Flowchart: Alternate Process: JUNE 2008

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  and  represent conjugate complex numbers then the value of x and y is  

 

                   (A)  .                           (B)  .

(C)    .                          (D)  .

       

b.      Imaginary part of  is

 

(A)    – cos x cosh y                              (B)  – cos x sinh y

(C)  – sin x cosh y                               (D)  – sin x sinh y

            

             c.   Three vectors  are coplanar, the value of their scalar triple product is

                  

(A)    0                                                  (B) 1

(C) –1                                                 (D) i

 

             d.   If  is the angle between the vectors  and  such that  then  is

 

(A)                                                  (B) 

(C)                                             (D) 

 

             e.   The value of the determinant  is

                  

(A)     1                                                  (B)  2

(C)  –1                                                (D)  0

 

             f.    If the product of two eigen values of the matrix  is 16, then the third eigen value is

 

(A)     0                                                  (B)  5

(C)  2                                                  (D)  –2

 

             g.   If f(x) is defined in (0, L), then the period of f(x) to expand it as a half range sine series is

 

(A)     L.                                                (B)  0.

(C) 2L.                                                (D)  .  

 

             h.   The inverse Laplace transform  is possible only when n is 

 

(A)    0                                                  (B) –ve integer

(C) –ve rational number                       (D) +ve integer

 

             i.    The differential equation of a family of circles having the radius r and centre on the x axis is  

 

(A)                        (B)

(C)          (D)

 

             j.    If y satisfies  with  then Laplace transform  is

 

(A)                              (B) 

(C)                                (D)

 

 

Answer any FIVE Questions out of EIGHT Questions.

Each question carries 16 marks.

 

 

  Q.2     a.   Find the moment of the force  about a line through the origin having direction of , due to a 30 Kg force acting at a point (– 4, 2, 5) in the direction of .                                (8)

       

             b.   Prove that the right bisectors of the sides of a triangle intersect at its circum centre.              (8)

 

  Q.3     a.   Show that the components of a vector  along and perpendicular to  in the plane of and  are  and .                (8)

                  

             b.   If show that  and .         (8)

                  

  Q.4     a.   If  then .                                         (8)

 

             b.   Show that the origin and the complex numbers represented by the roots of the equation , where a, b are real, form an equilateral triangle if .                                                 (8)

 

  Q.5     a.   Prove that

                   .                                     (8)

       

             b.   Determine the values of  when  is orthogonal.                (8)

                  

  Q.6     a.   Find the values of k such that the system of equations, ,  has non-trivial solution.                         (8)

 

             b.   Find the characteristic equation of the matrix  .  Hence find .            (8)

                               

 

 

Q.7             Find the Fourier series for .                                    (16)   

       

Q.8       a.   Find .                                                                                       (8)

                                                                                                                                                                                                                                                                                                                        

             b.   Find the inverse Laplace transform of .                                       (8)

                

 

Q.9       a.   Using Laplace transformation, solve the following differential equation:

                    if x(0) = 1, .                                                       (8)

 

 

             b.   Solve                                                                           (8)