olutios to RPL/. < F < F< Applig C C + C, we get F < 5 F < F< F, $. f() *, < f( h) f( ) h Lf () lim lim lim h h " h h " h h " f( + h) f( ) h Rf () lim lim lim h h " h h " h h " Lf () Rf (). Hee, differetile t. os( log ) log. d os tdt si tg Let log t d dt si(log ) si() he, t d whe, t log si(log ). Give ples re + + z 8, i.e. + + z 8 log d 5 + + z + 5, i.e. + + z + \ Diste 5 + 8 5+ 6 7 uits. 6 + + 5. + Y Y Y R T 6. Cosider os(si ) d d si(si ) (si ) si(si ) os d ( ) d d d si(si ) os si(si ) os d d F sisi mos si ( ) Mthemtis ()
7. Let side of equilterl trigle e m. \ Are A d da m/s d m /s dt dt Hee, side is m. 8. Give f() ( ) / i [, ] da d da dt dt dt Cotiuous i [, ] s lgeri epressio with positive epoet is otiuous. f () ( ) ( ) f () does ot eist t ot differetile t [, ] Hee, Rolle s Theorem is ot pplile. 9. Give lie is \ DR s re,, 6 + + Dividig + 9+ 6 7 7 6 5 6 z 7 z 5 6 6 \ Diretio osies of the lie re,,. 7 7 7. P(A B) P(A) P(A B) P(A B) Now, P(A). P(B) P(A B) 8 8 8 \ A d B re idepedet evets. P(A B) 8. Let lrge vs d smll vs e used. LPP is to miimise ost Z + ujet to ostrits, + 8 + Mthemtis ()
. Cosider + d + ( ) + d + e + d o + + t + C ( )( + ) + d +. Cosider t ( + ) + si + +...(i) t ( + ) si + + os + + Let t ( + ) q t q \ os q From (i) d (ii), we get os q os + + e + + o os + + + + + + + + ( + ), Both the vlues stisf (i). Hee,,.. Cosider + + + + + ( + + ) + + ( + + ) + e + + o...(ii) + + ( + + ) + + + + [B performig C C + (C + C )] [B tkig ( + + ) ommo from C ] [B performig R R R ] ( + + )[( ) ( + ) ( ) ( + )] ( + + )( + + + + ) ( + + )( + + ) + +. q + Mthemtis ()
5. Let t Let si q e o q si d t se Let os q e o q os si q t d t se e se os q o d os θ os θ t (t q) q se (se q) q si t os \ \ d d...(i) d d dt ' dt d d ( ) OR dt d...(ii) [From (i) d (ii)] Cosider log d log...(i) O differetitig oth sides w.r.t., we get d log d d log d d + d [From (i)] Agi differetitig w.r.t., we get d d + d + d d d d d d d Dividig throughout, we get 6. Cosider d d + d d d d 5+ os t t dt + 5+ f p + t ( + t ) 5+ 5t + t 9 t dt + t t + C t d t + C t dt + Let t t t t d t dt + t os + t Mthemtis ()
7. Cosider R / t t d se d d + os T / / < t F t d < t log se F d t log se ( log ) log log OR ( )( + ) For < <, ( )( )(+) > For < <, (+)( )(+) < For < <, (+)(+)(+) > d ( ) d + ( ) d + ( ) d e o + e o + e o 6 m+ m + m m + + + 8. + + + + d ( )( d + + ) d d For For + d, d + + d d + se q se qdq t q d + + d... (i) t dt t C # + + + C...(ii) d siq os q q si q si q os dt t ( t ) log log + os θ + + C os θ os θ + + + dq q e + dt o log t t + + + C...(iii) Let + t d dt Let t θ se θ dθ d Let os q t si q dq dt + t + + C t t + q t q Mthemtis (5)
ustitutig from (ii) d (iii) i (i), we get + log where C (ostt) C + C + + + + + C is the required solutio. 9. Let it+ j t + zkt...(i) ( it+ tj + kt ) ( it+ j t + zkt ) + + z...(ii) Also # i + j 8k it tj kt ti+ tj 8kt z (z )i t (z ) t j + ( )k t ti+ tj 8kt z...(iii) z...(iv) d 8...(v) From (ii) d (v), + z 6 From (iii) d (vi), we get (z ) + z 6 8z + z 6 9z 8 z From (iv) d (vi),, 5 \ 5it+ tj + kt OR it + tj + kt ^ it si α h + _ tj siβi + ^ kt si γh. Cosider reltios l + m + 5 _ si α+ si β+ si γi _ os α+ os β+ os γi ( os os os g) ( )...(vi) m l 5...(i) d 6m l + 5lm...(ii) 6( l 5) l + 5l( l 5) 8l l 5l 5l 5l + 5l + l + l + d l l + d + d l l + d + Mthemtis (6)
l l + or + l or l From (i), m 5 l m From (i), m 6 5 l m DR s :,, DR s :,, \ DR s re,, or,, \ os q + + + + + 6 q os d 6. Cse I: white lls d other oloured lls i g Cse II: white lls d other oloured ll i g Cse III: white lls d o other oloured ll i g E: white lls re drw P (I ) P (II ) P (III) P (E/I ) C C ; P (E/II) C C 6 ; P (E/III) C 6 C Usig Bes Theorem, proilit of drwig white lls from g whe ll the lls re white P (III/E ) PIII ( ). PEIII ( / ) PI (). PEI ( / ) + P( II). PEII ( / ) + P( III). PEIII ( / ) + + 6 6 6 6 5 + + 6. A: perso hs TB B: perso does ot hve TB ( i persos who hve TB) 999 P(A), P(B) E: perso is digosed to hve TB P(E/A).99; P(E/B). Mthemtis (7)
