RI P2

Section A: Pure Mathematics [40 marks] 1 Prove by mathematical induction that 1 1sin sin 22cos , 2sin 2 n r n rn        . [5] 2 A patient in a hospital receives a course of treatment. At the start of the treatment, 80 mg of drug A is injected into the blood stream of the patient. It is known that x mg, the amount of drug A remaining in the blood stream at time t hours after an injection decreases at a rate proportional to the amount of drug A present at that instant in the blood stream. It is also found that 3 hours aft er an injection, only a quarter of the drug A injected remains in the blood stream of the patient. (i) Write down a differential equation involving x and t. Hence find an expression for x in terms of t, where 06 t . [6] The treatment is such that the same amount of drug A is injected to the patient at regular intervals of 6 hours. Let mgnu be the amount of drug A present immediately after the nth injection, 1n . (ii) Find constants a and b such that 1nnu a bu  . [3] If the patient continues w ith this treatment indefinitely, use a calculator to determine what you can expect about the amount of drug A present in the blood stream of the patient immediately after each injection in the long run. [1] 3 The points A and B have coordinates  0, 9,c and  , 5, 2d  respectively, where c and d are constants. The line l has equation 3 1 5 1 4 3 x y z   . (a) Given that 22 7d  and the line AB intersects l , find the value of .c [3] Find also, the coordinates of the foot of perpendicular from A to l . [3] (b) Given instead that the lines AB and l are parallel, state the value of and cd . [2] (i) By using a cross product, or otherwise, find the shortest distance between the lines AB and l . [3] (ii) The plane 1p contains , AB and l . Find an equation of 1p in scalar product form. [2] (iii) The plane 2p is obtained by first translating 1p 2 units in the positive z- direction and then reflecting in the xz -plane. Obtain an equation of 2p in scalar product form. [2]

H2 MA 9740/ 2012 RI Year 6 Preliminary Examination [Turn Over 2 4(a) Solve the equation 6 1 0,z  giving the roots in the form , where 0 and .ire r       [2] (b) Let the points 1P , 2P , , nP represent complex numbers satisfying 1 0, 5nzn   , as shown on the Argand diagram below. The points 2Q , 3Q , , nQ are formed by dropping perpendiculars from 1P to 2OP , and kQ to 1kOP  for 2k  , 3 , , 1n . (i) State 1kkP OP , where 11 kn   . Find the length of 2OQ and 12PQ . [2] (ii) Show that 23, ,..., nOQ OQ OQ are in geometric progression. Show that 2 3 3 4 1, , ..., nnQ Q Q Q Q Q  are also in geometric progression with the same common ratio. [2]