Code: A-20

Code: A-01/C-01/T-01 Subject: MATHEMATICS-I

Time: 3 Hours Max. Marks: 100

NOTE: There are 9 Questions in all.

· Question 1 is compulsory and carries 20 marks. Answer to Q. 1. must be written in the space provided for it in the answer book supplied and nowhere else.

· Out of the remaining EIGHT Questions answer any FIVE Questions. Each question carries 16 marks.

· Any required data not explicitly given, may be suitably assumed and stated.

Q.1 Choose the correct or best alternative in the following: (2x10)

a. The value of limit is

(A) 0 (B) 1

(C) (D) does not exist

b. If , the total differential of the function at the point (1, 2) is

(A) (B)

(C) (D)

c. Let then equals

(A) 0 (B) 2u

(C) u (D) 3u

d. The value of the integral , over the domain E bounded by planes x = 0, y = 0, z = 0, x + y + z = 1 is

(A) (B)

(C) (D)

e. The value of so that is an integrating factor of the differential equation is

(A) (B) 1

(C) (D)

f. The complementary function for the solution of the differential equation is obtained as

(A) (B)

(C) (D)

g. Let , , be elements of . The set of vectors is

(A) linearly independent (B) linearly dependent

(C) null (D) none of these

h. The value of for which the rank of the matrix is equal to 3 is

(A) 0 (B) 1

(C) 4 (D)

i. Using the recurrence relation, for Legendre’s polynomial , the value of equals to

(A) 1.5 (B) 2.8

(C) 2.875 (D) 2.5

j. The value of Bessel function in terms of and is

(A) (B)

(C) (D)

Answer any FIVE Questions out of EIGHT Questions.

Each question carries 16 marks.

Q.2 a. Show that for the function , partial derivatives and both exist at the origin and have value 0. Also show that these two partial derivatives are continuous except at the origin. (8)

b. In a plane triangle ABC, if the sides a, b be kept constant, show that the variations of its angles are given by the relation

. (8)

Q.3 a. Find the shortest distance from (0, 0) to hyperbola in XY-plane. (8)

b. Express , as a single integral and then evaluate it. (8)

Q.4 a. Obtain the volume bounded by the surface and a quadrant of the elliptic cylinder , z > 0 and where a, b > 0. (8)

b. Solve the following differential equations:

(i) .

(ii) . (8)

Q.5 a. Solve the following differential equation by the method of variation of parameters.

. (9)

b. Solve . (7)

Q.6 a. Show that non-trivial solutions of the boundary value problem are , where D_n are constants. (9)

b. Show that the matrices A and have the same eigenvalues. Further if , are two distinct eigenvalues, then show that the eigenvector corresponding to for A is orthogonal to eigenvector corresponding to for . (7)