1 (a)
Find the characteristic equation of the matrix A given below and hence, find the matrix represented by A

where,

^{8}-5A^{7}+7A^{6}-3A^{5}+A^{4}- 5A^{3}+ 8A^{2}- 2A+Iwhere,

5 M

1 (b)
Find the orthogonal trajectory of the family of curves x

^{3}y-xy^{3}=c.
5 M

1 (c)
Evaluate:

where C is a circle |z|=1.

where C is a circle |z|=1.

5 M

1 (d)
Construct the dual simplex to solve the following LPP

Minimise z = x

subject to 2x

-x

x

Minimise z = x

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

_{1}+ x_{2}≥ 2;-x

_{1}- x_{2}≥ 1;x

_{1}, x_{2}≥ 0.
5 M

2 (a)
Find the eigen values and eigen vectors of the matrix.

6 M

2 (b)
Find the imaginary part of the analytic function whose real part is -

e

e

^{2x}(xcos 2y - sin 2y). Also verify that it is harmonic.
6 M

2 (c)
Use penalty method to solve the following LPP

Minimize z = 2x

subject to the constraints:

x

x

x

Minimize z = 2x

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

x

_{1}+ x_{2}≥ 5,x

_{1}+ 2x_{2}≥ 5,x

_{1}, x_{2}≥ 0
8 M

3 (a)
Use Lagrangian Multiplier method to optimize

z=2x

subject to x

x

z=2x

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

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

_{1}, x_{2}, x_{3}≥ 0
6 M

3 (b)
Evaluate \[\displaystyle\int\limits_c\dfrac{z^2}{(z-1)^2(z-2)}dz\]

6 M

3 (c)
Show that A is derogatory.

8 M

4 (a)
Show that A is diagonaisable. Also find the transforming and diagonal matrix.

6 M

4 (b)
Show that f(z)= √(|xy|) is not analytic at the origin although Cauchy-Reimann. Equations are satisfied at that point.

6 M

4 (c)
Using duality solve the following LPP.

Minimize z = 430x

subject to x

2x

x

x

Minimize z = 430x

_{1}+ 460x_{2}+ 420x_{3}subject to x

_{1}+ 3x_{2}+ 4x_{3}≥ 32x

_{1}+ 4x_{2}≥ 2x

_{1}+ 2x_{2}≥ 5x

_{1}, x_{2}, x_{3}≥ 0
8 M

5 (a)
Consider the following problem

Maximize z = x

Subject to x

2x

Determine:-

(i) All basic solutions.

(ii) All feasible basic solutions.

(iii) Optimal feasible basic solution.

Maximize z = x

_{1}+ 3x_{2}+ 4x_{3}Subject to x

_{1}+ 2x_{2}+ 3x_{3}= 42x

_{1}+ 3x_{2}+ 5x_{3}= 7Determine:-

(i) All basic solutions.

(ii) All feasible basic solutions.

(iii) Optimal feasible basic solution.

6 M

5 (b)
Obtain Taylor

^{'}s and Laurent^{'}s expansion of f(z)= [(z-1) / (z^{2}- 2z -3)] indicating regions of convergences.
6 M

5 (c)
Verify caley-hamilton theorem for the matrix A and hence find A

where,

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

8 M

6 (a)
If u = -r

^{3}sin 3θ,find the analytic function f(z) whose real part is u.
6 M

6 (b)
Prove that 3 tan A=A tan 3.

6 M

6 (c)
Use simplex method to solve the LPP

Max z = 3x

2x

2x

3x

x

Max z = 3x

_{1}+ 5x_{2}+ 4x_{3}subject to the constraints:2x

_{1}+ 3x_{2}≤ 8,2x

_{2}+ 5x_{3}≤ 103x

_{1}+ 2x_{2}+ 4x_{3}≤ 15,x

_{1}, x_{2}, x_{3}≥ 0
8 M

7 (a)
Find the bilinear transformation that maps the points ∞, i, 0 onto the points 0, i, ∞.

6 M

7 (b)
Find the laurent

f(z) = 2/[(z-1)(z-2)]

When (i) |z| < 1

(ii) 1 < |z|< 2

(iii) |z|> 2

^{ }s series which represents the functionf(z) = 2/[(z-1)(z-2)]

When (i) |z| < 1

(ii) 1 < |z|< 2

(iii) |z|> 2

6 M

7 (c)
Using Kuhn-Tucker conditions:

Minimize z = 2x

subject to x

5x

x

Minimize z = 2x

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

_{1}+ 3x_{2}≤ 65x

_{1}+ 2x_{2}≤ 10x

_{1},x_{2}≥ 0.
8 M

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