QUESTION PPER CODE 65// EXPECTED NSWERS/VLUE POINTS SECTION - -.. 5. { r ( î ĵ kˆ ) } ( î ĵ kˆ ) or Mrks ( î ĵ kˆ ) r. /. 5. 6. 6 7. 9.. 8. 5 5 6 m SECTION - B. f () ( ) ( ) f () >, (, ) U (, ) m f () <, (, ) U (, ) m f() is stritl inreg in (, ) U (, ) nd stritl dereg in (, ) U (, ) Point t θ is, d d os θ θ; θ os θ m dθ dθ slope of tngent t θ d is d θ os θ θ θ os θ θ ot m
Eqution of tngent t the point : m Eqution of norml t the point :. 6 6 os os d ( os ) [( os ) os ] os d os d os os d ( se ose ) d ( ) 8 d tn ot (ept ot lso) 9 ( ) 8 d 8 d 9 ( 8) ( ) 9 d m ( 8) 9 8 8 8 log 8 5
or ( 8) 9 8 8 { ( ) 8 log 8. Given differentil eqution n e written s d d m ( ) log ( ) Integrting ftor e e d m Solution is ( ) ( ) d ( ) m ( ) d ( ) log m. log log, Tking log of oth sides d log, Diff. w r t d d d, Diff. w r t d d d d d d 5.,, m 6
7,,, m m & etween eing ngle θ 9, θ os 5 9 m θ 5 5 θ os m 6. ot ot os os os os
ot os ot ot LHS tn 5 tn 8 se 5 7 5 8 tn tn ½ 7 tn tn tn 7 tn 7 m tn tn tn 7 5 5 tn () RHS m 7. (, ) (, ) R (, ) R is refleive m For (, ), (, d) If (, ) R (, d) i.e. d d then (, d) R (, ) R is smmetri m For (, ), (, d), (e, f) If (, ) R (, d) & (, d) R (e, f) i.e. d & f d e dding, d f d e f e 8
then (, ) R (e, f) R is trnsitive m R is refleive, smmetri nd trnsitive hene R is n equivlne reltion [(, 5)] {(, ), (, 5), (, 6), (, 7), (5, 8), (6, 9)} 8. let, g e ounger o nd girl nd, g e elder, then, smple spe of two hildren is S {(, ), (g, g ), (, g ), (g, )} m Event tht ounger is girl {(g, g ), (, g )} B Event tht t lest one is girl {(g, g ), (, g ), (g, )} E Event tht oth re girls {(g, g )} (i) P(E/) P (E I ) P () (ii) P(E/B) P (E I B) P (B) 9. LHS ( ) ( ) ( ) Ug, C C C C m ( ) Ug, R R R; R R R m ( ) {( ) } Epnding long C ( ) RHS m 9
. let u tn ; v ( ); θ θ u tn θ θ tn ( tn θ) θ ( ) ( ) m & du d v θ θ m dv, m d du dv m ( In se, if os θ then nswer is ) d log. ose log d d d m log Integrting oth sides we get [ os os ] d [ os ( )] d log os os m. Equtions of lines re : 5 7 z ; 5 8 7 z 5 m Here, 5,, 7,, z ; 5 ; 8, 7,,, z 5
z z 7 8 5 (7) (7) 8 ( ) lines re o-plnr SECTION - C. Let nd e eletroni nd mnull operted sewing mhines purhsed respetivel L.P.P. is Mimize P 8 sujet to 6 < 576 or < 8 < >, > For orret grph verties of fesile region re (, ), B(8, ), C(6, ) & O(, ) m m. Let E : Event tht lost rd is spde E : Event tht lost rd is non spde P() 6, P(B) 9, P(C) 5 For Mimum P, Eletroni mhines 8 Mnul mhines : Event tht three spdes re drwn without replement from 5 rds m P(E ), P(E ) m 5 C P(/E ) C, P(/E 5 ) 5 C C
P(E /) 5 C 5 C C C 5 C C m m 9 X No. of defetive uls out of drwn,,,, m 5 Proilit of defetive ul 5 Proilit of non defetive ul Proilit distriution is : : P() : P() : 6 8 8 8 8 8 8 8 8 8 8 8 8 Men P() or m 8 5. Here z ½ z 5 z 6 z 5 6 or X B ( ) () () 5 X B Co-ftors re
,, 5,,, 5, 5 z 5 5 5 5 5 6,, z i.e. Rs. for disipline, Rs for politeness & Rs. for puntulit One more vlue like erit, truthfulness et. m 6. Eqution of plne through points, B nd C is 5 8 z 8 6 z 56 i.e. z 7 m Distne of plne from (7,, ) (7) () () 7 9 6 m 9 m Generl point on the line is ( ) ( ) ( ) kˆ Putting in the eqution of plne; we get λ î λ ĵ λ m ( λ) ( λ) ( λ) 5 λ m Point of intersetion is î ĵ kˆ or (,, ) Distne ( ) ( 5) ( ) 69 m
7. Corret Figure m The line nd irle interset eh other t m re of shded region d ( ) d 6 8 8 sq.units m 8. Let I tn d I se ose se ( ) tn( ) ( ) ose ( ) d m I ( ) tn se ose d dding we get, I tn d se ose d os d m I m 9. For orret fiqure let rdius, height nd slnt height of one e r, h nd l respetivel r h l
V (volume) r h, [V is onstnt] r l, z r l r ( r h ) 9v r r r 9v r r m dz dr r 8v r m dz dr r 6 9v 9v d z 5v t r 6 ; r > dr r m 6 orved surfe re is minimum iff r 9v 6 i.e. r r h h r ot α h r α ot ( ) 5