AJC_H2_MATH_P2

Page 2 of 9 Anderson Junior College Preliminary Examination 2011 H2 Mathematics Paper 2 Section A: Pure Mathematics [40 marks] Answer ALL questions 1. At a water treatment plant, sewage water is being pumped into a processing tank, with an unknown capacity, at a constant rate of 100 litres per hour. The tank processes the water and discharges it at a rate proportional to the volume of sewage water currently in it. Due to a crack in the tank, sewage water is leaking out of the tank at a constant rate of 3 litres per hour. At time t hours, the volume of sewage water in the tank is u litres. (i) By setting up and solving a differential equation, show that the general solution is 1 97 e ktuA k  where A and k are constants, and k > 0. [3] (ii) The processing container is initially empty. After an hour, the volume of sewage water in the container is 70 litres. Given that the sewage water that leaked out of the tank is not processed, find the volume of water processed by the container when t = 2. [4] (iii)Sketch, on a single diagram, the graph of the solution found in part (ii) and find the minimum tank capacity for the model to be valid. Give your answer to the nearest litre. [2] 2. (a) The function f is defined by 1f( ) ( 1 )s i n 1x xx   , 10 2x  (i) By using differentiation or otherwise, show that the function f -1 exist. Hence find the value of f -1(1). [3] (ii) Find the equation of the tangent to the curve y = f(x) at the point where x = 0. [2] (iii) Write down the equation of the tangent to the curve y = f -1(x) at the point where x = 1. [1]

Page 3 of 9 (b) The functions g and h are defined by 2 3 11 , for , 0 4g( ) 5 ( 6) , for , 4 7 xx xx xx x      

1h( ) 1 for ,x xx axa   (i) Find the range of g. [1] (ii) Given that the composite function hg exists and a is an integer, find the greatest value of a such that the range of hg is a subset of (1, ) . [2] 3. With reference to the origin O, the position vectors of the points A and B are 23ij k and 2 ik whereis a negative integer. The cosine of the angle between the position vectors of A and B is 3 70  . The point M lies on the line segment AB such that 3AMM B , and the point C lies on OM produced such that 43OM OC . (i) Show that 4  . [2] (ii) Find the position vector of C. [2] (iii)Geometrically, what is the significance of p , where ACA B p AB       ? By finding the value of p , deduce the distance between O and the line passing through A and B.