S.Y. B.Sc. (IT) : Sem. III. Applied Mathematics. Q.1 Attempt the following (any THREE) [15]

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1 S.Y. B.Sc. (IT) : Sm. III Applid Mahmaics Tim : ½ Hrs.] Prlim Qusion Papr Soluion [Marks : 75 Q. Amp h following (an THREE) 3 6 Q.(a) Rduc h mari o normal form and find is rank whr A Ans.: Givn : A R R R, R 3 R 3 5R 3 6 A C C + C, C 3 3C, C 6C A R 3 R A 6 R A 3 5 C 3 + 3C, C + 5C A R, C 3 C A A I 3 Rank of A 3 (ordr of idni mari) Q.(b) Invsiga for wha valus of and, h quaions z 9, z 8, z. hav () no soluion () uniqu soluion (3) infini no. of soluions Ans.: Abov ar Homognous quaion AX B z Augmnd mari [A B] [A B] R 7R, R 3 R [A B]

2 Vidalankar : S.Y. B.Sc. (IT) Mahs III (i) Ssm will hav no soluion, if ssm of quaions ar inconsisn, P(A) P(A B) i.. 5, 9 Ssm will hav soluion if ssm of quaions ar consisn P(A) P(A B) r (ii) Uniqu soluion if r n n : no. of unknowns 3 5 can ak an valu (iii) Infini soluion if r < n 5, 9 Q.(c) Find h Eign valus and Eign vcors of mari A h Eign valu of A + 3A + I. Ans.: A Characrisic quaion, A I ( + 3 ) A ( + () + 6) B snhic division, ( ) ( + ) ( ) ( ),, Eign valus ar,, 3 Eign valus of A + 3A + I Eign valus of A :,, 3 Eign valu of I Eign valu of A :,, 3 :,, for Eign valu of A + 3A + I 3 () Also find 5

3 Prlim Qusion Papr Soluion For ign valus of A + 3A + I 3 () Eign valus of A + 3A + I ar 9,,,. 3() Q.(d) Solv h quaions, 7 cosh + 8 sinh for ral valus of. Ans.: 7 cosh + 8 sinh 7 8 (7 + 8 ) + (7 8 ) Quadraic quaion in, Comparing wih a + b + c a 5, b, c ( ) 5( )() (5) 8 8, 3 3 3, 5 For ral valus of, ak log on boh sids log Q.() Prov ha, an a ib ilog a ib a ab b Ans.: LHS log a ib log (a ib) log (a + ib) a ib log z log r + i whr r, an log a ib a ib b b log a b ian log a b ian a a ian log a ib i an a ib i log a ib i ian a ib b b ian a a b a a b b ian a a ib an ilog an an a ib a b () () (3) 3

4 Vidalankar : S.Y. B.Sc. (IT) Mahs III From sp no. 3 a ib an ilog a ib an an an a ib an ilog a ib a ib an ilog a ib an an an an a ab b b b an a a an an (b/a) (a b )/a b a an(an (b/a)) (an(an (b/a))) a ab b (b/a) (b/a) an an ( ((b/a)(b/a)) ab an an a b RHS b/a (b/a) a ab b RHS Q.(f) Using D-Moivr s horm prov ha : ( + i) 8 + ( i) 8 3 Ans.: L Z + i (Carsion Form), In polar form, r an i Z r i( /) Z cos isin L Z i (Carsion Form), In polar form, r an IV quadran i Z r i( /) cos isin () Considr, LHS ( + i) 8 + ( i) 8 Z Z 8 cos isin cos isin using Dmoivrs Thorm, cos isin cos isin 8 (cos ) (/) 8 () ( i cos + i sin ) [ cos ] 5 3 RHS ( + i) 8 + ( i) 8 3

