SRJC_H2_MATH_P2_STUDENT_S

1 SERANGOON JUNIOR COLLEGE 2015 JC2 PRELIMINARY EXAMINATION MATHEMATICS Higher 2 9740/2 Tuesday 25 Aug 2015 Additional materials: Writing paper List of Formulae (MF15) TIME : 3 hours READ THESE INSTRUCTIONS FIRST Write your name and class on the cover page and on all the work you hand in. Write in dark blue or black pen on both sides of the paper. You may use a soft pencil for any diagrams or graphs. Do not use staples, paper clips, highlighters, glue or correction fluid. Answer all the questions. 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 a graphic calculator. Unsupported answers from a graphic calculator are allowed unless a question specifically states otherwise. Where unsupported answers from a graphic 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. At the end of the examination, fasten all your work securely together. Total marks for this paper is 100 marks. This question paper consists of 6 printed pages (inclusive of this page) and 2 blank pages. [Turn Over

2 Section A: Pure Mathematics [40 marks] 1 Functions f and g are defined as below. 1f : , 2 1xx x  g : ln(2 ) , 2x x x x    (i) Sketch the graph of y = g(x) and state its exact range. [3] (ii) Determine whether the composite function s fg and g f exist, justifying your answer. Find the range of the composite function if it exists. [4] (iii) On the same diagram as part (i), sketch the graph of y = g −1(x), indicating on your sketch the line in which the graph of y = g(x) must be reflected in order to obtain the graph of y = g−1(x). [2] 2 A tank contains water which is heated by an electric water heater working under the action of a thermosta t. When the water heater is first switched on, the temperature of the water is 35 C . The heater causes the temperature to increase at a rate Cr per minute, where r is a constant, until the water temperature hits 75 C .The heater then switches off. (i) Write down, in terms of r, the time taken for the temperature to increase from 35 C to 75 C . [1] The temperature of the water then immediately starts to d ecrease. The temperature of the water at time t minutes after the heater is switched off is C . It is known that the temperature of the water decreases at a variable rate ( 25) Ck  per minute, where k is a positive constant, until 35  . (ii) Write down a differential equation involving  and t, to represent the situation as the temperature is decreasing