Prime DSE Series - Mathematics Mock Exam Papers (Extended Part) Module 2 Set 3

HKDSE-MATH-EP(M2)(Set 3) − 10 10 © 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. 7. Consider the curve 2 : 1 x y a Γ = − , where 2 2 x − ≤ ≤ and 1 a > . The figure shows the shaded region which is enclosed by Γ and the horizontal line y h = , where 0 h ≥ . A craftsman makes a pendant in the form of the solid of revolution generated by revolving the shaded region about the y -axis. Let V be the volume of the pendant. (a) Prove that [( 1) ln( 1) ] ln V h h h a π = + + − . (b) The craftsman adjusts the size of the pendant by two different methods. (i) In Method I , it is given that the rate of change of a is ln 3 units per second. Find the rate of change of V when a = 3 . (ii) In Method II , it is given that the rate of change of h is ln 3 units per second. The craftsman claims that the rate of change of V in Method II when h = 2 is equal to the rate of change of V in (i). Is the claim correct? Explain your answer. (7 marks) Method I Fix the value of the height of pendant at h = 2 , then increase the value of a . Method II Fix the value of a at 3, then increase the value of h . 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.