H2_Math_Paper_2_Question

Name:____________________________________________ Class:____________ JURONG JUNIOR COLLEGE Preliminary Examination MATHEMATICS 9740/02 Higher 2 11 September 2012 3 hours Additional materials: Answer Paper List of Formulae (MF15) Cover Page READ THESE INSTRUCTIONS FIRST Write your name and civics class 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, with the cover page in front. This document consists of 6 printed pages. [Turn over

2 Section A: Pure Mathematics [40 marks] 1 An offshore oil well is located in the ocean at point W, which is 8 km from point A on land as shown in the figure above. Oil is to be piped to point B, which is 12 km from point A, using a straight pipe laid underwater from W to some point P between A and B, and then on to point B via another straight pipe laid on land. If the cost of laying a pipe is $60 000 per km underwater and $45 000 per km on land, how far should point P lie from point A to minimise the cost of laying the pipe? [5] 2 In the theory of learning, the amount of material memorised by a person, y units, in time t minutes satisfies the differential equation t y d d = k(10 – y), where k is a positive constant. Initially, zero units of material were memorised. (i) Find y in terms of t, given that five units of material were memorised in two minutes. [6] (ii) Sketch the graph of y against t and explain what will eventually happen to the amount of material memorised. [3] (iii) The rate at which material is forgotten is known to be proportional to the amount memorised in time t. Write down a differential equation involving y when forgetfulness is taken into account. [2] 8 km La