Note : Attempt all questions. The marks for each question are indicated against it. Symbols have their usual meanings. You may use log tables.

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1 No. of Printed Pages : E-41 BACHELOR OF SCIENCE (B.Sc.) N-1 Term-End Examination C'") CO June, 2012 PHYSICS PHE-4 : MATHEMATICAL METHODS IN PHYSICS-I Time : 11/2 hours Maximum Marks : 25 B.Sc. EXAMINATION, PHE-4 : MATHEMATICAL METHODS IN PHYSICS-I PHE-5 : MATHEMATICAL METHODS IN PHYSICS-II Instructions : 1. Students registered for both PHE-4 & PHE-5 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. 2. Students who have registered for PHE-4 or PHE-5 should answer the relevant question paper after entering their enrolment number, course code and course title on the answer book. Note : Attempt all questions. The marks for each question are indicated against it. Symbols have their usual meanings. You may use log tables. 141 PHE-4 1 P.T.O.

2 1. Attempt any three parts : 4x3=12 (a) Determine the angle between the vectors A = 6i 4j+ 2kand B = k (b) Calculate the volume of the parallelepiped formed by the vectors : -> A A A A = i+2j k - A A B = j + k - A A C = (c) Determine the unit vector normal to the surface x2 + y2 + z2 =3 at (1, 1, 1). (d) A frictionless bead slides down a vertical helix of radius R such that its position vector at time t is given by : 1 2 A - (t) = a (coscot +sinwt j) 2 gt k Determine its velocity and acceleration. (e) The cylindrical coordinates u1 = p, u2 = 4, u3 = z are related to the Cartesian coordinates x, y and z as follows : x = p cos (I) y=p sin st, z= z Show that the cylindrical coordinate system is orthogonal, i.e., gij (i#j)= 0 for all i and j. PHE-4 2

3 2. A force acting on a particle is given by : 5 A A F = kx i ky j Calculate the work done in moving the particle from (1, 1) to (4, 4) along the path x=y. OR Calculate the surface integral of a vector -4 A A A A= x i +2y j +3zk over the surface of a sphere of radiul, 2 by using Gauss's divergence theorem. 3. The average number of calls received in a BPO is 3 4 per minute. Calculate the probability that not more than one call is received during one minute OR A continuous random variable X has the probability distribution 1 P(x) Tr(1+x2 ) 1 < x < 1 = 0 otherwise Calculate its mean. 4. The extension in a material is measured as a 5 function of load in appropriate units and is given by : Load (x) Lxtension (y) Obtain the least square fit to the data OR PHE-4 3 P.T.O.

4 Derive an expression for the mean of the normal distribution n (x; x, o-) = 1 e xp [- 1 X-x 271- Q 2 o- - 2 PHE-4 4

5 f479. 1"11C1ch ( A1%71111 t.m.1 ITNIT, (41r6W r ii 1:11%7W1.- : " ru fearce- I ART : 11/2 IEFQ- 31fTWM7 37T : 25 t.71-1*. tiftag, th : %tif9-4- iiruirilti farvert4 "cr4 TITRW4-5 : uli Pf41 : 1. OT7 tr7:7-ae_4 3 rir<21wr1 f-c-n vi-4-7tff 0.1/ siyrly dot 31T-37-7T drh gr V71 31P-747* Iliqq5#' 41 ff21t (110M 41 7r4 ql`h- L/1(17 r(1{ich #.7-ae,4 qt ftrg kfch ric' ci Mq-1(1,1 art, dru-syrtf/w arrit- 37-P-7:17* (ingshsi RT1T (INd x'llth-t(11(h c I 7: ch I TIF:IT 51Y-f aiw ufw TE-474 rqg TR t q/41/-q 3-T 2 f 311V Wig TITI"ITT W/- fich g/ PHE-4 5 P.T.O.

6 1. m--?e a c : 4x3=12 (a) 1: rod TifTA.11.1 chlui : A= 6i 4j+ 2k AT B = 2) k (b) f9r-iffftid Triqt : -) A A A A=i+ 2 j k j+k c- =i- i (c) (1, 1, 1) -ER x2 +y2 +z2 =3 Trkvr fic.t I (d) ) 71fuT ict) ITT t I TITRE t ITT H1c-I1 TItVE cf t : 1-4r (t) = a (coscot 1 + simot j) 2 gt` k Int (41 ccrul,4 I (e) AW11. f#71it u1= p, u2 = u3 = z3 --Ed-T4 x, y, z ca-cr F-P irrifigcl 44g t : x = p cos 41 y= p sin (1) z = z f-- TQ I A---dt f ri es t, qi-11 : gtj (i#j)=0,tfift 3117 ji kit; I PHE-4 6

7 =bur WIT Tr{ c141 A-F r14ircirod : A kx i ky 9. x=y 72T t 31-1t7T Ris (1, 1) -4 (4, 4) Clch f-*--zrr TRIT *14 Hricnrcid TrkT ' -514z Wt-lc "F:112q1 A A A= x i +2yj +3zk T ti+ilchcl LIRchroci ct-) I 3. -) 'clf at ark 3th:M 3 4s,e4 4 t 1 IRIchdr LIRchfild ch rct) 3-1f9"W ebto 31-M i 31'4-4T -crr9,11txt (11d-I rOdt: P(x) 1 1 < x <1 Tr(l+x2 ) T -Ppzi 4 rich fri d cta I 4. r) Liqiq r(m-rui 31-qc-fT*Ti'zifTl 4TI7t tboit 5 tc\ \Midi t 3t1{1:14--IrCirOdt : ITT{ (x) '0R-do' (y) )'er r -ri FT I PHE-4 7 P.T.O.

