## PART A

( Answer All Questions Each carries 15 Marks )

1 a) If f (z) is an analytic function with constant modulus , then prove that f (z) is a constant.
b). If w=Φ+iΨ represents the complex potential of an electric field and determine the function Φ .
2.

3   a. Find the bilinear transformation which maps the point -1, i, 1 of the z-plane  on to the points 1,i,-1  Of  the w-plane respectively
b. Show that under the transformation
the real axis in the z-plane is mapped into the  circle │w│=1.

PART B
(Answer any TWO, Each carries 15 Marks)

4.
5.

6.

PART C

(Answer All Questions Each Carries 20 Marks)

7  a. Express v=(2,7,-4) in R3 as a linear combination of the u1=(1,2,0), u2=(1,3,2)  and u3=(0,1,3)  (10 mark)
b.Diagonalize the matrix (10 mark)
8 a. Find the rank and then find the basis for the row space and column space
(15 mark)
b. Are the following sets of vectors linearly independent?
[4  -1   3], [ 0   8    1],  [ 1   3   -5],[2   6   1]       (5 mark)

9 a. Solve the linear system by Gauss elimination method. (10 mark)
b. Find Eigen values and Eigenvectors of the following Matrix (10 mark)