5 Prlim Qusion Papr Soluion Q. Amp h following (an THREE) Q.(a) Solv : ( + / ) d + / d. Ans.: ( + / ) d + / d M d + N d M + / N / M / N / / N / M N DE is ac. Soluion is givn b Md Nd C + ra consan rms fr of / ( )d d C consan / / C + / C Q.(b) Solv : d d Ans.: Th abov is linar D.E in d P Q d Whr P Q 3 ( ) Ingraing Facor (I.F) d I.F. ( + ) Soluion is givn b.(i.f) Q.(IF)d C.( + ) ( + ) ( + ) ( ) d c 3 ( ) d c. an () + c 3 d log( ) Q.(c) Solv : (D 3D + ) sin Ans.: To find complimnar funcion (C.F) Auillar quaion f(d) D 3D + D, 5

6 Vidalankar : S.Y. B.Sc. (IT) Mahs III Roos ar ral and disinc C.F. C + C To find Paricular Ingral, P.I.Q sin f(d) D 3D a a P.I V V f(d) f(d a) P.I. sin (D ) 3(D ) P.I sin D D D sin D D D sin D sin D sin D Dsin sin P.I. cos sin Gnral Soluion G.S. C.F. + P.I. C C c cos sin cos sin Q.(d) Solv : (D 3 + D + D) 3 + sin + Ans.: To find complimnar funcion Auiliar Equaion F(D) D 3 + D + D D(D + D + ) D (D + ) D, D -, - C.F C (C C ) 3 To find P.I P.I.Q f(d) 3 sin D(D D ) 3 log P.I ( ) sin D(D D ) D(D D ) D(D D ) 3 cos (D 3) (D 3) (D 3) D(D D ) D(D D log 6

7 3 cos 3 (D 3)(D 8D 6) (D D D) (log) (log) log o 3 cos Prlim Qusion Papr Soluion log D D D8 D D D D D D log log (log) log 3 3 o D D D cos D D D 8 D log (log) (log) log D D D D D D log cos 3D 8 log (log) log 3 5 D () log (3D 8)cos 9D 6 log (log) log 3 5 ( 3sin 8cos) log (log) log 3 5 (6sin38cos PI (log) log log Gnral Soluion Y C.F + P.I 3 6sin 8cos 5 c (c c ) (log)[(log) log] Q.() Solv : p + p ( + ) +. Ans.: a, b ( + ) c ( ) ( ) ( ) ( ) p () P -, - p - d d d d log - + c log c () P - d d d d c () 7

8 Vidalankar : S.Y. B.Sc. (IT) Mahs III Final soluion is produc of individual soluion. log c c Q.(f) Solv : ( + p) + P. d dp p dp d Ans. : ( p) p d d d d p dp p p ( ) d dp p d d dp p ( ) d p dp Linar diffrnial quaion in p. d p Q dp P Q - p P & Q f(p) pdp Ingraing facor I.F dp Soluion is (IF) Q (IF)dpc p p p dp c p Q.3 Amp h following (an THREE) Q.3(a) Find Laplac ransform of f() 5 cosh. Ans.: f() 5 cosh cosh 5 f() 5 ( ) L{f()} L L{ 5 cosh } 5 5 L[ ] L[ ] (A) (Linari Propr) L{ 5 5! n n! } 6 L[ ] n s s L[ 5 ] s s (s ) Also L[ 5 ] s s + 6 (s ) Subsiuing qs. () and () in q. (A) L[ 5 cosh ] 6 6 (s ) (s ) 6 () () ( s shifing horm) 6 (s ) 6 (s ) 6 8

9 Q.3(b) Evalua b using Laplac ransform Ans.: L I sin d Compar wih dfiniion of L[f()] L[f()] s f()d W g s, f() sin I L[sin ] sind L[f()] s s L[ sin ] () d ds s Prlim Qusion Papr Soluion sin d. {muliplicaion b } d s (s ) ( ) s(s )s (s ) 8s 3 ds (s ) (s ) (s ) 6s 3 (s ) Now, L[ 6(s ) sin ] s s + (firs shifing horm) 3 ((s ) ) 6( ) sind L[f()] s 6 3 (( ) ) 8 sind Q.3(c) Find L - s a log s b Ans.: L f() L s a log s b f() L [log (s + a ) log s b ] W know, L[ n f()] () n n d n F(s) ds To ak s drivaiv w.r.. s in RHS W nd o mulipl b in LHS d d f() L log(s a ) log(sb) ds ds f() s L L s a s b f() cos a f() b cosa b f() b cosa s L s a sb 9