8 It q n (x; x, o-) 1 tap 1 x-x 2 V2 Tr o- 2 Trit7f -uf-act, 0-AccF- 0 I PHE -4 S

9 PHE-5 BACHELOR OF SCIENCE (B.Sc.) Term-End Examination June, 2012 PHYSICS PHE-5 : MATHEMATICAL METHODS IN PHYSICS-II Time : 11/2 hours Maximum Marks : 25 Note : Attempt all questions. The marks for each question are indicated against it. Symbols have their usual meanings. 1. Attempt any three parts : 4x3=12 (a) Show that the following equation is exact (b) and solve it : ey dx + (xey + 2y) dy =0 Obtain the integrating factor of the following equation and solve it : dy -r I y = 3x dx (c) Determine all the first and second order partial derivatives of the function f (x, y) = x2-5xy3 (d) Separate the following PDE into two ODEs. (E p ) a (x, t)= (x, t) at a x PHE-5 9 P.T.O.

10 (e) Obtain the particular integral of d2y + dx + 2y = 3x dx2 2. Expand the square wave V(x) given by : 6 V(x) = 0 - rr < x < 0 Vo 0< x < ir in Fourier series. OR Obtain the singular point of the following ODE and specify its nature : dy ' dy _ 2 + xy u x d x2 dx Determine the indicial equation and its roots. 3. The electric potential in a power transmission line 7 along the x axis satisfies the equation : a 2 V RC av.x2 at Using the method of separation of variables, solve this equation for V under the following conditions : (x, t) = 0 at x = 0 (x, 0) = Vo at t = 0 (x, t) = 0 at x = L OR PHE-5 10

11 A particle of mass m falls freely under gravity in a liquid that offers a resistive force proportional to its velocity : d X Tres -Y dt Set up the equation of motion and solve it. PHE-5 11 P.T.O.

12 c1011 f--11cich *.711:11% i-1,1ic1 TR* fad * : iiruidlq fear-at-ii Wig : 11/2 '11J2 alft-177:1 aft : 25 Fit# so-i cal Tr v-d-vt- 34-q4 \vpip-q 3-fw afr- TITO 1. c 2I7 T Wt 4x3=12 (a) f< f- tqcf W-Ichkui 7P194 t 3f err (b) (=IA : ey dx + (xey +2y) dy = 0 F-14-1 itgki ti4-1chtuf "WI t14-i chei-t 111Th*fair. dy 1 : = 3x dx x (c) t 'P.m cfli 3iffkrW 3rd c 1 I cr)- : f (x, y) = x2-5xy3 (d) 31-ftrT a:rtit -q-ittp-trot 314WF -1111chuif mpt-ff-ci : p ---)kij(x,t)= rn0 (x, t) (E at ax PHE-5 13 P.T.O.

13 (e) *Of r fad chC*1 d2 y dy + 2y = 3x dx2 dx 2. V(x) fr IFF7 9t-IR 71FT c : (-1 44 ch(uf V(X) = 0 -Tr < x < 0 Vo 0 < x <7r 3111WT x d2y 2 dy + v - 0 dx2 dx x- fqfq7 f4s fiqfftff ch'k, 71-ff c qttitu 011. I -0-wr 3. arlt-vr 1 ti 1 I oirr 1--tp fq-qa 7 F-P-Ircirtgc-1 r t: a2 v axe _ RC a v at liqq-chut PART i w-fl cbtui art (gcr atrti-tff v -9-ru : (x, = o, x = o (x, = v0, t = o (x, = o, t = 3111"41' PHE-5 14

14 « im am' f.).6u1 Tq titcatchtfur chrur fi r6rt I w:rrt 6,4( : alit TrfdN-.i(1 dx Tres 7 dt +I cf ( ti{ ci.; PHE-5 15

BACHELOR OF SCIENCE (B.Sc.) Term-End Examination June, B.Sc. EXAMINATION CHE-1 : ATOMS AND MOLECULES AND CHE-2 : INORGANIC CHEMISTRY

No. of Printed Pages : 15 00881 CHE-1 BACHELOR OF SCIENCE (B.Sc.) Term-End Examination June, 2010 CHEMISTRY CHE-1 : ATOMS AND MOLECULES Time : 1 hour Maximum Marks : 25 B.Sc. EXAMINATION Instructions :