MTEMTICL PHYSICS SOLUTIONS GTE- Q. Considr an ani-symmric nsor P ij wih indics i and j running from o 5. Th numbr of indpndn componns of h nsor is 9 6 ns: Soluion: Th numbr of indpndn componns of h nsor Q. Th valu of h ingral ns: N N 5 = N 5 5 C sin d, whr h conour C is h uni circl:, is πi 4πi πi sin Soluion: Pol is a, Circl d i. C Q. Th ignvalus of h marix ns: ar 5,, - -5, -, - 5,, - -5,, Soluion: Th characrisic quaion of h marix, I I 5 5 5,, Q4. If x for x, f hn h Laplac ransform of f(x) is x for x sx s sx s s s sx fiiks, H.No., G.F, Jia Sarai, Nar IIT, Hau Khas, Nw Dlhi 6 Phon: 6865455/+9 98745498 8 B/6, Jia Sarai, Nar IIT Hau Khas, Nw Dlhi 6 Email: fiiks.physics@gmail.com
ns: Soluion: L sx sx sx f x f xdx f xdx f xdx x sx dx L sx s sx s sx s s sx sx f x x dx dx s d y Q5. Th soluion of h diffrnial quaion for y : y cosh( ), subjc o h d ns: dy iniial condiions y and d cosh sinh sinh cosh, is cosh sinh D m Soluion: For C.F y P I. D s C. F. C C cosh. y C dy d dy d C D D C y C C C C C C C SincC C and C C C, C. Thus y y sinh D fiiks, H.No., G.F, Jia Sarai, Nar IIT, Hau Khas, Nw Dlhi 6 Phon: 6865455/+9 98745498 8 B/6, Jia Sarai, Nar IIT Hau Khas, Nw Dlhi 6 Email: fiiks.physics@gmail.com
GTE- Q6. Two marics and B ar said o b similar if B = P - P for som invribl marix P. Which of h following samns is NOT TRUE? D = DB Trac of = Trac of B and B hav h sam ignvcors and B hav h sam ignvalus ns: Soluion: If and P b squar marics of h sam yp and if P b invribl hn marics and B = P - P hav h sam characrisic roos Thn B I P P P IP P I P whr I is idniy marix. B I P I P P I P I P P I PP I Thus h marics and B (= P - P) hav h sam characrisic quaion and hnc characrisic roos of ign valus. Sinc h sum of h ign valus of a marix and produc of ign valus of a marix is qual o h drminan of marix hnc hird alrnaiv is incorrc. Q7. If a forc F is drivabl from a ponial funcion V(r), whr r is h disanc from h origin of h coordina sysm, i follows ha F F V V ns: Soluion: Sinc F is drivaiv from ponial V(r) and F V V F. Q8. marix has lmns such ha is rac is and is drminan is 6. Th ignvalus of h marix ar all known o b posiiv ingrs. Th largs ignvalus of h marix is 8 9 6 ns: Soluion: W know ha for any marix. Th produc of ignvalus is qual o drminan of ha marix. r fiiks, H.No., G.F, Jia Sarai, Nar IIT, Hau Khas, Nw Dlhi 6 Phon: 6865455/+9 98745498 8 B/6, Jia Sarai, Nar IIT Hau Khas, Nw Dlhi 6 Email: fiiks.physics@gmail.com
.... Trac of marix and 6. Hnc h largs ign valu of h marix is 9. Q9. Th uni vcor normal o h surfac x + y = a h poin P(,, ) is ns: iˆ ˆj kˆ iˆ ˆj kˆ 6 iˆ ˆj kˆ 6 Soluion: Th quaion of h sysm is f x, y, x y x iˆ ˆj kˆ Th gradin of h abov funcion is ˆ ˆ ˆ f i j k x y y xiˆ yj ˆ lˆ Hnc uni normal vcor a (,, ) f f iˆ ˆj kˆ. Q. Considr a cylindr of high h and radius a, closd a boh nds, cnrd a h origin. L r ix ˆ ˆjy kˆ b h posiion vcor and nˆ a uni vcor normal o h surfac. Th surfac ingral S r nˆ ds ovr h closd surfac of h cylindr is O y x πa (a + h) πa h πa h ro ns: Soluion: r nds rd d a h S. ˆ. V V Q. Th soluions o h diffrnial quaion circls wih diffrn radii circls wih diffrn cnrs dy x dx y ar a family of fiiks, H.No., G.F, Jia Sarai, Nar IIT, Hau Khas, Nw Dlhi 6 Phon: 6865455/+9 98745498 8 B/6, Jia Sarai, Nar IIT Hau Khas, Nw Dlhi 6 Email: fiiks.physics@gmail.com 4
