Code: D-23 / DC-23                                                                       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  are vectors then  is equal to

(A)

                   (B)

(C)

(D) none of above.

b. If  A, B are square matrices of the same size then

(A) (B)

(C) (D)

             c.   If  are two complex numbers then   is

(A) (B)

(C) (D)

             d.   The value of  is equal to

(A) 3a²x                                             (B)  a² (3x - a)

(C)  a² (3x + a)                                   (D)  3ax²

             e.   If I+A+A²+…+A^K=0, then A^-1 is equal to

(A) A^K (B)  A^K-1

(C)  A^K+1 (D)  I+A

             f.    If  A is any real square matrix then A+A^t is

(A) Hermitian.                                    (B)  Skew-hermition.

(C)  Symmertic.                                   (D)  Skew-symmertic.

             g.   The Laplace transform L(tⁿ) is

(A) .                                             (B) .

(C) .                                                (D)

             h.   The solution of differential equation   is

(A) (B) .

(C) .                       (D)

             i.    The value of a₀ in the Fourier series is given by

(A) (B)

(C) (D) 0

             j.    The inverse Laplace transform  is

(A) (B)

(C) (D)

Answer any FIVE Questions out of EIGHT Questions.

Each question carries 16 marks.

Q.2     a.   Express  in the form x+iy.                                                        (8)

             b.   Write down all the values of .                                                               (8)

Q.3     a.   Using vector method prove that the altitudes of a triangle are concurrent.             (8)

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

Q.4     a.   Prove that (8)

b.   Find the angle between two vectors  and  if = .                     (8)

Q.5     a.   Let A be a square matrix. Prove that A can be written the sum of a symmetric and a skew-symmetric matrix.                                                           (8)

     b.   State Cayley Hamitton theorem and use it to find the inverse of, if the inverse exists.                                                            (8)

  Q.6     a. Prove that .                        (8)

             b.   Give condition under which we can find  so that the following system of linear equations has a non-trivial solution.

(8)

Q.7     a.   Find the Fourier series of the function defined by

(8)

b.   Find the Fourier series representing the function

(8)

Q.8     a.   If F(t) is piecewise continuous and satisfies  for all  and for some constants  a and M then

(8)

             b.   Define Inverse Laplace Transform of a function F(t). Prove that

                                                       (8)

Q.9     a.   Solve (D⁴ +2D²+1) y = 0.                                                                                  (8)

b.   Solve .                                                                      (8)