Usig Bes Theorem, proilit tht perso tull hs TB whe he is digosed to hve TB. P(A/E) PA ( ) PE ( A) PA ( ) PE ( A) + P( B) P( E B) 99. 999 99. +. 99 99 99 + 999 989 er msk or use loth whe oughig. Mediies must e tke s direted dotor.. Drw grph of iequlities +,, + ;,. The fesile regio determied the ostrits, +,, +, d is show; ABCDA is the fesile regio. The orer poits of the fesile regio re A(, 5), B(, ), C(5, ) d D(, ). The vlues of Z t these orer poits re s follows: Corer poits Z + A(, 5) Miimum B(, ) Miimum C(5, ) 5 D(, ) Mimum The mimum vlue of Z is t (, ) d the miimum vlue of Z is t ll the poits o the lie segmet joiig the poits (, 5) d (, ). Mthemtis (8)
R os si. Cosider A si os T os si A si os os + si Mtri formed oftors of eh elemet i A R A os α A si α A A si α A os α A A A A T R os si l os si \ Adj A si os si os T R os α si α R os α si α \ A Adj A si α os α si α os α A R T T os si R os si Cosider AA si os si os T R T os α+ si α osαsi α siαos α + osαsi α siαos α si α+ os α + + + + + + + + TR I T OR Give A G; B G (A + B) (A + B) (A + B) AA + AB + BA + BB (A + B) A + AB + BA + B Give (A + B) A + B \ AB + BA O...(i) AB G G G BA G G + G + From (i), G + + G G + Mthemtis (9)
+ G G + ; ; Hee,, stisfies ll four equtios. 5. For refleive: Let for (, ) A A (, ) R (, ) + +, true. Hee, refleive. For smmetri: Let for (, ), (, d) A A For trsitive: (, ) R (, d) + d + + d + (, d) R (, ) Hee, smmetri. Let for (, ), (, d), (e, f) A A (, ) R (, d) d (, d) R (e, f) + d + d + f d + e + d + + f + + d + e + f + e (, ) R (e, f). Hee, R is trsitive As reltio R is refleive, smmetri d trsitive. Hee, R is equivlee reltio. Equivlee lss [(, )] (, ) R (, ) + + Give A {,,,..., } \ Ordered pirs e (, ), (, ), (, ), (, 5), (5, 6), (6, 7), (7, 8), (8, 9), (9, ) \ [(, )] {(, ), (, ), (, ), (, 5), (5, 6), (6, 7), (7, 8), (8, 9), (9, )} OR Give f : N Y where Y { : N} defied f(). To show f is ivertile, we hve to show tht it is ijetive. For oe-oe: Let for, N For oto: f( ) f( ) Hee, oe-oe Let for some Y, there eists N suh tht N. Hee, oto. As futio is oe-oe d oto. Hee, f is ivertile. e defie futio g : Y N s g(). Cosider fog() f(g()) f ( ) ( ) d gof() g(f()) g( ) As fog I Y d gof I N Mthemtis ()
f is ivertile d g is iverse of f Hee, f () 6. Let squre of side m e ut off from ll orers. The l 5,, h \ olume () (5 ) ( ) (5 ) ( ) d (5 ) ( ) + ( ) ( ) d 8 8 8 + 8 8 + 76 + 8 ( + 9) 5 m m ( 8) ( 5) For mimum, d 8 or 5 5, 8, ( 8 ot possile) d d d [( 8) + ( 5) ] d H ( + ) < d 5 \ For 5, volume is mimum. Hee, squre of side 5 m must e ut off to mke se of mimum volume. 7. Give the irle + d the lie. O plottig the urve d lie, we otie we hve to fid re of shded portio. As the urve is smmetril to the -is. \ Are re (LBA) d d + si G > e + si o * + si e oh f p e o e o sq uits. 8 d 8. Cosider equtio t d d O Y L B A(, ) d d t d Fd Hee, homogeeous. Mthemtis ()
λ λ λ As Fd td td Fd λ λ λ d dv Let v v + d d v + dv v t v d t v dv d ot vdv d log si v log + log log si v log C si is the required solutio. 9. If lies re oplr, the z z. Here d,, z + d; α δ, α, α + δ d,, z + ; β γ, β, β + γ. + d + d α δ β γ α β α+ δ β+ γ B performig C C + C, we get + d α α α+ δ β β β+ γ + d α α α+ δ β β β+ γ, True, s C d C re idetil. Hee, the lies re oplr. OR Ple determied the poits A(, 5, ), B(,, 5) d C(5,, ) is 5 z + 5 5 5 5+ + 5 z + 8 8 ( )(6) ( 5)( ) + (z + )() Mthemtis ()
6 + + z + 96 6 + + z 56 + + z 7 Diste of the poit P(7,, ) from the ple + + z 7 is 7+ + 7 + 9+ 6 + 6+ 6 7 9 9 uits. Mthemtis ()