ns: sraigh lins wih diffrn slops sraigh lins wih diffrn inrcps on h y-axis Soluion: dy dx x x y xdx ydy dy y C x y x y C C which is family of circls wih diffrn radii. y y C Q. Which of h following samns is TRUE for h funcion ns: Soluion: f f has a ro a f has a pol of ordr a f has a simpl pol a f is analyic vrywhr in h complx plan sin has a pol of ordr a sin f? Q. Considr a counrclockwis circular conour abou h origin. L ns: sin f, hn h ingral f d ovr his conour is iπ ro iπ iπ fiiks, H.No., G.F, Jia Sarai, Nar IIT, Hau Khas, Nw Dlhi 6 Phon: 6865455/+9 98745498 8 B/6, Jia Sarai, Nar IIT Hau Khas, Nw Dlhi 6 Email: fiiks.physics@gmail.com 5
GTE- Q4. Idnify h corrc samn for h following vcors a iˆ ˆj Th vcors a and b ar linarly indpndn Th vcors a and b ar linarly dpndn Th vcors a and b ar orhogonal and b iˆ ˆj ns: Soluion: If Th vcors a and b ar normalid a iˆ ˆ, j b iˆ ˆ j ar linarly dpndn a mb, for som valus of m bu + m = and + m = do no hav any soluion. So hy ar linarly indpndn. a b (No orhogonal); a b (No normalid) Q5. Th numbr of indpndn componns of h symmric nsor ij wih indics i, j,, is 6 9 ns: For symmric nsor,, ij, hnc hr ar six indpndn componns. Q6. Th ignvalus of h marix ar,,,,,,,, ns: Soluion: I,, fiiks, H.No., G.F, Jia Sarai, Nar IIT, Hau Khas, Nw Dlhi 6 Phon: 6865455/+9 98745498 8 B/6, Jia Sarai, Nar IIT Hau Khas, Nw Dlhi 6 Email: fiiks.physics@gmail.com 6
GTE- B r is B B r ro xˆ yˆ ˆ B B xˆ yˆ ˆ and r xxˆ yyˆ ˆ. Q7. If and B ar consan vcors, hn ns: Soluion: L, B r xˆ Q8. For h funcion yb yˆ xb ˆ y xb B r 6 f, h rsidu a h pol is (your answr should b an ingr). ns: Soluion: pol is of ordr so rsidu is d 6 d. = 4 Q9. Th dgnra ignvalu of h marix 4 4 is (your answr should b an ingr) ns:,5,5 4 4 (4 ) 5 4 5 = (4 )(5 ),5,5. Q. Th numbr of disinc ways of placing four indisinguishabl balls ino fiv disinguishabl boxs is. ns: Soluion: 5 4 C 4 = ways fiiks, H.No., G.F, Jia Sarai, Nar IIT, Hau Khas, Nw Dlhi 6 Phon: 6865455/+9 98745498 8 B/6, Jia Sarai, Nar IIT Hau Khas, Nw Dlhi 6 Email: fiiks.physics@gmail.com 7
GTE-4 Q. Th uni vcor prpndicular o h surfac x y a h poin (,, ) is xˆ yˆ ˆ xˆ yˆ ˆ xˆ yˆ ˆ ns: Soluion: L f x y f xxˆ yyˆ ˆ f xˆyˆˆ xˆ yˆˆ nˆ a,, f xˆ yˆ ˆ Q. Th marix i is i orhogonal symmric ani-symmric Uniary uniary uniary I Q. Th valu of h ingral whr C is h circl 4, is C d i i 4 i 4 i ns. Soluion: Pol Rsidu Similarly, Rs i i m whr m,,,... i,rs I i 4 i i fiiks, H.No., G.F, Jia Sarai, Nar IIT, Hau Khas, Nw Dlhi 6 Phon: 6865455/+9 98745498 8 B/6, Jia Sarai, Nar IIT Hau Khas, Nw Dlhi 6 Email: fiiks.physics@gmail.com 8
d y Q4. Th soluion of h diffrnial quaion y, subjc o h boundary condiions d y and y is cos sin cosh sinh cos sin cosh sinh ns: Soluion: y c c D D pplying boundary condiion c c and yc c y c, c y cosh sinh y Q5. Considr a complx funcion f samns is corrc? ns.: Soluion: For f has simpl pols a and f has scond ordr pol a f has all simpl pols GTE-5 cos f has infini numbr of scond ordr pols f cos h n ordr pol a n f a x lim fini and. Which on of h following fiiks, H.No., G.F, Jia Sarai, Nar IIT, Hau Khas, Nw Dlhi 6 Phon: 6865455/+9 98745498 8 B/6, Jia Sarai, Nar IIT Hau Khas, Nw Dlhi 6 Email: fiiks.physics@gmail.com 9
