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

HKDSE-MATH-EP(M2)(Set 3) − 18 18 © 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. 10. (a) (i) Using integration by substitution, find 2 2 1 1 d 1 x x x − ∫ . (ii) Using (a), find 1 2 2 0 1 d (1 ) 2 x x x + + ∫ . (4 marks) (b) Let f( x ) and g( x ) be continuous function defined on R . It is given that f( x ) is an even function and ( ) ( ) 1 g x g x − ≡ . Suppose there are no real roots for the equation g( x ) + 1 = 0 . Prove that 0 f ( ) d f ( ) d g( ) 1 a a a x x x x x − = + ∫ ∫ , where a is a constant. (4 marks) (c) Evaluate 3 3 1 2 2 1 d ( 1)(1 ) 2 x x e x e x x − + + + ∫ . (4 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.