The Institution of Engineers,India 2005 A.M.I.E.T.E Electronics & Communication Engineering Code: A-01/C-01/T-01 - Subject: MATHEMATICS-I - Question Paper

Code: A-01/C-01/T-01 Subject: MATHEMATICS-I
Time: three Hours Max. Marks: 100

NOTE: There are 11 ques. in all.

• ques. one is compulsory and carries 16 marks. ans to Q. 1. must be written in the space given for it in the ans book supplied.
• ans any 3 ques. every from Part I and Part II. every of these ques. carries 14 marks.
• Any needed data not explicitly given, may be suitably presumed and said.


Q.1 select the accurate or best option in the following: (2x8)

a. The value of limit

(A) equals 0. (B) equals .
(C) equals . (D) does not exist.

b. The total differential of the function at the point (1, 1) is
(A) dx + dy. (B) .
(C) . (D) .

c. Let then equals

(A) 0. (B) two u.
(C) . (D) .

d. The value of so that is an integrating factor of differential formula is

(A) . (B) - 2.
(C) . (D) .

e. If method of undetermined coefficients is used for finding a particular integral of differential formula then the solution to be tried is

(A) . (B) .
(C) . (D) .

f. Let A be a non-singular matrix. Then the inverse of the matrix

(A) is symmetric. (B) is skew – symmetric.
(C) does not exist. (D) equals .

g. The linear transformation represents

(A) reflection about -axis.
(B) reflection about -axis.
(C) clockwise rotation through angle .
(D) Orthogonal projection on to -axis.

h. For the Legendre’s polynomial of order n, then equals

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

PART I
ans any 3 ques.. every ques. carries 14 marks.

Q.2 a. If where c is a constant, then obtain in terms of x, y, z. (6)

b. Expand in powers of x and as far as third degree terms using Taylor’s series expansion. (8)

Q.3 a. Let u and v be 2 functions of x, y. Show that where denotes the Jacobian of u, v with respect to x, y. (6)

b. obtain points of local minima and local maxima and saddle points for the function . (8)

Q.4 a. Solve the subsequent system of equations by using the Cramer’s Rule
(7)

b. obtain the value of so that the vectors (1, 2, 9, 8), are linearly independent. (7)

Q.5 a. Prove that similar matrices have the identical eigenvalues. Also provide the relationship ranging from the eigenvectors of 2 similar matrices. (6)

b. obtain the eigenvalues and the eigenvectors for the matrix (8)

Q.6 a. If a matrix obtain the matrix using Cayley Hamilton theorem. (8)

b. Let a matrix A have eigenvalues and matrix obtain
(i) determinant of matrix B.
(ii) trace of matrix B. (6)


PART II
ans any 3 ques.. every ques. carries 14 marks.

Q.7 a. change the order of integration in integral and then evaluate the integral. (7)

b. obtain the quantity of the solid bounded by the surfaces z = 0, and . (7)

Q.8 a. Solve the differential formula by transforming the formula by substitution . (7)

b. obtain the differential formula whose general solution is where a, b are arbitrary constants. (7)

Q.9 a. Using method of variation of parameters, show that A can always be determined so that is a solution of the differential formula . (7)

b. obtain the general solution of the formula . (7)

Q.10 a. Solve the differential formula . (6)

b. Using Frobenius method, show that the differential formula has a solution near the origin. Suggest the form of second solution linearly independent of . (8)

Q.11 a. Show that under change of dependent variable y described by the substitution the Bessel’s formula of order becomes . Hence show that for large values of t, the solutions of Bessel’s formula are defined approximately by the expression of the form . (7)

b. Using Rodrigues formula for Legendre polynomials show that

where f is any function integrable on interval . Hence show that (7)

Earning:   Approval pending.