lim f fini is a simpl pol. lim lim cos lim.cos. cos sin lim cos sin fini f has scond ordr pol a Q6. Th valu of 6d is (upo on dcimal plac) ns.:. 4 d d d Soluion: 6 x Q7. If f x and gx x x, hn f and g ar diffrniabl vrywhr f is diffrniabl vrywhr bu g is no g is diffrniabl vrywhr bu f is no g is disconinuous a x ns. Soluion: f ( x) x is diffrniabl bu gx ( ) x x is no diffrniabl. x gx ( ) x x x ; x ; x lim g xh x h Lf hand Limi h x h fiiks, H.No., G.F, Jia Sarai, Nar IIT, Hau Khas, Nw Dlhi 6 Phon: 6865455/+9 98745498 8 B/6, Jia Sarai, Nar IIT Hau Khas, Nw Dlhi 6 Email: fiiks.physics@gmail.com
Righ hand Limi lim h lim lim g xh g x h h ho x h g x h x h Q8. Considr w f ux, y ivx, y on of h following opions is NOT corrc? u x, ysaisfis Laplac quaion in D v x, ysaisfis Laplac quaion in D ns.: f o b an analyic funcion in a domain D. Which f d is dpndn on h choic of h conour bwn and in D can b Taylor xpndd in D Soluion: w f( ) uxy, ivxy, o b an analyic funcion in a domain D, f ( d ) is indpndn on h choic of h conour bwn and in D. for Q9. Th Havisid funcion is dfind as H and is Fourir ransform is for givn by i /. Th Fourir ransform of H / H / is ns.: sin cos sin H f f h d i Soluion: For a funcion h, H f i fiiks, H.No., G.F, Jia Sarai, Nar IIT, Hau Khas, Nw Dlhi 6 Phon: 6865455/+9 98745498 8 B/6, Jia Sarai, Nar IIT Hau Khas, Nw Dlhi 6 Email: fiiks.physics@gmail.com
For h, Fourir Transform is i f H f Shifing horm For i i i i i i h h i i sin. Th Fourir ransform of H / H / Q. funcion y saisfis h ordinary diffrnial quaion m,,,,... Considr h four samns P, Q, R, S as givn blow. m y y y, whr P: m and m ar linarly indpndn soluions for all valus of m Q: m and m ar linarly indpndn soluions for all valus of m R: ln and ar linarly indpndn soluions for m S: m and ln ar linarly indpndn soluions for all valus of m Th corrc opion for h combinaion of valid samns is P, R and S only P and R only Q and R only R and S only ns.: Soluion: m y y y y y m y d m,,,,... x, D dx If m ; y y D D D y D DD y D m y y c cx D m y c c ln mx c c mx R is corrc. c c mlog mlog fiiks, H.No., G.F, Jia Sarai, Nar IIT, Hau Khas, Nw Dlhi 6 Phon: 6865455/+9 98745498 8 B/6, Jia Sarai, Nar IIT Hau Khas, Nw Dlhi 6 Email: fiiks.physics@gmail.com
or if m, m y c cosh mlog ic sinh mlog x m GTE-6 dy Q. Considr h linar diffrnial quaion xy. If y a x, hn h valu of y a dx x is givn by ns.: Soluion: dy xy dy xdx dx y x ln y ln c y c x / If y a x c y x /. Th valu of y a x is givn by y Q. Which of h following is an analyic funcion of vrywhr in h complx plan? * ns.: Soluion: xiy x y ixy u x y and v xy Cauchy Rimann quaions u v v u x, y x y x y saisfis. Q. Th dircion of f ns.: ˆj kˆ 5 Soluion: ˆ ˆ ˆ f x y i xjk ˆj kˆ 5 for a scalar fild f x, y, x xy a h poin,, ˆj kˆ 5 f ˆj kˆ nˆ f 5,, ˆj kˆ 5 P is fiiks, H.No., G.F, Jia Sarai, Nar IIT, Hau Khas, Nw Dlhi 6 Phon: 6865455/+9 98745498 8 B/6, Jia Sarai, Nar IIT Hau Khas, Nw Dlhi 6 Email: fiiks.physics@gmail.com
Q4. priodic funcion f xof priod is dfind in h inrval x ns.: Soluion:, x f x, x Th appropria Fourir sris xpansion for f 4 f x sin x sin x/ sin 5x 4 f x sin x sin x/ sin 5x 4 f x cos x cosx/ cos5x 4 f x cos x cosx/ cos5x f x,, x x L f xa a cos nxb sin nx n a f x dx n n x / 5... / 5.. / 5... / 5... a f x dx dx dx x x This can also b sn wihou ingraion, sinc h ara undr h curv of o is ro. a f x cos nxdx n!is f x bwn an cos nxdx cos nxdx b f x sin nxdx n sinnx sinnx n n fiiks, H.No., G.F, Jia Sarai, Nar IIT, Hau Khas, Nw Dlhi 6 Phon: 6865455/+9 98745498 8 B/6, Jia Sarai, Nar IIT Hau Khas, Nw Dlhi 6 Email: fiiks.physics@gmail.com 4
