The Institution of Engineers,India 2004 A.M.I.E.T.E Electronics & Communication Engineering 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 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 2 linearly independent solutions of the differential formula . Then , where are constants is a solution of this differential formula for



(A) . (B) .

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



f. If A, B are 2 square matrices of order n such that AB=0, then rank of



(A) at lowest 1 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 lowest 1 of A, B is zero.



g. A real matrix has an eigenvalue i, then its other 2 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
ans any 3 ques.. every ques. carries 14 marks.



Q.2 a. calculate and for the function

(6)

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



where Show that . (8)

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



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



Q.4 a. obtain the rank of the matrix



(6)



b. Let


be a linear transformation from to
and



be a linear transformation from to .

obtain the linear transformation from to by inverting improper 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. obtain eigenvalues and eigenvectors of the matrix . (8)



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



b. Prove that a general solution of the system


can be written as

+ + where are arbitrary. (6)















PART II
ans any 3 ques.. every ques. 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. calculate the quantity of the solid bounded by the surfaces and . (7)



Q.8 a. Let be an integrating factor for differential formula 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)



b. Solve the initial value issue . (8)



Q.9 a. obtain general solution of differential formula . (7)



b. Solve the boundary value issue

. (7)

Q.10 a. Solve the differential formula . (5)



b. Using power series method obtain a 5th degree polynomial approximation to the solution of initial value issue

. (9)



Q.11 a. Let denote the Bessel’s function of 1st type. obtain the generating function of the sequence . Hence prove that (7)



b. Show that for Legendre polynomials
(7)

Earning:   Approval pending.