2006 The Institution of Engineers (India) A.M.I.E.T.E University QUestions

[sams.raghu]

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.

Answer any FIVE Questions out of EIGHT Questions.

Each question carries 16 marks.


Q.2 a. Consider the function f (x, y) defined by

Find and .

Is differentiable at (0, 0)? Justify your answer. (8 )

b. Find the extreme values of subject to the constraints and . (8 )

Q.3 a. Find all critical points of and determine relative extrema at these critical points. (8 )

b. Find the second order Taylor expansion of about the point . (4)

c. Change the order of integration in the following double integral and evaluate it : . (4)

Q.4 a. Solve the differential equation . (4)

b. Solve the differential equation . (6)

c. Find the general solution of the differential equation . (6)

Q.5 a. Find the general solution of the differential equation (8 )

b. Find the linear Taylor series polynomial approximation to the function about the point (1, 2). Obtain the maximum absolute error for the polynomial approximation in the region , . (8 )

Q.6 a. Find the general solution of the differential equation . (9)

b. Show that the eigenvalues of a Hermitian matrix are real. (7)

Q.7 a. Using Frobenius method, find two linearly independent solutions of the differential equation . (10)

b. Solve the following system of equations by matrix method:(6)

Q.8 a. Express the polynomial in terms of Legendre polynomials. (8 )

b. Let be the Bessel function of order . Show . (8 )

Q.9 a. If A is a diagonalizable matrix and f (x) is a polynomial, then show that f (A) is also diagonalizable. (7)

b. Let. Find the matrix P so that is a diagonal matrix. (9)