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

Time: 3 Hours Max. Marks: 100

NOTE: There are 11 Questions in all.

· Question 1 is compulsory and carries 16 marks. Answer to Q. 1. must be written in the space provided for it in the answer book supplied.

· Answer any THREE Questions each from Part I and Part II. Each of these questions carries 14 marks.

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

Q.1 Choose the correct or best alternative in the following: (2x8)

a. The value of limit

(A) equals 0. (B) equals .

(C) equals 1. (D) does not exist.

b. If then equals

(A) . (B) .

(C) . (D) .

c. The function has

(A) a minimum at (0, 0).

(B) neither minimum nor maximum at (0, 0).

(C) a minimum at (1, 1).

(D) a maximum at (1, 1).

d. The family of orthogonal trajectories to the family , where k is an arbitrary constant, is

(A) . (B) .

(C) . (D) .

e. Let be two linearly independent solutions of the differential equation . Then , where are constants is a solution of this differential equation for

(A) . (B) .

(C) no value of . (D) all real .

f. If A, B are two square matrices of order n such that AB=0, then rank of

(A) at least one of A, B is less than n.

(B) both A and B is less than n.

(C) none of A, B is less than n.

(D) at least one of A, B is zero.

g. A real matrix has an eigenvalue i, then its other two eigenvalues can be

(A) 0, 1. (B) -1, i. (C) 2i, -2i. (D) 0, -i.

h. The integral, n>1, where is the Legendre’s polynomial of degree n, equals

(A) 1. (B) . (C) 0. (D) 2.

PART I

Answer any THREE questions. Each question carries 14 marks.

Q.2 a. Compute and for the function

(6)

b. Let v be a function of (x, y) and x, y are functions of defined by

where Show that . (8)

Q.3 a. Expand near (1, 1) upto 3^rd degree terms by Taylor’s series. (7)

b. Find the extreme value of subject to the conditions and . (7)

Q.4 a. Find the rank of the matrix

(6)

b. Let

be a linear transformation from to

and

be a linear transformation from to .

Find the linear transformation from to by inverting appropriate matrix and matrix multiplication. (8)

Q.5 a. Prove that the eigenvalues of a real matrix are real or complex conjugates in pairs and further if the matrix is orthogonal, then eigenvalues have absolute value 1. (6)

b. Find eigenvalues and eigenvectors of the matrix . (8)

Q.6 a. Find a matrix X such that is a diagonal matrix, where . Hence compute . (8)

b. Prove that a general solution of the system

can be written as

++ where are arbitrary. (6)

PART II

Answer any THREE questions. Each question carries 14 marks.

Q.7 a. Let Recognise the region R of integration on the r.h.s. and then evaluate the integral on the right in the order indicated. (7)

b. Compute the volume of the solid bounded by the surfaces and . (7)

Q.8 a. Let be an integrating factor for differential equation Mdx+Ndy=0 and is a solution of this equation, then show that is also an integrating factor of this equation, G being a non-zero differentiable function of . (6)