AJC_H2_MATH_P2_Question

Page 1 of 6 AJC / 2015 Preliminary Examination / 9740 / P2 Anderson Junior College 2015 Preliminary Examination H2 Mathematics Paper 2 (9740/02) Section A: Pure Mathematics (40 marks) 1 (a) The variables w, x and y are connected by the following differential equations: 22 de d x w wx  and d d y wx  . (i) Verify that 2 2 e xw A  is the general solution of 22 de d x w wx  , where A is an arbitrary constant. [2] (ii) Hence find y in terms of x. [3] (b) A tank initially contains 50 grams of salt dissolved in 100 litres of water. Brine that contains 2 grams of salt per litre of brine fl ows into the tank at a rate of 5 litres per minute. The solution is kept thoroughly mixe d and flows out from the tank at a rate of 5 litres per minute. Given that the amount of salt in the tank at time t minutes is given by S , show that d 200 d2 0 SS t  . Hence find the time, in minutes, at whic h the concentration of salt in the tank reaches 1 gram per litre. [7] 2 Solve the equation 5 32 0z  , expressing your answers in the form ier  , where 0r  and   . [2] 1z , 2z and 3z are three of the roots of 5 32 0z   such that 1230 arg arg argzz z  . (i) Find the smallest positive integer n such that 1 2 n z z     is real and positive. [3] (ii) The points A and B represent the roots 1z and 3z respectively in the Argand diagram. The line segment 'BA is obtained by rota ting the line segment BA through 2  clockwise about the point B. Find the real part of the complex number represented by point 'A , giving your answer in exact trigonometric form. [4]

Page 2 of 6 AJC / 2015 Preliminary Examination / 9740 / P2 3 Referred to the origin O, the position vectors of points A, B and C are 6i , 424i j k and 326i j k respectively. The plane p1 is given by the equation 1x yz . (i) Find the equation of the plane ABC in scalar product form. [3] (ii) The perpendicular from the point D(3, 1, 4) to the plane p1 meets the plane ABC at the point S. Find the coordinates of S. [3] (iii) Find the equation of the plane p2 such that every point on p2 is equidistant from points S and D. [3] 4 Betty needs to decorate a wall of length 5 metres for a party. She attaches hooks, starting from the extreme left end of the wall, and numbers each hook “1”, “2”, “3” and so on. The spacing between the 1 st and 2nd hook is 50 cm and each subsequent spacing is 2 cm shorter than the previous spacing. This will c ontinue till she is unable to place the next hook due to insufficient space. (i) How many hooks can she attach in total? [4] Betty also cuts a 6 metre roll of ribbon into pieces of varying lengths. The first piece cut off is of length 80 cm and each s ubsequent piece cut off is 10 % shorter than the previous piece. (ii) If the length of the remaining roll of ribbon is less than 1 metre after n cuts, find the smalles