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 & m 5 os θ θ 5 m. Given differentil eqution n e written s d d m ( ) log ( ) Integrting ftor e e d m Solution is ( ) ( ) d ( ) m ( ) d ( ) log m. 6 6 sin os sin os d ( sin os ) [(sin os ) sin os ] sin os d sin os d sin os d sin os 7
( se ose ) d ( ) d tn ot (ept ot lso) O 9 ( ) d d 9 ( ) ( ) 9 d m ( ) 9 log or ( ) 9 { ( ) log. f () ( ) ( ) f () >, (, ) U (, ) m f () <, (, ) U (, ) m f() is stritl inresing in (, ) U (, ) nd stritl deresing in (, ) U (, ) O Point t θ is,
d d os θ sin θ; sin θ os θ m dθ dθ slope of tngent t θ d is d θ os θ sin θ sin θ os θ θ ot m Eqution of tngent t the point : m Eqution of norml t the point : 5. (, ) (, ) (, ) is refleive m For (, ), (, d) If (, ) (, d) i.e. d d then (, d) (, ) is smmetri m For (, ), (, d), (e, f) If (, ) (, d) & (, d) (e, f) i.e. d & f d e dding, d f d e f e then (, ) (e, f) is trnsitive m is refleive, smmetri nd trnsitive hene is n equivlne reltion [(, 5)] {(, ), (, 5), (, 6), (, 7), (5, ), (6, 9)} 9
6. ot sin sin sin sin ot sin os sin os sin os sin os ot ot sin os ot O 7 5 se tn 5 tn LHS 7 tn 5 tn ½ 7 tn tn 7 tn tn m HS () tn 5 5 tn 7 tn tn m 7. log log, Tking log of oth sides d d log, Diff. w r t
, d d d d Diff. w r t d d d d. 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 () ) P (E I (ii) P(E/B) P (B) B) P (E I 9. LHS, m m
; m HS (Epnd long ). let tn θ θ tn tn θ θ θ u tn tn tn tn tn θ m tn θ v sin sin ( sin θ) θ tn tn θ m du d du dv dv ; m ( ) d m ( ) ( In se, if ot θ then nswer is ). Differentil eqution n e written s : (sin. os ) d. (. log ) d m Integrting oth sides we get os sin os log m sin log t nd, solution is : sin log ½
. Generl points on the lines re ( λ) î ( λ) ĵ kˆ & ( μ) î ( μ )kˆ m lines interset if λ μ...(); λ...(); μ...() for some λ & μ m From () & () λ, μ sustituting in eqution () Sine, () () is true lines interset m Point of intersetion is : î kˆ or (,, ). Let nd e eletroni nd SETION - mnull operted sewing mhines purhsed respetivel L.P.P. is Mimie P sujet to 6 < 576 or < < >, > For orret grph verties of fesile region re (, ), B(, ), (6, ) & O(, ) m m P() 6, P(B) 9, P() 5 For Mimum P, Eletroni mhines Mnul mhines m. Let E : Event tht lost rd is spde E : Event tht lost rd is non spde
: Event tht three spdes re drwn without replement from 5 rds P(E ), P(E ) m 5 P(/E ), P(/E 5 ) 5 P(E /) 5 5 5 m m 9 O 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 Men P() or m 5. orret Figure m The line nd irle interset eh other t m
re of shded region d ( ) d 6 sin sq.units m 6. Eqution of plne through points, B nd is 5 6 56 i.e. 7 m Distne of plne from (7,, ) (7) () () 7 9 6 m 9 m O 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 5
7. Here ½ 5 6 5 6 or X B ( ) () () 5 X B o-ftors re,, 5,,, 5, 5 5 5 5 5 5 6,, i.e. s. for disipline, s for politeness & s. for puntulit One more vlue like sinerit, truthfulness et. m. le I ( ) d ; I sin os os sin os os sin d sin m dding we get, I sin os tn se d ; d sin os ( tn ) m tn ( tn ) m I 6 m 6
9. let r nd h e the rdius nd height of the linder then, Volume of linder (V) r h h r r m Surfe re of linder r rh (r rh) m r r ds dr r r S ds dr r 6 or r d s 56 56 t r ; > m dr r 6 surfe re is minimum t r m ; h m m 7