29T MATH.pmd

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1 9T MATH Totl numer of pges MATHEMATICS Full Mrks : 100 Pss Mrks : 0 Time : Three hours The figures in the mrgin indite full mrks for the questions. Q. No. 1 ( j) rries 1 mrk eh 1 10 = 10 Q. Nos. 1 rr 4 mrks eh 4 1 = 48 Q. Nos rr 6 mrks eh 6 7 = 4 Totl = 100 Contd.

2 1. Answer the following questions : 1 10=10 Ó 1 õ ùüà ı±11 Î M 1 ø ± : () Let A : 1 10, is n odd nturl numer nd B : , is prime numer. Write the numer of reltions from A to B. 1 Ò1± í A : 1 10, È ± ±ˆ ±øªfl ± ±1n B : , º A 1 Û1± B Δ Œ ±ª± g 1 ± ø ±º È ± Œ Ãø fl ± () Write down the rnge of f ot. 1 f 1 ot Ù Ú1 Ûø1 1 ø ±º 1 () Find ll the positive vlues of determinnts whose entries re from the set 1, 0, 1. 1 øó øúì«fl 1±º 1, 0, 1 1 Œ à À À1 øí Ó ±1øÌfl À ı±11 fl À ± ÒÚ±Rfl ±Ú (d) Let A e skew-smmetri mtri of odd order. Write the vlue of A. 1 Ò1± í A È ± ±S±1 ø ı ø Ó Œ à fl é º A 1 ±Ú ø ±º 9T MATH [ ]

3 (e) Let A e mtri of order, suh tht A 9. Find the vlue of A 1. 1 Ò1± í A È ± ±S±1 Œ à fl é íó A 9 fl í1±º º 1 A 1 ±Ú øúì«(f) If d, then find. 1 øpple d, ŒÓ ÀôL øúì«fl 1±º (g) Evlute f. 1 f øúì«fl 1±º (h) Find the order nd degree of the differentil eqution d 7 d d 7 d 6 0 ıfl œfl 1Ì1 Sê ±1n ±Ó øúì«fl 1±º (i) Write the intervl in whih the funtion f os is stritl deresing. 1 f os Ù ÚÀÈ ± Œfl ±ÚÀÈ ± ôl1± Ó Ó Ó ò± ±Ú ø ±º 9T MATH [ ] Contd.

4 (j) Write the eqution of the plne pssing through,, nd prllel to -plne. 1,, ø ıμ 1 ±ÀÊÀ1 Œ ±ª± ±1n - Ó 1 ±ôl1± Œ ±ª± Ó Ú1 œfl 1Ì ø ±º. Let the mpping f, 0, mps 1, 1 onto 0, ; show tht ot ot 7 ot 8 ot 18 f. 4 Ò1± í Ù Ú f, Œpple ª± Œ, ot ot 7 ot 8 ot 18 f ±1n 1, 1 1 ±Â±ppleÚ õ øó ø S í. 0, ; Find the vlue of os os, os os, 11 ±Ú øúì«fl 1±º. Let f : R R is defined f nd g : R R is defined g. Show tht f g g f. 4 9T MATH [ 4 ]

5 9T MATH [ 5 ] Ò1± í R R f : fl f Œ1 :± ıx fl 1± Δ À ±1n R R g : fl g Œ1 :± ıx fl 1± Δ Àº Œpple ª± Œ, f g g f. 4. Show tht 4 Œpple ª± Œ, Without epnding show tht 0 ø ıô ±1 Úfl 1±Õfl Œpple ª± Œ, 0 Contd.

6 5. Show tht the funtion f defined f 1, R is ontinuous funtion. 4 Œpple ª± Œ, f 1, R Œ1 :± ıx Ù Ú f, È ± ø ıøâiß Ù Úº Disuss the ppliilit of Rolle s theorem to the funtion f 1 on,. Ù Ú f 1, ôl1±, Ó 1 ƒ Î Û Û±pple 1 õ À ± ±À ± Ú± fl 1±º 6. If e d, find. 4 øpple e d, ŒÓ ÀôL øúì«fl 1±º If os, 0, 1 4 d find , øpple os, 0 ŒÓ ÀôL d 1 ±Ú øúì«fl 1±º 9T MATH [ 6 ]

7 Determine the set of ll points where the funtion f is differentile. Ù Ú f ªfl Úœ Œ ±ª±1 ø ıμ 1 øó øúò«±1ì fl 1±º Evlute ±Ú øúì«fl 1±º Evlute os 8 1 tn ot. os 8 1 tn ot 1 ±Ú øúì«fl 1±º 9. Evlute ±Ú øúì«fl 1±º 9T MATH [ 7 ] Contd.

8 Evlute tn tn 1 ±Ú øúì«fl 1±º 10. Solve the differentil eqution 4 d log ªfl œfl 1Ì d log ±Ò±Ú fl 1±º 11. If os log 4sin log, d d show tht 0. 4 øpple os log 4sin log, d d º Œpple ª± Œ, 0 1. If 6î 8 ĵ nd ĵ 4kˆ then determine the vetor omponent of long. 4 øpple 6î 8 ĵ ±1n ĵ 4kˆ ŒÓ ÀôL 1 øpple Ó 1 Œˆ " 1 Î Û± øúì«fl 1±º 9T MATH [ 8 ]

9 Find unit vetor perpendiulr to eh of the vetors, where î ĵ kˆ nd î ĵ kˆ. î ĵ kˆ, î ĵ kˆ Œ ±ª± fl fl Œˆ " 1 øúì«fl 1±º ŒÓ ÀôL nd ±1n pple À ±È ± Œˆ " 11 Û1Ó 1. A nturl numer is seleted t rndom from the set A : Find the proilit suh tht the numer stisfies the ineqution øó A : Û1± ±pple ø fl ˆ ±À ı È ± ±ˆ ±øªfl ± øú ı«± Ú fl 1± í º ±ÀÈ ±Àª Ó ± œfl 1Ì 1 0 ±Ò±Ú fl 1±1 ±øªó ± øúì«fl 1±º 14. If A 0 tn tn, then 0 os sin show tht I A I A, sin os where I is the identit mtri of order. 6 9T MATH [ 9 ] Contd.

