MATHEMATICS PAPER & SOLUTION

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Kelley Barnett
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1 MATHEMATICS PAPER & SOLUTION Code: SS--Mtemtis Time : Hours M.M. 8 GENERAL INSTRUCTIONS TO THE EXAMINEES:. Cndidte must write first is / er Roll No. on te question pper ompulsorily.. All te questions re ompulsory.. Write te nswer to e question in te given nswer-ook only.. For questions ving more tn one prt, te nswers to tose prts re to e written togeter in ontinuity.. If tere is ny error / differene / ontrdition in Hindi & Englis versions of te question pper, te question of Hindi version sould e treted vlid. 6. Setion Q. Nos. Mrks per questions A B C 6 7. Tere re internl oies in Q. Nos.,,, 7, 9 nd. You ve to ttempt only one of te lterntives in tese questions. 8. Drw te grp of Q. No. on te grp pper. Find, if ot. SECTION - A ot ot ot. Construt mtri A [ ij ], wose elements re given y ij i j. A 8 6. If [ ], ten find te vlue of. 6 [ ] [ 8] ± /

2 . Find. ot ot (se ) dy y. Find te generl solution of te differentil eqution. dy y dy y lny ln lny ln ln y y e k y k 6. If î ĵ kˆ nd k î ĵ λkˆ ( î ĵ kˆ) k(î ĵ λkˆ ) k; k; kλ k λ λ su tt, find te vlue of λ. 7. Find te diretion osine of te line y z d.r.s, 7, 7 d..s ±, ± ±, ±, ± y z. 7 7, ± Find te ngle etween plnes r.(î ĵ kˆ) nd r.(î ĵ kˆ) 7. Angle etween plnes ngle etween teir normls ( î ĵ kˆ) (î ĵ kˆ) osθ 6 8 θ 9º /

3 9. Sow te region of fesile solution under te following onstrints y 8,, y in nswer ook. y (, 8) O (, ) y 8 ; ; y y 8. If A nd B re independent events wit P(A). nd P(B)., ten find te vlue of P(A B). P(A B) P(A) P(B) P(A B) P(A) P(B) P(A) P(B) (Q A & B re independent).. (.) (.).7..6 SECTION - B. Prove tt te reltion R in set of rel numers R defined s R {(, ) : } is refleive nd trnsitive ut not symmetri. Consider f : R R given y f(). Sow tt f is invertile. Find lso te inverse of funtion f. R R {(, ) : } R R true so refleive reltion R true ut R not true. so not symmetri R R R. so trnsitive reltion Hene proved f : R R f() f( ) f( ) y so f() is one-one funtion. f() R O y R Rnge R odomin onto funtion f() is one-one & onto f() is invertile funtion /

4 / y y y Inverse funtion f (). Prove tt. sin 9 Solve (sin ) ( se ), < <. LHS RHS sin Hene proved (se )) ( (sin )) ( (se ) (sin ) < < sin sin se sin se.os sin os. Epress te mtri A s te sum of symmetri nd skew symmetri mtri. A A A A A T T

5 / / / / / / / / / symmetri mtri Skew-symmetri mtri. Find te vlue of K so tt te funtion is ontinuous t te point. ; ; K os () f ; ; K os () f RHL f K os lim put K os lim K sin lim K () K sin K lim for funtion to e ontinuous t f f K K. Find te intervls in wi te funtion f given y f() 6 is () Stritly inresing () Stritly deresing Find te eqution of te gent to te urve / y / t te point (, ). f() 6 f ' () 6 ( ) () stritly inresing (, ) () stritly deresing (, )

6 / y / diff. / / y dy / y / / dy dy y dy t point (, ) gent y ( ) y y 6. Te rdius of irle is inresing uniformly t te rte of m/se. Find te rte t wi te re of te irle is inresing wen te rdius is 6 m. dr m/se dt re A r r da dr r dt dt da (6) () 6 m/se dt r 6m ( )( log ) 7. Find. Find log( ). ( )( log ). put ln t dt t ( ln) t dt I l n( ). l n( ) II I ln( ) ln( ) ln( ) ln( ) ( ) ln( ) 6 /

7 8. Find ( ) ( ) l n l n 9. Find te re ounded y te prol y nd te strigt line y. (Drw te figure in nswer ook) y y ; y y Are ( ) / / / 6 ( ) 8 sq. units. Using integrtion find te re of tringulr region wose sides ve te equtions y, y nd. (Drw te figure in nswer ook.) y y y Sded Are y) ( y ( ) ( ) sq. units 7 /

