PHE-05 : MATHEMATICAL METHODS IN PHYSICS-II Instructions : (i)

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1 No. of Printed Pages : 12 B.Sc. Examination June, 2014 PHE-04 : MATHEMATICAL METHODS IN PHYSICS-I PHE-05 : MATHEMATICAL METHODS IN PHYSICS-II Instructions : (i) Students registered for both PHE -04 and PHE-05 courses should answer both the question papers in two separate answer books entering their enrolment number, course code and course title clearly on both the answer books. (ii) Students who have registered for PHE-04 or PHE-05 should answer the relevant question paper after entering their enrolment number, course code and course title on the answer book. lt. trftvr T4, 2014 : ORO 4 t4.cnsi 41: (i) evo tome ?- t7w.e. -05 q livelthwy * q?.ii sgrarrgt- drff ard77-37-rv.itv sipwari 31RT aisthwiw, zuggsbq ca5 R2IT tileshif 3771 ATilWf *' I (ii) 177:7V.i T 7d. e. -05 f47:17 * it ff, NO. 3-e W * URT,,art jrawt aprat arim--47*, tfavismi las ffeir victershy 7N- typh-f1/y7 fkrwi- **. PHE-04/05 1 P.T.O.

2 PHE-04 BACHELOR OF SCIENCE (B.Sc.) Term-End Examination June, 2014 PHYSICS PHE-04 : MATHEMATICAL METHODS IN PHYSICS-I Time : 11 hours Maximum Marks : 25 2 Note : Attempt all questions. The marks for each question are indicated against it. Symbols have their usual meaning. You may use a log table or calculator. 1. Answer any three parts : 3x4=12 (a) > --> Three vectors a, b and c are given by >!1/4 A A a =2i + 4 j + k A A A b =7i +4j +2k -> A A A c = j + k Determine the angle between the vectors > > --> ( b a ) and ( c a ). (b) Obtain the unit vector normal to the surface of a cone z2 = 4 (x2 + y2) at the point (1, 0, 2). PHE-04 2

3 (c) The spherical polar coordinates u1 = r, u2 = and u 3 = are related to the cartesian coordinates x, y, z as follows : x = r sin 0 cos 4) y = r sin 0 sin 4) z = r cos 0 Show that the spherical polar coordinate system is orthogonal. -> (d) Compute ( A x B). x D) for the vectors A A = i + 2 j A A B = 3 i + 2 j -> A A A C =2i +3j +4k -4 A A A D =6i 7j +2k -4 (e) The position vector r (t) of a particle of mass m is moving on a curve given by > A A A r (t) = 3t` i + 6t j +7k Determine the angular momentum of the particle about the origin. 2. Obtain the work done by the force -4 A A A F =zi +xj +yk in moving a particle along the curve x = cos t, y = sin t, z = 3t for (0 t < 2n). OR PHE-04 3 P.T.O.

4 State Gauss's divergence theorem and use it to --+ evaluate the surface integral A. d S over a S spherical surface of radius 2 units, for A A A = 7xi -zk The probability of a successful brain operation is 0.2. What is the probability that at least one person out of 10 undergoing the operation will survive? OR The probability distribution for a continuous random variable X lying between 0 X < co is e-x. Calculate the mean < X > and the variance a Resistance of a coil at different temperatures is measured. The data is given below : T ( C) : R (Q) : Obtain the equation R = a + bt for the best fit. 5 OR Obtain the value of E(X2) for the Poisson distribution : 5 3 p(x; m) = -m e x! mx. PHE-04 4

5 1 qui t1inrn (*.7111:11.) W4, gllda* Id iii ift.7w1.-04 : Tfifttt As it 1 ffhel :1 Eiu2 361Wffir : 25 2 We: Frsti W# w 7 a. 4 a*.i jq Hlif 1TR. g / 37-4e 3m-4 apf ff aim C/N/ 29T17* z1t 4"-- e?" meng friod 1. Wē ITTTII*TffK tra : 3x4=12 () 144-iili1 sii 417 Vitg 7>t, 13> -+ A AA a =2i +4j + k - + A A A b =7i +4j +2k - -> A AA c =4i +3j + k Tftt41 (-1)-> - a)3 (-c* - 71> ) fqui s i *e4r I N) its (1, 0, 2) z2 =' 4(x2 + y2) t,ct) Tff *rp4r I PHE-04 5 P.T.O.

6 (TT) ill d14 Vrai f4ti-4ul = r, u2 = 0 IT u3 = 3-11T 1-47TiO x, y, z t9. ')01 : 4rz 1 (1:1) TrW x = r sin 0 cos (1) y = r sin 0 sin 4) z = r cos 0-3 A A A = i +2j ftt4dtzr gtzt AkkTk - A A B =-3i + 2 j -3 A A A C =2i +3j +4k --> A A A D =6i 7j +2k * x ( c* x i)*) Lachrocilf* I (3,) i'4wi arf-qr TA RR c m aic -17r Trr-qr /7>(t)CIHRifigo : ---> A A r (t) = 3t + 6t j + 7 k ito-f4s*npi f wr.1) ullti A'rr-A-R 2. R"'" WTI o4i am! PHICiRgo : -3 A A A F =zi +xj + y k WIT -0 (0 t < 27r) x = cos t, y = sin t, z = 3t pia 14)til Trm WTI 3T2T4T PHE-04 6

