AJC_H2_MATH_P1

Page 2 of 7 Anderson Junior College Preliminary Examination 2011 H2 Mathematics Paper 1 Answer ALL questions. 1. Find 12cos dx xx  . [3] 2. The usual selling price of a T-box game console, a Kinect sensor and a game DVD is $499 in total. During the Great Singapore Sale, two companies, P and Q, offered the following discounts to their customers: Company Discounts given for each item Total price after the discount T-box game console Kinect sensor Game DVD P 10% 15% 10% $439.15 Q 5% 25% 20% $426.30 The employees from Company P are offered a further 5% discount on the original price of both the Kinect sensor and the Game DVD. Determine if this additional discount will make it more attractive for the employees to purchase all the 3 items from their own company than from Company Q. [4] 3. Solve the inequality      2 0xaxb xxc    , where ,,abc  and 0 cba  . Hence, find the range of values of x which satisfy       2 ln ln 0 ln ln xa xb xc x    . [5] 4. (i) By using the substitution tantx , show that  1 2 11 d tan 2 tan1s i n 2 x xcx   . [3] (ii) Find the exact volume of revolution when the region bounded by the curve 2 12 1s i ny x  , the lines 4x  , y = 2 and the y-axis is rotated completely about the line y = 2. [3]

Page 3 of 7 5. (a) Find sin 2 cos dx xx . [2] (b) The diagram below shows the curve defined by the parametric equations sinx tt , sin 2y t for 0 t  . The curve has a maximum point at A. (i) Find the equation of the tangent to the curve at the point A. [3] (ii) Find the exact area bounded by the curve, the tangent at A and the line x  . [3] 6. Given that y =  1 2s i n 2 x , show that 22 2 2 d2 d 4s i n 2dd yy yxxy x   . [3] (i) Hence find the Maclaurin’s series for y up to and including the term in x3. [3] (ii) By considering the standard series for sin x, verify that the series obtained in (i) is correct. [2] 7. The above 48 cm by 18 cm plastic sheet is cut out to form a shape represented by the shaded region. The shape is folded into a box with width x, length y and height z as y x A Base Side Side Side Side Top cover

Page 4 of 7 shown in the diagram. (i) Express the volume of the box, V, in terms of y. [2] (ii) Using differentiation, find the maximum possible volume of the box. [3] Two small robots of negligible dimensions, A and B, are initially placed on the edge of the base of the box where m = 0 cm and n = 0 cm respectively as shown in the diagram. Robot A starts to travel along the length of the base with speed 2 cm