10 Vidalankar : S.Y. B.Sc. (IT) Mahs III Q.3(d) Find L (s ) using Convoluion Thorm. (s s 8) Ans.: L f() L (s ) (s s 8) Convoluion Thorm, L[F(s) G(s)] OR L [F(s) G(s)] f() g( u) du f( u) g() du f() g( u) du s s L F(s), G(s) s s8 s s 8 f() L s [F(s)] L s s8 s f() L s L s L s s (s ) s f() cos Similari, g() L s s [G(s)] L L s s8 (s ) g() cos (s ) L (u) cos cos( u) du (s s 8) u u cos cos( u) du [coscos( u)] du u [cos( u) cos(u)] du b a cos(b c)d a u u cos( u) du cosudu a [acos(b c) bsin(b c)] a b In firs ingral in q. (A), a, b In scond ingral in q. (A) a, b (A) u (cos( u) ( )sin( u)) () ( ) u (cosu sinu) () ( ) (s ) L (s s 8) (cos sin) (cos sin ) (cos sin) ( ) cos (cos sin )

11 Prlim Qusion Papr Soluion Q.3() Solv h diffrnial quaion using Laplac Transform. (D + D + 5) sin () () Ans.: Givn (D + D + 5) sin () () L D d d, d D d d d 5 d d sin Appling Laplac Transform on boh sids, d d L 5 d d L[ sin ] n d L f() n d s n F(s) s n f() s n f() L L[()] Y(s) d d L L L[5] L[ sin ] d d [s Y(s) 5() ()] + [s Y(s) ()] + 5Y(s) L[ sin ] L [sin ] s L[ sin ] s s + (s ) s s (s + s + 5) Y(s) () Y(s) s s s s3 s s5 s s3 L s s5 L L [] s s () L s () () sin (s s 5) L L s s5 s s5 L [] L (s ) () sin Q.3(f) Find Laplac ransform of f() ( ) u( ) 3 ( ) Ans.: f() ( ) u( ) 3 ( ) L[f()] L[( ) u( ) 3 ( )] L[( ) u( )] L[ 3 ( )] s L[ ] () 3 F(s) s! 8 3 s s F(s) 8 3 s Q. Amp h following (an THREE) Q.(a) Evalua : I Ans.: I / d d / / d d d / ( )d

12 Vidalankar : S.Y. B.Sc. (IT) Mahs III () () I 3 Q.(b) Chang h ordr of ingraion and valua Ans.: limi, LL () (Lin passing hrough origin) UL () + (quaion of circl) C : (, ) R : Poin of inrscion of qs. () and () A(, ), B(, ) B dd -ais ( ) (, ) R R A (,) -ais ( ) + X-limi : o Chang h ordr of ingraion, Considr srip paralll o -ais Rgion (R ) -limi : LL UL -limi : LL UL Rgion (R ) -limi LL UL -limi : LL UL I dd R R dd I + I I I pu + d d d d (/) d () dd / d () d Limi : : : d ( ) ( ) I 3

13 Prlim Qusion Papr Soluion I + d d d d dd (/) () dd d /) () d ( ) d I I + I (3 ) ( ) Q.(c) Chang o polar coordinas and valua I Ans.: Chang o polar coordinas I r cos, r sin + r dd dd B -limi : LL : () UL : () ( ) + C : (, ) R : C (, ) + Poin of inrscion of qs. () and () ( ), A (, ), B(, ), -limi : : o Chang o polar r-limi (r-srip) r : LL : r UL : + r r cos r cos r : o cos 3