8 . If,, re unit vetors su tt, find te vlue of ( ) ( ) ( ) (... ) (... )... /.. Find unit vetor perpendiulr to e of te vetors î ĵ kˆ. î ĵ kˆ î ĵ kˆ î ĵ kˆ î ĵ kˆ î ĵ kˆ î ĵ kˆ ( î ĵ kˆ ) î kˆ î ĵ kˆ Required vetor nd, were î ĵ kˆ, î( ) ĵ( 9 ) kˆ ( ) î ĵ kˆ ( î ĵ kˆ ) ( î ĵ kˆ) Perpendiulr unit vetor ± 9 ± ( î ĵ kˆ). By grpil metod solve te following liner progrmming prolem for mimiztion. Ojetive funtion Z 6y Constrints y y,, y. y (, ) (, ) (, 8) (, 8) (, 6) (, ) y y 8 /

9 Sded region is required region Now, Z 6y y z ,8, 6,6, Q Z m,6, Wen, y 6. Bg A ontins red nd lk lls wile noter g B ontins red nd lk lls. One ll is drwn t rndom from one of te g nd it is found to e red. Find te proility tt it ws drwn from g B. Bg A Bg B Red Blk Red Blk Event E Bg A is seleted P(E ) / Event E Bg B is seleted P(E ) / Event X drwn ll is Red P(X) P(E ) P(X/E ) P(E ) P(X/E ) P(X) 7. P(E )P(X / E) P(E /X) 7 P(X) From lot of uls wi inlude 6 defetives, smple of uls re drwn t rndom wit replement. Find te proility distriution of te numer of defetive uls. uls 6 defetive let no. of defetive uls X Xi Pi C 9 C 6 C. C 8 C 6 C C 9 9 /

10 SECTION - C 6. Prove tt ( ) ( ) ( ) ( ). C C C ( ) Apply R R R R R R ( ) ( ) [( )( ) ( )( )] ( ) [( ) ( ) ( ) ( )( )( )] ( ) ( ) ( ) [( ) ( )] ( ) ( ) ( ) ( ) Hene Proved 7. If y (sin ), ten sow tt ( d y dy ). y (sin ) dy sin () ( ) sin d y ( ) ( d y (sin ) ( d y ) ( d y ) dy ) (using ) dy Hene proved /

11 sin 8. Evlute. os sin I os ( )sin( ) I os ( ) ( )sin I os Add () & () sin I os I / sin os I / sin os Let os t sin dt dt I t I [ t] I [ ] () () f () f ( ) f () f () if f ( ) f () dy 9. Solve te differentil eqution y y. dy solve te differentil eqution y ot ot. dy y y dy. y y dy y y Let y v dy dv v Put in () dv v v v dv v () /

12 dv v Integrting ot sides dv v loge v y loge Were is onst of integrtion dy y ot ot () ot sin log e I.F. e e sin Multiply ot sides of y I.F. & ten integrte we get y.sin sin ot.sin y sin sin os sin sin sin y sin sin. y z y z Find te sortest dise etween te lines nd. Prove tt if plne s te interepts,, nd is t dise p units from te origin, ten prove tt p. y z r (r (î ĵ kˆ)). (î ĵ kˆ) r r p y z r (r (î ĵ kˆ)). (î ĵ kˆ) r r q r r r r ( ).(p q) Sortest dise r r p q î ĵ kˆ p q î ( ) ĵ ( ) kˆ( ) î kˆ r ( î ĵ kˆ ) ( î ĵ kˆ ) î ĵ kˆ Sortest dise ( î ĵ kˆ).( î kˆ) 6 9 units /

13 / Plne z y Perpendiulr dise from origin (,, ) P P p Hene Proved

EXPECTED ANSWERS/VALUE POINTS SECTION - A

EXPECTED ANSWERS/VALUE POINTS SECTION - A 6 QUESTION PPE ODE 65// EXPETED NSWES/VLUE POINTS SETION - -.... 6. / 5. 5 6. 5 7. 5. ( ) { } ( ) kˆ ĵ î kˆ ĵ î r 9. or ( ) kˆ ĵ î r. kˆ ĵ î m SETION - B.,, m,,, m O Mrks m 9 5 os θ 9, θ eing ngle etween