Code: DE-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. Modules of is  

(A) (B)

(C) (D)

b. If  then the value of cos x cos hy is

(A) –1                                                (B)  0

(C)  1/2                                               (D)  1

             c.   The two non-zero vectors  and  are parallel if

(A) (B)

(C) (D)

             d.   The volume of the parallelopipid with sides , A is

(A)  5 cubic units                                (B)  10 cubic units

(C)  15 cubic units                              (D)  20 cubic units

             e.   If  then eigen value of A^–1 are

(A) (B)  1, 2, 3

(C)  0, 1, 2                                          (D)

             f.    The sum and product of the eigen values of   are

(A) Sum = 5,   Product = 7                (B)  Sum = 7,   Product = 5

(C)  Sum = 5,   Product = 5                 (D)  Sum = 7,   Product = 7

             g.   If     then the value of f(0) is

(A) 0                                                  (B)

(C) (D)

             h.   The inverse Laplace transform of (s+2)^–2

(A) e^–2t (B) e^2t

(C)  te^2t (D) te^-2t

             i.    The solution of the differential equation  satisfying the condition y(0)=1,  is

(A) (B)

(C) (D)

             j.    Fourier Sine transform of 1/x is

(A)  S                                                  (B)  S/2

(C)  S²/2                                             (D) –S²/2

Answer any FIVE Questions out of EIGHT Questions.

Each question carries 16 marks.

Q.2     a.   A rigid body is spinning with angular velocity 27 radians per second about an axis parallel to passing through the point . Find the velocity of the point of the body whose position vector is .                           (8)

             b.   Find the sides and angles of the triangle whose vertices are ,  and .                                                                       (8)

Q.3     a.   Find the volume of the tetrahedron formed by the point (1,1,1) (2,1,3) (3,2,2,), (3,3,4).                  (8)

             b.   The centre of a regular hexagon is at the origin and one vertex is given by  on the Argand diagram. Determine the other vertices.                (8)

Q.4     a.   Prove that the general value of   which satisfies the equation

where m is any integer (8)

b.   Use De Moivre’s theorem to solve the equation x⁴–x³+x²–x+1=0                       (8)

Q.5     a.   Show that

(8)

     b.   Express the following matrix as a sum  of a symmetric matrix and a skew symmetric matrix.

.                                                                                                    (8)

  Q.6     a. Find the values of l, for which following system of equations has non-trivial solutions. Solve equations for all such values of l.

(8)

             b.   Find the characteristic equation of the matrix  and hence evaluate the matrix equation A⁸–5A⁷+7A⁶–3A⁵+A⁴–5A³+8A²–2A+I.   (8)

Q.7           Expand in a Fourier Series.

Hence evaluate (16)

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

             b.   Find the Inverse Laplace transform of (8)

  Q.9     a.   Solve the initial value problem

Using Laplace transforms.                                                                                  (8)

b.   Solve

(8)