14 Vidalankar : S.Y. B.Sc. (IT) Mahs III limi : LL: UL : I / cos (r )(rdrd ) ( /) r 6 / cos d ( /) / cos d ( /) / ( cos cos )d ( /) / sin sin I 3 8 cos 3 r drd ( /) r / (cos ) d ( /) cos cos / / / ( /) r / ( /) cos cos cos d d log Q.(d) Evalua I Ans.: L I log logz dz d d. logzdz z [log z ] log log z dz d d. I z[logz ] dd log [ [ ] ]dd [(log ) log ( )]d log [ [ ] [log ]dd log [ ( ) () ] d log log ( )d log d ( ) (log ) ( ) 83 I 3 Q.() Find b doubl ingraion h ara includd bwn h curvs a. Ans.: (i) a (parabola, righ) (ii) a (parabola, up) a and Poin of inrscion of qs. () and () 6a a

15 Prlim Qusion Papr Soluion 6 a 3 ( 3 6a 3 ) a a A(, ), B(a, a) Considr ship paralll o -ais -limi : LL : 9 UL : a A a (a, a) B a -limi : LL : UL : a a a a a a A dd d 3/ /a a d a a 3/ a a 3 / 3/ (a) 6a 6 (a) (a) (a) a 3 a a A 6 a 3 (unis) Q.(f) Find h volum boundd b h clindr + and h plans z and + z. Ans.: Volum of boundd b clindr, + C : (, ) R : Plan + z A (,, n) B (,, ) Considr z-srip paralll o z-ais LL : Z UL : Z Considr clindrical coordinas r cos r sin z z : o r sin dddz rdrddz Considr r-srip r-limi LL : r UL : + r r r o -limi : v : o rsin d ddz r z rsin rdr ddz rz r z r rsin drd 5

16 Vidalankar : S.Y. B.Sc. (IT) Mahs III rrsindr d r 8 8 cos 3 v 6(uni) 3 r r sin d sin d 3 Q.5 Amp h following (an THREE) n Q.5(a) Evalua : I log d Ans.: I( ) d log (A) di d d log logd d log di d di d I(α) log(α+) + C (B) Pu α I() from q(a) C From q. (B) I( ) log( ) Q.5(b) Evalua : I Ans.: 5 () d 5 I () d () d 5 5 I 5 5 Q.5(c) Show ha : Ans.: d () d 5 Limis / / log( asin log a sin d sin di d da a sin / sin d sin asin / cosc d co a Pu co cosc d d cosc d -d Limi / a a >. / d asin / cosc d co ( a) / cosc d a cosc 6

17 Prlim Qusion Papr Soluion ( a) () d d a d da a d da a an a a ( a) c I(a) a c (B) pu a I() from q(a) From q(b) I(a) a Q.5(d) Show ha : a b log a log b using DUIS. a b Ans.: I d log Considr a as paramr. a b I(a) d log Using DUIS a b a a di.log d da d a log log di da di da a I(a) log(a ) c () I(a) log(a ) c (B) Pu a b in q(a) I(b) C -log(b + ) I(a) log (a + ) + (-log (b + )) a I(a) log b Q.5() Dfin rror funcion and prov ha rror funcion is an odd funcion. Ans.: Error Funcion arf() d and ar indpndn of ach ohr. rf( ) d Pu -u d -du 7

18 Vidalankar : S.Y. B.Sc. (IT) Mahs III Limis - X rf( ) ( du) -rf() rf() is an odd funcion. Q.5(f) Prov ha : d [rf C (a)] + a d [rf (a)] d da Ans.: W hav, rf c (a) a u.du Appling Libniz rul of D.U.I.S w.r., d u a d [rf (a)] du (a) c d d a a a a a a u Again, rf(a) du a d d u rf(a) du da da () a u a du a n n From Equaions () and () d d (rf(a) a [rf(a)] c d da () a a a a 8