Einstein Classes, Unit No. 102, 103, Vardhman Ring Road Plaza, Vikas Puri Extn., Outer Ring Road New Delhi , Ph. : ,

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1 Dte :... IIT/AIEEE APPEARING 0 MATRICES AND DETERMINANTS PART & PART Red the following Instrutions very refully efore you proeed for PART Time : hrs. M.M. : 40 There re 0 questions in totl. Questions to rry one mrk eh, questions to 7 rry four mrks eh, questions 08 to 0 rry si mrks eh. Red the following Instrutions very refully efore you proeed for PART Time : hrs. M.M. : 4 There re questions in totl. For eh orret nswer (+) will e wrded nd for inorret nswer (-) mrk will e wrded PART -. Prove, using the properties of determinnts Show tht ( )( )( )( ).. For wht vlues of nd y re the following mtries equl A 0 y,b y y 0 y. If A, B, C re three mtries suh tht A = [ y z] B h g h 4. Epress the mtri f g f, C y, find ABC. z 4 A 7 s the sum of symmetri nd skew-symmetri mtri.. Prove tht the determinnt independent of. sin os sin os 7. If,, re positive nd re the pth, qth nd rth terms respetively of G.P., Show without epnding tht log log log 8. Find the inverse of 9. If A A = I. A 0 p q r 0. A 4 nd verify tht 4 0, find A. Using A, solve the system of liner equtions y = 0, + y + z = 8, y + z = 7. Einstein Clsses, Unit No. 0, 0, Vrdhmn Ring Rod Plz, Viks Puri Etn., Outer Ring Rod New Delhi 0 08, Ph. : 990, 87

2 0. For wht vlues of nd, the system of equtions + y + z = 8, + y + z =, + y + z = 4 hs (i) (ii) (iii) unique solution infinitely mny solutions no solution. PART -. The numer of vlues of stfying 0 () () (). If,, re positive nd re the pth, qth nd rth terms respetively of G.P. The vlue of determinnt depends on. If log log log () () p () n n ( ) ( ) () 0 () () n n n p q r 4n n depends on depends on n n, t depends on nd n oth 4. If A, then 4 () A A 4 I = 0 () A 4A 4 I = 0 () A A 4 I = 0 A A 4 I = 0 then the vlue of. The inverse of the mtri () () () A 4. If A, then A + A T, where A T denotes 8 the trnspose of mtri A. () () () symmetri mtri skew symmetri mtri oth n e possile 4 7. If A, then A A T, where A T denotes the trnspose of mtri A. () () () 8. If () () symmetri mtri skew symmetri mtri oth n e possile A nd B = [ 4], then ( AB) BA () ( AB) AB ( AB) BA Einstein Clsses, Unit No. 0, 0, Vrdhmn Ring Rod Plz, Viks Puri Etn., Outer Ring Rod New Delhi 0 08, Ph. : 990, 87

3 For questions (9-0) From the following eqution, y 4 9. The vlue of () () () 9 0. The vlue of y () () () 9. If A, find k suh tht A = ka I 4. () () () 9. The vlue of. () ( ) () ( + ) () ( ) (). The vlue of () () () 4 4. The vlue of determinnts y 4y 0 8y 4 8 () () () depends on only depends on y only depends on nd y oth independent on oth. The vlue of the determinnt y y z z y z y z depends on () only () y only () z only Einstein Clsses, Unit No. 0, 0, Vrdhmn Ring Rod Plz, Viks Puri Etn., Outer Ring Rod New Delhi 0 08, Ph. : 990, 87

4 4 RESPONSE SHEET AIEEE TEST Dted :... M.M. : 4 Sujet Phys Chemtry Mths Totl NAME OF THE STUDENT :... No. of Corret Mrks Otined Inorret Negtive Mrks Net Mrks () () () () () () () () () Einstein Clsses, Unit No. 0, 0, Vrdhmn Ring Rod Plz, Viks Puri Etn., Outer Ring Rod New Delhi 0 08, Ph. : 990, 87

5 IIT/AIEEE APPEARING 0 MATRICES AND DETERMINANTS ANSWERS Dte :... PART - TIME : HR. M.M. : 4. d. d d... d 4.. d Einstein Clsses, Unit No. 0, 0, Vrdhmn Ring Rod Plz, Viks Puri Etn., Outer Ring Rod New Delhi 0 08, Ph. : 990, 87