Mathematical Physics I — DU B.Sc. (Hons) Physics Previous Year Question Papers

Name of the Department Question Paper Set No B.Sc. Hons.-(Physics)_NEP: UGCF-2022 Mathematical Physics-I Attempt five questions in total. Q1. Answer any six of the following: (a) A temperature field in a room is given by I(x,y,z) = 20 -x2 -2y2 -z2 (in °C). Find the direction of maximum temperature increase at the point (1, -1, 2). What is the rate of increase in that direction? (d) Find JJ i. i? ds for the vector field j?(x,y,z) = z i + y i + x fi , over the unit sphere x2 + y2 + z2 = 1. (e) Given the vectors a = (3,4) and a = (5, -2), compute their dot product before and after rotating the axes counterclockwise by 450, and verify invariance. ® Find the Wronskian of the set {sZ7i x, 2 cos x, 3 sZ7t% + cosx }, and determine whether the set is linearly dependent or independent. (g) From the probability distribution Find the v.alue of k, and evaluate E(X). (b) If A+ is an irrotational vector field, show that A+X F is solenoidal, where F = xi + Q3. (a) Solve by the Method of variation of parameters: (b) Solve the differential equation: Q4. (a) Solve by Method of undetermined Coefficients: (b) A resistor of A = 2000 n and a capacitor of capacitance C = 5 X 10-6 F are connected in series to a 100 Volts DC source. What is the current J(I) as a function of time for this RC charging circuit, given that I = 10-2 A at the time the switch is closed (I = 0). Also, determine the initial charge on the capacitor at I = 0. 9 Q5. (a) Verify Green's theorem in the plane for Sc (xy +y2) cZx + %2 czy where c is the closed curve of the region boundedby y = x and y = x2. 9 ® Consider the scalar function I(%, y,z) = z2 defined on the tetrahedral domain enclosed by the three coordinate planes and the plane x + y + z = 1. The vertices of this tetrahedron are (0,0,0), (1,0,0), (0,1,0), and (0,0,1). Compute the volume integral of T over this region. 9 Q6. (a) Obtain the expression for the mean and variance of the Binomial distribution. 8 @) In a radioactive decay process, the number of decays per minute is Poisson-distributed with a mean of4. Find the probability that exactly 3 decays occur in one minute. 5 (c) Find the scalar potential function dy for i = y2i + 2xy i - z2ft. 5