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 best alternative in the following:                                         (2x10)

a. The complex numbers Z = x + iy, which satisfy the equation  lie on  

(A)  the x-axis.                                   

                   (B)  the line y = 5.

(C)   A circle passing through the origin.

(D) None of these.

b. If , then

(A) (B)

(C)  Z=0                                             (D) , with x real

             c.   If  and  are two unit vectors and  is the angle between them, then  is equal to

(A) (B) 0

(C) (D)

             d.   A vector which makes equal angles with the vectors ,  and   is

(A) (B)

(C) (D)

e.   If  is a cube root of unity and , then 

(A) x = 1                                            (B) 

(C)  (D)  none of these

             f.    If , then  is equal to 

(A) (a+b) (b+c) (c+a)                        (B)  bc + ca + ab

(C)  2abc                                            (D)  none of these

             g.   If A is a skew-symmetric matrix and n is a positive integer, then  is

(A) a symmetric matrix.                     

(B) skew-symmetric matrix for even n only.

(C) diagonal matrix.                           

(D) symmetric matrix for even n only. 

             h.   The period of the function sin x + sin 2x + sin 3x is 

(A) (B)

(C) (D)

             i.    The Laplace transform of  is

(A) (B)

(C) (D)

             j.    The solution of the differential equation  is

(A) (B)

(C) (D)

Answer any FIVE Questions out of EIGHT Questions.

Each question carries 16 marks.

Q.2     a.   Simplify .                                                                        (8)

             b.   Find all the values of  .                                                                          (8)

Q.3     a.   If  and  are two complex numbers, prove that

If and only if  is purely imaginary.   (8)

             b.   A vector  satisfies the equation .  Prove that  provided and  are not perpendicular.                          (8)

Q.4     a.   Using vector methods prove that the diagonals of a parallelogram bisect each other.            (8)

b.   The constant forces 2i – 5j + 6k, -i+2j-k and 2i + 7j act on a particle which is displaced from position 4i – 3j – 2k to position 6i + j – 3k.  Find the total work done.                                                   (8)

Q.5     a.   Show that

(8)

     b.   Write the following equations in the matrix form AX = B and solve for X by finding .

(8)

  Q.6     a. Test the consistency of the following equations and if possible, find the solution

(8)

             b.   Obtain the characteristic equation of the matrix  and use Cayley-Hamilton theorem to find its inverse.                                                (8)

Q.7             Find the Fourier series expansion for the function

.                                                                           (16)

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

             b.   Find the Inverse Laplace transform of (8)

Q.9       a.   Solve the differential equation

.                                                                               (8)

b.   By using Laplace transform solve the differential equation

 with initial conditions , when t = 0.                 (8)