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

HKDSE-MATH-EP(M2)(Set 3) − 26 26 © 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. Let cos sin ( ) sin cos A θ θ θ θ θ −   =     , where θ is real. (a) Prove that (i) ( ) A θ is non-singular and 1 [ ( )] ( ) A A θ θ − = − ; (ii) ( ) ( ) ( ) A A A α β α β = + , where α and β are real; (iii) [ ( )] ( ) n A A n θ θ = for all positive integers n by mathematical induction. (6 marks) (b) Let X and Y be two square matrices of the same order and XY = YX . It is given that for all positive integers n , 0 ( ) n n n n k k k k X Y C X Y − = + = ∑ , where 0 X and 0 Y are by definition the identity matrix. By considering 1 { ( ) [ ( )] } n A A θ θ − + , prove that ( ) 0 2 cos 0 ( 2 ) 0 2 cos n n n n k n n k C A n k θ θ θ =   − =     ∑ for all positive integers n . (4 marks) (c) Using (b), evaluate (i) 5 3 16cos 10cos 5cos 20 20 20 π π π − − ; (ii) 3 5 0 cos d π θ θ ∫ . (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.