1 (a)
Check if the following function is harmonic.

f(γ,θ) = [γ + (a

f(γ,θ) = [γ + (a

^{2}/γ)] cosθ
5 M

1 (b)
Integrate function f(z) = x

^{2}+ iy from A (1,1) to B (2,4) along the curve x=t, y=t^{2}
5 M

1 (c)
Prove that the eigen values of an orthogonal matrix are +1 or -1.

5 M

1 (d)
Construct the dual of the following LPP.

Maximise z = x

subject to 3x

5x

x

Maximise z = x

_{1}+ 3x_{2}- 2x_{3}+ 5x_{4};subject to 3x

_{1}- x_{2}+ x_{3}-4x_{4}= 6;5x

_{1}+ 3x_{2}- x_{3}- 2x_{4}= 4;x

_{1}, x_{3}≥ 0; x_{3}, x_{4}unrestricted.
5 M

2 (a)
Evaluate:

where C is a circle |z|=1.

where C is a circle |z|=1.

6 M

2 (b)
Diagonalise the hermitian matrix

6 M

2 (c)
Use simplex method to solve the LPP

Maximize z = 1000x

x

x

x

Maximize z = 1000x

_{1}+ 4000x_{2}+ 5000x_{3}subject to the constraints:x

_{1}+ 2x_{2}+ 3x_{3}≤ 14x

_{1}+ 2x_{1}≤ 14x

_{1}, x_{2}, x_{3}≥ 0
8 M

3 (a)
Evaluate
using contour integration:

6 M

3 (b)
State caley-hamilton theorem. Use it to find A

where,

^{-1}and A^{4}where,

6 M

3 (c)
Use penalty method to

Minimize z = x

subject to x

(3/2) x

x

Minimize z = x

_{1}+ 2x_{2}+ x_{3}subject to x

_{1}+ (x_{2}/2) + (x_{3}/2) ≤ 1(3/2) x

_{1}+ 2x_{2}+ x_{3}≥ 8x

_{1}, x_{2},x_{3}≥ 0
8 M

4 (a)
Find A

where,

^{100}where,

6 M

4 (b)
If f(z) is analytic function, prove that

6 M

4 (c)
Use dual simplex method to solve the LPP.

Minimize z = 3x

subject to 2x

3x

5x

x

Minimize z = 3x

_{1}+ 2x_{2}+ x_{3}+ 4x_{4};subject to 2x

_{1}+ 4x_{2}+ 5x_{3}+ x_{4}≥ 103x

_{1}- x_{2}+ 7x_{3}- 2x_{4}≥ 25x

_{1}+ 2x_{2}+ x_{3}+ 6x_{4}≥ 15x

_{1}, x_{2}, x_{3}, x_{4}≥ 0
8 M

5 (a)
Find the bilinear transformation that maps the points 1, -1, 2 in z-plane onto the points 0, 2, -i in w-plane.

6 M

5 (b)
Is A derogatory?

6 M

5 (c)
Evaluate:

where 0 < b < a.

where 0 < b < a.

8 M

6 (a)
If A = where a, b, c are positive integers, then prove that

(i) a + b + c is an eigen value of A and

(ii) if A is non-singular, one of the eigen values is negative.

(i) a + b + c is an eigen value of A and

(ii) if A is non-singular, one of the eigen values is negative.

6 M

6 (b)
Find the image of region bounded by x=1, y=1, x+y=1 under the transformation w=z

^{2}.
6 M

6 (c)
Use Lagrangian Multiplier method to optimize (8)z=2x

subject to x

x

_{1}^{2}+ x_{2}^{2}+ 3x_{3}^{2}+ 10x_{1}+ 8x_{2}+ 6x_{3}- 100;subject to x

_{1}+ x_{2}+ x_{3}= 20x

_{1}, x_{2}, x_{3}≥ 0
8 M

7 (a)
Obtain two Laurent s series for 1/[(z-1)(z-2)] in the regions:(i) 1 < |z-1|< 2

(ii) 1 < |z-3|< 2

(ii) 1 < |z-3|< 2

6 M

7 (b)
Find the analytic function f(z) whose imaginary part is

e

e

^{-x}{2xy cos y + (y^{2}- x^{2}) sin y}.
6 M

7 (c)
Solve the following N.L.P.P. using Kuhn-Tucker conditions.

Optimize z = 2x

subject to x

2x

x

Optimize z = 2x

_{1}+ 3x_{2}- (x_{1}^{2}+ x_{2}^{2}+ x_{3}^{2})subject to x

_{1}+ x_{2}≤ 12x

_{1}+ 3x_{2}≤ 6x

_{1}, x_{2}≥ 0.
8 M

More question papers from Applied Mathematics 4