7 si4ci WiR 'PAR Hcni ctit; "R1,3-04i 2 4.nii ctit ) ITIOtzt 7 A= 7x1 - z k NR Wilch0 A. ds S ~Rchrrlcl*rrAR I 5 3. TIMfRF 'i atichfq.lii chat 0.2 t I ITit atfq-1-4r 10 ftall xr{ %en 7R, t * waa 51Acbc11 TEIT )411? 312T1T -Pfd" # I 4tichi NiNchai te'q ex t I 1:1TuEr < X> Nkitul a sirch rrli*ar I 3 4. WilA trk 174 TR'crW t4"-4t * 31%tT T ( C) : R (a) : aft-*-0 kiachtul R = a + bt 4 317iftff *e4r I 5 arateit 1:4TN1 '1 p(x; m) = APT WAR e -m mx *rci E(X2) 117 x! 5 PHE-04 7

8 BACHELOR OF SCIENCE (B.Sc.) Term-End Examination June, 2014 PHE-05 PHYSICS PHE-05 : MATHEMATICAL METHODS IN PHYSICS-II Time : 1-1 hours 2 Maximum Marks : 25 Note : Attempt all questions. The marks for each question are indicated against it. Symbols have their usual meaning. 1. Answer any three parts : 3x4=12 (a) Show that the solution of the ODE : (y + 4) y + x = 0 is a family of concentric circles centred at (0, 4). (b) Solve the ODE : y" + 3y + 2y = ex. (c) If z = In (x2 + cy2), what should be the value of c so that z satisfies the equation axz. axe a 2 z + o =0 (d) Solve the initial value problem : y" + 5y + 6y = 0, y(0) = 1, y'(0) = 4 (e) Is the periodic function f(x) = x, 1 < x < 1 f(x + 2) = f(x) odd or even? Obtain its Fourier series expansion. PHE-05 8

9 2. Answer any one part : (a) Determine the roots of the indicial equation around the origin for the following differential equation : 6 x2y, xyl x 2 y = 0 9 Also obtain the recurrence relation. (b) A conductor of resistance R and inductance L is connected in series with an alternating voltage source E = E0 sin wt. Show that the current i(t) in the circuit is given by i(t) = E sin (wt - 0) 11R where 0 = tan-l(wl/r). 3. Answer any one part : (a) Obtain the Fourier series of the periodic function 0, if -T/2 <t<0 E(t) = E 0 sin cot, if 0 < t < T/2 where T = 27c/o.). (b) The steady-state temperature distribution, T(x, y), of a rectangular plate is governed by the following equation : a2t(x, y) a2t(x, 3r) = 0; ax 2 ay 2 0<x<L;0<y<B Determine T(x, y) if the boundary conditions are (i) T(0, y) = 0, r(l,y) =0, 0<y<B ax T(x, 0) = 0, T(x, B) = To, 0 < x < L PHE-05 9

10 411.1 wiffitnit, 2014 ftsidi : 11 2 Oft oil 1:11.7Ās.t -05 : * Ai At' *: FS?, -WV e/f4v / SR* 1R4f iff Ws/ f0. Trdie 370 foini-d, Wit' ff : ik4i 1-1Tril art tf* : 3x4=12 f*-47i orgd Trrtm-Err 31-4-*-F cnv.,1 (y+4)y'+x= 0 7" 1:4-4P4 r t "k4k (0, - 4) TRt I P-14-1R-iRgi,i-mrt-aT 31-4T-a TO. : y" + 3y' + 2y = ex (TO 71R z = In (x2 + cy2) t, c c fir ITT174 qikr -N44 z PHR-160 ki41.nkui kige : n ar. (Er) srrtrmitii err : y" + 5y' + 6y = 0, y(0) = 1, y'(0) = 4 Atil4R I PHE-05 10

11 ;Err f(x) = x, 1 < x < 1 f(x + 2) = fkx) Nor zr1f4erq? Flg11 1#17 NRR 3ra:r *1-NR I 2. ITT *38T tr-4 () x = 0 ( ) ' -Erri44-T 31-4-*-F lzfrdiq. ' TO Ta" *rl'a : xyt + (x2 y = 0 9 F(1-1%*-4u 31-FT trrqr (w) -5PAtT R ck T I tl ch 3 k 1 kcb L =-CdT c tii1d E = Eo sin cot * WaT auft 1TFMT Mimi TRIO fkg *1NR -I% tritemi > tim i(t) : i(t) = E sin (cot 0) VR 2 + co2l2 TAT 0 = tan-1(0)l/r)t 3. Qi 7-w 11Fr zuttf* : 144-1R-iRsid c for 11(7 aft Sur 6 7 E(t) = 0, * T/2 <t<0 E 0 sin cot, Nft 0 < t < T/2 7-4 T = 27* t I PHE P.T.O.

12 7i 3TRMTWR titg -if wrzft-3t4r-rr frdttit T(x, y) PIHICirtgcl : Tireq ctaii a2t(x, y) a2t(x, 0; ax2 ay 2 0 <x<l; 0 <y<b T(x, y) Tff mil, zft Trfttp-r sr1w-4tt zi (.10 : (i) (ii) T(0, y) = 0, at(l, y) =0, 0 < y < B lax T(x, 0) = 0, T(x, B) = T0, 0 < x < L PHE ,000

PHE-5 : MATHEMATICAL METHODS IN PHYSICS-II Instructions : 7. Students registered for both PHE-4 & PHE-5 courses

PHE-5 : MATHEMATICAL METHODS IN PHYSICS-II Instructions : 7. Students registered for both PHE-4 & PHE-5 courses No. of Printed Pages : 15 00 (.0 BACHELOR OF SCIENCE (B.Sc.) Term-End Examination June, 2010 PHYSICS PHE- I PHE- : MATHEMATICAL METHODS IN PHYSICS-I Time : 1 1/2 hours Maximum Marks : 25 B.Sc. EXAMINATION,