DipIETE – ET / CS (OLD SCHEME)

 

Code: DE23/DC23                                                                         Subject: MATHEMATICS - II

Flowchart: Alternate Process: DECEMBER 2009Time: 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 the best alternative in the following:                                   (2x10)

                                

a.       If then x, y equal to   

 

                   (A)                                   (B)  

(C)                                    (D) 

       

b.      The value of  is 

 

(A)                                    (B) 

(C)                              (D) 

            

             c.   If  &  be two vectors indefined at an angle , then  is:

                  

(A)                                    (B)

(C)                                      (D) -

 

d.   If  and  then  is 

(A)                                      (B) 

                    (C)                                   (D) 

 

             e.   The values of x, y, z if

                  

                  

(A)                       (B) 

(C)                  (D) 

 

             f.    is equal to 

 

(A)     -1                                                (B)  0

(C)  (b-a) (c-d)                                   (D)  (a-b) (b-c) (c-a)

             g.   The characteristic equation of  is

 

(A)                               (B) 

(C)                             (D)          

 

             h.   The period of  is 

 

(A)                                                   (B)

(C)                                              (D)

 

             i.    The laplace transform of the function  is

  

 

(A)                                          (B)

(C)                                        (D)

 

             j.    The solution of differential equation is 

 

(A)      

(B) 

(C)   

(D)

 

 

Answer any FIVE Questions out of EIGHT Questions.

Each question carries 16 marks.

 

 

  Q.2     a.   If , where c is real, prove that  and .        (8)

 

             b.   If n is a positive integer, prove that .           (8)

 

  Q.3     a.   For what value of x and y are the numbers  and  conjugate complex?                                                                     (8)

       

             b.   The adjacent sides of a parallelogram are represented by the vectors  and .  Find unit vectors parallel to the diagonals of a parallelogram.                                                      (8)       

 

  Q.4     a.   Prove that the points having position vectors ,  form a right angled triangle.                                     (8)

                  

             b.   Find the area of the triangle formed by the points whose position vectors are , , .                                                                                                                          (8)

                  

  Q.5     a.   Let , find f(A) if .                                       (8)

 

             b.   Prove that = .       (8)   

 

  Q.6     a.   Solve the system of equations by matrix method.

                                                                                                                    (8)

       

             b.   Verify Cayley-Hamilton theorem for the matrix A and find its inverse.

                                                                                                            (8)

                  

  Q.7     a.   Find the Laplace transform of .                                                            (8)

 

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

                               

 

 

Q.8       a.   Solve .                                                                         (8) 

                  

             b.   Solve the differential equation , given that x=1 &  when t =0.                                                             (8)

            


 

Q.9       a.   Determine the period of the following functions:

                   (i)                                           (ii)  

                   (iii)                                          (iv)                                      (8)

                                                                                                                                                                                                                                                                                                                        

 

             b.   Obtain the fourier series for

                                       

                   Hence, prove that                                 (8)