DipIETE – ET / CS (OLD SCHEME)

Code: DE23/DC23 Subject: MATHEMATICS - II

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.

Q.1 Choose the correct or the best alternative in the following: (210)

a. The principal argument of –2i is

(A) (B)

(C) (D)

b. The value of is

(A) (B)

(C) (D)

c. The value of (2i+3j+4k) (i+j+k) is

(A) 8 (B) 7

(C) 9 (D) 6

d. The projection of the vector (i – 2j+k) on (4i – 4j+7k) is

(A) (B)

(C) (D)

e. The value of is

(A) 0 (B) a

(C) b (D) c

f. If , then the eigenvalues of is

(A) 2, 55, 5 (B) 2, 100, 5

(C) 4, 110, 10 (D) 4, 100, 15

g. If the is orthogonal, then is

(A) (B)

(C) (D)

h. The inverse Laplace transform of is

(A) (B)

(C) (D)

i. In the half-range series the period 0 to, b_n is equal to _________

when

(A) 0 (B) 2

(C) 1 (D) 3

j. The solution of differential equation is

(A) (B)

(C) (D)

Answer any FIVE Questions out of EIGHT Questions.

Each question carries 16 marks.

Q.2 a. Find the real and imaginary parts of sec(x + iy). (8)

b. Use De Moivre’s theorem to solve the equation =1, for general value of, which satisfies the equation. (8)

Q.3 a. If and are two complex numbesr then show that. (8)

b. Using vector method, prove that the angle in a semi-circle is a right angle. (8)

Q.4 a. For any two vectors and prove that, . (8)

b. A particle acted on by two forces and is displaced from the point to the point . Find the work done by the forces. (8)

Q.5 a. Prove that . (8)

b. Use Cramer’s rule to solve the equations

(8)

Q.6 a. For what values of k the system of equations

(8)

has a non- trivial solutions.

b. Use Cayley-Hamilton theorem to find inverse of . (8)

Q.7 a. Find the Laplace transform of cos t cos 2t cos 3t. (8)

b. Find the inverse Laplace transform of . (8)

Q.8 a. Solve the differential equation, where D =. (8)

b. Use Laplace transform technique to solve the equation, (8)

Q.9 a. Find the value of. (8)

b. Find the Fourier series to represent the function for. (8)