DipIETE – ET / CS (OLD SCHEME)

Code: DE23 / DC23                                                                       Subject: MATHEMATICS - II

Time: 3 Hours                                          klklk                                                  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:                                   (2x10)

a. Argument of  is  

(A)   0                                                 (B)

(C) (D) 

b.  is equal to

(A)  1                                                  (B)  –1

(C)  0                                                  (D)  None of these

             c.     is equal to

(A) 0                                                  (B) 1

(C)  2 (D) 3

d.   If  is perpendicular to the sum of the vectors  and  then m is equal to

(A)  1                                                  (B)  2

                    (C)  3                                                  (D)  4

             e.  

 is equal to

(A) a (B)  b

(C)  c (D)  0

             f.    The sum and product of eigen values of  are

(A) 7, 8                                              (B)  7, 3

(C)  3, 0                                              (D)  3, 1

             g.   If   A =  and I is the unit matrix of order 3, then A³ is equal to

(A) (B)

(C) (D)

             h.   The inverse Laplace transform of  is

(A) 1 + sin t                                        (B) 1 – sin t

(C)  1 + cos t                                      (D) 1 – cos t

             i.    The period of the function  is  

(A)  2                                                  (B)

(C) n                                                   (D)

             j.    The solution of the differential equation = Sin 2x 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 tan (x + iy).                                                  (8)

b.   Use De-Moivre’s Theorem to solve the equation.                               (8)

  Q.3     a.   If  and  are two complex numbers, show that .           (8)

             b.   ABCDEF is a regular hexagon whose centroid is 0.  Show that                              

                                                                   (8)

Q.4     a.   Show that .                                                   (8)

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

Q.5     a.   Evaluate  .                                                 (8)

b.   Use Cramer’s rule to solve the equations   

(8)

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

has a solution and solve them completely in each case.                                        (8)

     b.   Use Cayley-Hamilton theorem to find inverse of

(8)

  Q.7     a. Find the Laplace transform of .                                                           (8)

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

Q.8       a.   Solve the differential equation 

.                                                           (8)

             b.   Use Laplace transform to solve the initial value problem

                   Given that   at x = 0.                                                                     (8)

Q.9       a.   Find all values of .         (6)

             b.   Find a Fourier series expansion of the function

(10)