Time: 3 Hours                                           DECEMBER 2006                                                        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 (A)  0                                                  (B) (C) (D)  does not exist

b.      Let a function f(x, y) be continuous and possess first and second order partial derivatives at a point (a, b).  If is a critical point and , , then the point P is a point of relative maximum if

(A) (B) (C) (D) c.   The triple integral gives

(A)    volume of region T                        (B)  surface area of region T

(C)  area of region T                            (D)  density of region T

d.   If then matrix A is called

(A)    Idempotent Matrix                      (B) Null Matrix

(C)  Transpose Matrix                        (D)  Identity Matrix

e.   Let be an eigenvalue of matrix A then , the transpose of A, has an eigenvalue as

(A) (B) (C) (D) f.    The system of equations is said to be inconsistent, if it has

(A)     unique solution                             (B)  infinitely many solutions

(C)  no solution                                    (D) identity solution

g.   The differential equation is an exact differential equation if

(A) (B) (C) (D) h.   The integrating factor of the differential equation is

(A) (B) (C) xy                                                 (D) i.    The functions defined on an interval I, are always

(A)   linearly dependent                         (B) homogeneous

(C) identically zero or one                    (D) linearly independent

j.    The value of , the second derivative 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 the function is continuous at (0, 0) but its partial derivatives and do not exist at (0, 0).                (8)

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

Q.3     a.   Find the shortest distance between the line and the ellipse .                   (8)

b.   Evaluate the double integral , where R is the region bounded by the x-axis, the line y = 2x and the parabola .                      (8)

Q.4     a.   Evaluate the integral where R is the parallelogram with successive vertices at , , and .      (8)

b.   Show that where is the Bessel function of order.                                                                     (8)

Q.5     a.   Show that .                    (6)

where are the Legendre polynomials.

b.   Find the power series solution about x =2, of the initial value problem .

Express the solution in closed form.                                                                  (10)

Q.6     a.   Solve the initial value problem y(0) = 0, , .              (8)

b.   Solve .                                                                 (8)

Q.7     a.   Show that set of functions forms a basis of the differential equation .  Obtain a particular solution when .                                                         (6)

b.   Solve the following differential equations:

(i) (ii) (2 5 = 10)

Q.8     a.   Let be a linear transformation defined by .  Taking as a basis in determine the matrix of linear transformation.                                                  (8)

b.   If then show that , for .     Hence find .             (8)

Q.9     a.   Examine whether matrix A is similar to matrix B, where , .                 (8)

b.   Discuss the consistency of the following system of equations for various values of  and if consistent, solve it.                                                                                    (8)