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. If  then n is equal to  

(A)   –1                                               (B)  1

(C)   2                                                 (D)  4

b. If then  is equal to

(A) (B)

(C) (D)

             c.   is equal to

(A) (B)

(C)  (D) 0

d.   If , the angle between the and is

(A) (B)

                    (C) (D)

             e.  

 is equal to

(A) (B) 

(C)  (D)

             f.    If inverse of   is  , then K is equal to

(A) 2                                                  (B)  4

(C)  6                                                  (D)  8

             g.   The characteristic equation of is

(A) (B)

(C) (D)

             h.   The period of  is  

(A) (B)

(C)  (D) 2

             i.    The inverse 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.   Show that

(8)

             b.   If show that

(8)

Q.3     a.   The centre of a regular hexagon is at the origin and one vertex is given by 1+i on the Argand diagram. Find the remaining vertices.                          (8)

             b.   Show that the vectors form a right angled triangle.                                                            (8)

Q.4     a.   The vertices of a quadrilateral are

 At the point A the forces of magnitudes 2, 3, 2 gm wt. act along the line AB, AC, AD respectively. Find their resultant.                           (8)

b.   Find a unit vector perpendicular to the plane of  and .                    (8)

Q.5     a.   Evaluate

.                                                            (8)

     b.   Use Cramer’s rule to solve the equations

(8)

  Q.6     a. Investigate for what values of  and , the equations

                   have (i) no solution, (ii) a unique solution, (iii) an infinite number of solutions.                       (8)

             b.   Find the characteristic equation of the matrix

 and hence find the inverse of the matrix A.                     (8)

  Q.7     a.   Find the Laplace transform of 

.                                                                 (8)

             b.   Find the inverse Laplace transform of

(8)

Q.8       a.   Use Laplace transform technique to solve

                   given that y = 0, at t = 0                                                                        (8)

             b.   Solve the differential equation

.                                               (8)

Q.9       a.   Define even and odd functions. Give two examples of each.                                (4)

b.   Find the Fourier series expansion for the function

   for . (12)