bn sin nxdx sin nxdx cosnx cosnx n n n bn n n n n n n n n If n is vn b and If n is odd n b n 4. n 4 Thus Fourir sris is f x sin x sin x sin 5 x... 5 GTE-7 d Q5. Th conour ingral valuad along a conour going from o along h ral axis and closd in h lowr half-plan circl is qual o.. (up o wo dcimal placs). ns. : Soluion: d dx d C x C Pols, i i is insid C Rs ilim i i i i ii i dx i i x (Sinc hr w us lowr half plan i.. w ravrsd in clockwis dircion hnc w hav o ak i ) fiiks, H.No., G.F, Jia Sarai, Nar IIT, Hau Khas, Nw Dlhi 6 Phon: 6865455/+9 98745498 8 B/6, Jia Sarai, Nar IIT Hau Khas, Nw Dlhi 6 Email: fiiks.physics@gmail.com 5
Q6. Th cofficin of ikx in h Fourir xpansion of u x sin x for k is 4 4 ns.: Soluion: W can wri sin x Hnc, sin x i x i i x i x Sinc, k, hnc 4 i x Hnc sin ikm x 4 ikx c k 8 sin x dx ikx ikx ikx ikx ikx dx dx dx 8 ikx ikx dx dx dx 8 Th firs wo ingrals ar ro and h hird ingral has h valu. Thus c k 8 4 Q7. Th imaginary par of an analyic complx funcion is fiiks, H.No., G.F, Jia Sarai, Nar IIT, Hau Khas, Nw Dlhi 6 Phon: 6865455/+9 98745498 v x, y xy y. Th ral par of h funcion is ro a h origin. Th valu of h ral par of h funcion a i is... (up o wo dcimal placs) ns. : Soluion: Th imaginary par of h givn analyic funcion is v x, y xy y. From h Cauchy Rimann condiion 8 B/6, Jia Sarai, Nar IIT Hau Khas, Nw Dlhi 6 Email: fiiks.physics@gmail.com 6
v u x y x Ingraing parially givs, u x y x x g y From h scond Cauchy Rimann condiion u v, w obain y x dg y dy Hnc, y g y y c u x y x x y c Sinc h ral par of h analyic funcion is ro a h origin. Hnc c c Thus ux, y x x y Thus f x x y ixy y Thus h valu of ral par whn i x y, ha is and is. Q8. L X b a column vcor of dimnsion n wih a las on non-ro nry. Th ns. : Soluion: numbr of non-ro ignvalus of h marix M T XX is n n L a X T hn X... a... fiiks, H.No., G.F, Jia Sarai, Nar IIT, Hau Khas, Nw Dlhi 6 Phon: 6865455/+9 98745498 8 B/6, Jia Sarai, Nar IIT Hau Khas, Nw Dlhi 6 Email: fiiks.physics@gmail.com 7
Hr X is an n column vcor wih h nry in h i h row qual o a. vcor having nry in h i h nry in h i h Hnc column qual o a. Thn row and i h column qual o T XX a.......... ih row......... ih row T X is a row T XX is an n marix having h Sinc his marix is diagonal is ignvalus ar a,,.... Hnc h numbr of T nonro ignvalus of h marix XX is. dy, y... (up o wo dcimal placs) ns.:.5 Th givn diffrnial quaion is a linar diffrnial quaion of h form dy p xy cos x dx Q9. Considr h diffrnial quaion y an x cosx dx. If y Ingraing facor pxdx is Thus ingraing facor an x dx ln sc x I F sc x Thus h gnral soluion of h givn diffrnial quaion is I is givn ha y sc x sc xcos xdxc ysc x x c -(i) y. Hnc sc c c fiiks, H.No., G.F, Jia Sarai, Nar IIT, Hau Khas, Nw Dlhi 6 Phon: 6865455/+9 98745498 8 B/6, Jia Sarai, Nar IIT Hau Khas, Nw Dlhi 6 Email: fiiks.physics@gmail.com 8
Thus h soluion saisfying h givn condiion is ysc x x y Thus h valu of x sc y is / / y 5 sc / 6 x fiiks, H.No., G.F, Jia Sarai, Nar IIT, Hau Khas, Nw Dlhi 6 Phon: 6865455/+9 98745498 8 B/6, Jia Sarai, Nar IIT Hau Khas, Nw Dlhi 6 Email: fiiks.physics@gmail.com 9