10 9T MATH [ 10 ] øpple 0 0 tn tn A, ŒÓ ÀôL Œpple ª± Œ, os sin sin os A I A I, íó I È ± ±S±1 fl fl Œ à fl é º If 4 1 A, then find 1 A ; nd hene solve the sstem of equtions z z z øpple 4 1 A, ŒÓ ÀôL 1 A øúì«fl 1± ; ±1n œfl 1Ì ÛX øó z z z ±Ò±Ú fl 1±º

11 15. Form the differentil eqution stisfied r, where nd re ritrr onstnts. 6 r œfl 1ÀÌ ø X fl 1± ıfl Ê œfl 1Ì Í Ú fl 1±, íó ±1n øâfl ÒËn ıfl º Find the mimum nd minimum vlues of the funtion f sin on 0,. 0, ôl1± Ó f sin Ù Ú1 À ı«± ±1n ı«øú ß ±Ú øúì«fl 1±º 16. Prove tht the re of right ngled tringle of given hpotenuse is mimum when the tringle is isoseles. 6 õ ±Ì fl 1± Œ õ pplem øó ˆ Ê ø ıø Ü È ± Àfl ±Ìœ øsˆ Ê1 fl ±ø À ı«± ı Œ øó ± øsˆ ÊÀÈ ± øx ı±u ıº Find the re of the smller portion enlosed the urves 9 nd 8. 9 ±1n 8 ısê ±&1± 1n 1 fl ±ø øúì«fl 1±º 9T MATH [ 11 ] Contd.

12 17. Find the shortest distne etween the lines r 6î ĵ kˆ î ĵ kˆ nd r 4î kˆ î ĵ kˆ. 6 r 6î ĵ kˆ î ĵ kˆ ±1n r 4î kˆ î ĵ kˆ Œ1 ± pple Î ± 1 ±Ê1 ı«øú ß pple 1Q øúì«fl 1±º Find the equtions of two lines through the origin whih interset the line z 1 1 t. ø ıμ 1 ±ÀÊÀ1 Û±1Õ Œ ±ª± pple Î ± 1 œfl 1Ì øúì«fl 1±º z 1 1 Œ1 ±Î ± fl Œfl ±ÌÓ ŒÂpple fl 1± Œ1 ±. Hene find the re of the 18. Prove tht prllelogrm whose digonls re the vetors î ĵ kˆ nd î ĵ 4kˆ. 6 9T MATH [ 1 ]

13 õ ±Ì fl 1± Œ î ĵ 4kˆ º ÀÓ Àfl Œˆ " 1 î ĵ kˆ ±1n fl Ì«ø ıø Ü ±ôlø1fl 1 fl ±ø øúì«fl 1±º Find the vetor eqution of the line pssing through 1,, nd prllel to the plnes r. î ĵ 5 nd. î ĵ 6 1, r., ø ıμ 1 ±ÀÊÀ1 Û±1Õ Œ ±ª± ±1n. î ĵ 5 î ĵ kˆ 6 kˆ r r kˆ. kˆ ±1n Ó 1 ±ôl1± Œ1 ±1 Œˆ " 1 œfl 1Ì øúì«fl 1±º 19. Solve the liner progrmming prolem grphill. 6 Δ ø fl øú À À1 Ó 1 Δ1ø fl õ À Ëø ±ÀÈ ±1 ±Ò±Ú Î ø ª±º Mimize z 0 15, sujet to the onditions 00, 150 nd 0, 0. z 0 15 íó, 1 À ıı«± ±Ú Î ø ª± 00, 150 ±1n 0, 0 º 9T MATH [ 1 ] Contd.

14 Mimize nd minimize z 5, sujet to the onditions,, 1, nd 0, 0. z 5 1 À ı«± ±1n ı«øú ß ±Ú Î ø ª± íó, 1, ±1n 0, 0 º 0. Two numers re seleted t rndom from set of first 90 nturl numers. Find the proilit tht the produt of rndoml seleted numers is divisile. 6 õ Ô 90È ± ±ˆ ±øªfl ±1 Û1± È ± ± ±pple ø fl ˆ ±À ı øú ıı«± Ú fl 1± í º ±pple ø fl ˆ ±Àª øú ıı«± Ú fl 1± ± pple È ±1 &ÌÙ Œ1 ø ıˆ ±Ê Œ ± ı±1 ±øªó ± øúì«fl 1±º In mtri, entries ij re seleted rndoml from the digits 0, 1,,, 4, 5, 6, 7, 8, 9 with replement where eh element ij is three digit numer. Find the proilit tht eh element in eh row is divisile 15. 9T MATH [ 14 ]

15 È ± ±S±1 Œ à fl é 1 Œ à ij fl fl 0, 1,,, 4, 5, 6, 7, 8, 9 1 Û1± Û Ú1 ö±ø ÛÓ ø ± ıó ±pple ø fl ˆ ±À ı øú ıı«± Ú fl 1± Δ ÀÂ, íó õ ÀÓ fl ij È ± øó øúè ± fl ø ıø Ü ±º õ ÀÓ fl ±1œ1 õ ÀÓ fl Œ à 15Œ1 ø ıˆ ±Ê Œ ±ª±1 ±øªó ± øúì«fl 1±º 9T MATH [ 15 ]

16 9T MATH [ 16 ] 45+