Prime DSE Series - Mathematics Mock Exam Papers (Extended Part) Module 1 Set 3

HKDSE-MATH-EP(M1)(Set 3) − 20 20 © United Prime Educational Publishing (HK) Ltd. Answers written in the margins will not be marked. Answers written in the margins will not be marked. Answers written in the margins will not be marked. 12. (a) Let M( t ) be a continuous function such that 20 (20 ) M ( ) 5 t k t ' t e t − = + , where k is a constant. It is given that M(0) = M ′ (0) = 10. (i) Using the substitution 20 1 5 t u te − = + , find 20 (20 ) 5 t t dt e t − + ∫ . (ii) Hence, find M( t ). (6 marks) (b) Tony starts a promotion campaign in his shop. Let S( t ) (in thousand dollars) be the sales of his shop at time t , where t is the number of months elapsed since the start of the promotion campaign. Tony models the rate of change of the sales of his shop by 1 S ( ) M ( ) 2 t ' t ' t   = +     . (i) Find the change of the sales from t = 0 to t = 4. (ii) The sales of the shop is 12 thousand dollars at the start of the promotion campaign. Using the fact that lim 0 t t te − →∞ = , estimate the sales of his shop after a very long time. (7 marks) Sample © United Prime Educational Publishing (HK) Limited All rights reserved; no part of this publication may be reproduced, photocopied, recorded or otherwise, without the prior written permission of the Publisher.