to deduce in exact form, the range of values of x that satisfy the inequality 222e 3e 1 0xx −− −  . [1] (iv) Deduce the range of values of p for which 2e 3e 1 0pp− − −  . [1] 4 A curve C satisfies the equation 2 d d ( 5) ya xx= − , where a is a constant. The equation of the tangent to C at the point P where the x-coordinate is 2 is 3 2 16yx+= . (i) Show that the equation of C is y = 161 5x  + − . [4] (ii) Sketch C, stating the equation(s) of the asymptotes and coordinates of any points of intersection with the axial axes. [2] (iii) Write down the equation of the line which passes through the origin and P. [1] (iv) Hence, find the exact area of the finite region bounded by C, the line in (iii) and the x-axis. [4]

3 @TJC 2022 8865 [Turn over DO NOT WRITE IN THIS MARGIN DO NOT WRITE IN THIS MARGIN DO NOT WRITE IN THIS MARGIN DO NOT WRITE IN THIS MARGIN DO NOT WRITE IN THIS MARGIN 5 A small high -end collectible manufacturer decides to produce nineth, sixth and quarter scale diecast figurings of superhero War-Robot. The total cost of making 20 nineth scale War-Robots is $30 more than the cost of making 4 sixth scale War -Robots. The tot al production cost of making 30 nineth scale, 16 sixth scale and 6 quarter scale War-Robots is $8517. Based on experience, if the manufacturer decides that the profit earned from selling a nineth, sixth and quarter scale War -Robot are 120%, 100% and 80% of their production cost respectively, the total profit earned from selling a set of 3 different scale War-Robots will be $661.50. (i) By writing down three linear equations, find the cost price of making each scale of War-Robot, correct to the nearest cent. [4] Due to high demand for the sixth scale War-Robot, the manufacturer decides to use a new manufacturing process with new molds. It is given that the manufacturer must produce at least 21 sixth scale War -Robots to be sold so that it will be operatio nally viable. The manufacturer is interested in modelling his manufacturing costs, C thousand dollars, for number of sixth scale War-Robot, w. The model the manufacturer uses is ln(3 60) 1120 wCw= − − + , for w > 20. (ii) Use differentiation to find the value of w which gives a stationary point on the graph of C against w. Justify whether C is a minimum or maximum. [4] (iii) Sketch the graph of C against w for 20w . Estimate the minimum cost C. [2] The manufacturer decides to sell each sixth scale War-Robot for $420. (iv) Write down an equation relating the profit earned, P thousand dollars, from selling a sixth scale War -Robot. What is the minimum order you would advise the manufacturer to accept? [2]

4 DO NOT WRITE IN THIS MARGIN DO NOT WRITE IN THIS MARGIN DO NOT WRITE IN THIS MARGIN DO NOT WRITE IN THIS MARGIN DO NOT WRITE IN THIS MARGIN @TJC 2022 8865 Section B: Probability and Statistics [60 marks] 6 The elected college student coun cillors consist of 9 boys and 5 girls. An exec utive committee consisting of 5 members is to be formed from these student councillors. (i) How many different committees can be formed if there must be at least one girl and at least one boy? [2] The 5 chosen members consist of 3 boys and 2 girls. They are to stand in a line to take a photoshoot. (ii) In how many ways can they stand in a line where Linda, one of the girls, must stand in between 2 boys? [2] 7 Events A and B are such that ( ) ( )P 0.8, P 0.4 and P( | ) 0.3.A B B A B = = = (i) Explain what is meant by ( )P AB and find its value. [3] (ii) Find P(A). [2] (iii) Complete the Venn diagram to show the probability in each of the four regions. [1] (iv) Explain whether A and B are independent. [1] A B

5 @TJC 2022 8865 [Turn over DO NOT WRITE IN THIS MARGIN DO NOT WRITE IN THIS MARGIN DO NOT WRITE IN THIS MARGIN DO NOT WRITE IN THIS MARGIN DO NOT WRITE IN THIS MARGIN 8 A fruit grower produces a large number of peaches every day. A small portion p of these peaches is infected. A check is carried out each day by taking a random sample of 60 peaches and examining them for infection. The number of infected peaches , X, in the sample may be well modelled by a binomial distribution. (i) It is given that for a particular sample, the probability that at most one peach is infected is 0.61. Form an equation in p and find the value of p numerically. [3] Using a special cultural method, p is now controlled at 0.02. (ii) Find the probability that, for a particular day, at least 2 peaches are infected in a random sample of 60 peaches