Code: D-23 / DC-23                                                                       Subject: MATHEMATICS - II

Time: 3 Hours                                           June 2006                                                           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. Let  and .  Express  in the form a + bi, a , b R. 

(A) (B)

(C) (D) 

b. The complex numbers ,  and  satisfying  are vertices of the a triangle which is

(A) acute-angled and isosceles            (B)  right-angled and isosceles

(C)  obtuse-angled and isosceles          (D)  equilateral

             c.   A unit vector parallel to 3i+4j-5k is

(A) (B)

(C) (D)

             d.   Let  = (1, 2, 0),  = (-3, 2, 0),  = (2, 3, 4). Then  equals

(A)  33                                               (B)  30

(C)  31                                               (D)  32

             e.   If  is complex cube root of unity, and , then  is equal to

(A) 0                                                  (B)  -A

(C)  A                                                 (D)  none of these

             f.    If  A and B are symmetric matrices, then AB + BA is a

(A) diagonal matrix                             (B)  null matrix

(C)  symmetric matrix                          (D)  Skew-symmetric matrix

             g.   The function  is 

(A) odd                                              (B)  even

(C)  neither                                          (D) none of these  

             h.   The function cos x + sin x + tan x + cot x + sec x + cosecx is

(A) both periodic and odd                  (B) both periodic and even

(C) periodic but neither even nor          (D) not periodic

       odd                                             

             i.    The Laplace Transform for  sin at  is

(A) (B)

(C) (D)

             j.    The Inverse Laplace Transform for  is

(A) (B)

(C) (D)

Answer any FIVE Questions out of EIGHT Questions.

Each question carries 16 marks.

Q.2     a.   If a, b, c are real numbers such that  and b + ic = (1 + a)z, where z is a complex number, then show that .            (8)

             b.   Given that  and  where  is a cube root of unity. Express  in terms of A, B, C and .                                                     (8)

Q.3     a.   Show that for all real , .   (8)

b.   For any four vectors prove that . Hence prove  that (8)

Q.4     a.   In  let , .  Then find the vector representing AB and OM, where M is the midpoint of AB.                                                 (4)

b.   Prove that the straight line joining the mid-points of two non-parallel sides of a trapezium is parallel to the parallel sides and is half their sum.        (12)

Q.5     a.   For reals A, B, C, P, Q, R find the value of determinant

(8)

     b.   Using matrix method find the values of  and  so that the system of equations:  

 has infinitely many solutions.                                                 (8)

  Q.6     a. Solve the system of equations

                   by using inverse of a suitable matrix.     (8)

             b.   Using Cayley-Hamilton theorem find  for .                                (8)

Q.7           State whether the function f(x) having period 2 and defined by

                   is even or odd.  Find its Fourier Series.                                                             (16)

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

             b.   Find the Inverse Laplace transform for .                                      (8)

Q.9       a.   Solve the differential equation

                   given that y = -0.9 and , when x=0                                                   (8)

b.   Using the Laplace transform solve the differential equation

                   with initial conditions . (8)