1 (a)
Construct dual penalty of the following LPP:

Max z = 8x

Subject to x

5x

Max z = 8x

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

_{1}- 6x_{2}≥ 25x

_{1}+ 7x_{2}= -4; x_{1}, x_{2}≥ 0
5 M

1 (b)
Find the orthogonal tracjectory of the family of curves given by:

2x - x

2x - x

^{3}+ 3xy^{2}= a
5 M

1 (c)
Evaluate ∫{(1/z)dz}, z=C; where C is the upper half of | z |= 1.

5 M

1 (d)
Show that every skew-hermitian matrix can be expressed in the form P+iQ where P is real skew-symmetric and Q is real Symmetric matrix.

5 M

2 (a)
Determine the analytic function f(z) = - r

^{3}sin 3 Θ.
6 M

2 (b)
Show that A = is derogatory.

6 M

2 (c)
Use simplex method to solve the LPP

Max z = 3x

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_{1}≤ 15,x

_{1}, x_{2}, x_{3}− 0
8 M

3 (a)
Evaluate

_{ -∞ }∫^{∞}{(x^{2}- x + 2)/(x^{4}+ 10x^{2}+ 9) dx}; using contour integration.
6 M

3 (b)
Find the eigen values and eigen vectors of A

^{3}where A =
6 M

3 (c)
Use penalty method (Big M) to solve Mini z = 4x + y subject to

3x + y = 3,

4x + 3y ≥ 6,

x + 2y ≤ 4,

x, y ≥ 0

3x + y = 3,

4x + 3y ≥ 6,

x + 2y ≤ 4,

x, y ≥ 0

8 M

4 (a)
If w = f(z) then prove that

(dw/dz) = (cosθ - isinθ) (∂w/∂r)

(dw/dz) = (cosθ - isinθ) (∂w/∂r)

6 M

4 (b)
If A is orthogonal, find a,b,c. Also find A

^{-1}.
6 M

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

Minimize z = 6x

subject to 5x

x

2x

x

Minimize z = 6x

_{1}+ 7x_{2}+ 3x_{3}+ 5x_{4};subject to 5x

_{1}+ 6x_{2}- 3x_{3}+ 4x_{4}≥ 12x

_{2}+ 5x_{3}- 6x_{4}≥ 102x

_{1}+ 5x_{2}+ x_{3}+ x_{4}≥ 8;x

_{1}, x_{2}, x_{3}, x_{4}≥ 0
8 M

5 (a)
Prove that 3tan A = Atan 3, where:

6 M

5 (b)
Find the image of the line y - x + 1 = 0 under the transformation w = 1/z. Also find the image of the line y = x under the same transformation. Draw rough sketches.

6 M

5 (c)
Solve the NLPP using the method of Lagrangian Multipliers.

Minimize z = x

subject to x

x

Minimize z = x

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

_{1}+ x_{2}+ 3x_{3}= 2, 5x_{1}+ 2x_{2}+ x_{3}= 5,x

_{1}, x_{2}, x_{3}≥ 0
8 M

6 (a)
Verify caley-hamiton theorem for A and hence find

A

A

where,

A

^{-1}A

^{3}- 5A^{2}.where,

6 M

6 (b)
Find the bilinear transformation that maps the points ∞, i, 0 onto the points 0, i, ∞. Find the fixed points.

6 M

6 (c)
State and prove Cauchy s integral theorem and hence evaluate

,

where,

,

where,

8 M

7 (a)
Obtain two Laurent s series for {[(z-2)(z+2)] / [(z+1)(z+4)]}

6 M

7 (b)
Evaluate

_{0}∫^{2π}{(cos 2θ) / (1 - 2acosθ + a^{2})) dθ}, using residues.
6 M

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

Minimize z = 7x

x

x

x

Minimize z = 7x

_{1}^{2}+ 5x_{2}^{2}- 6x_{1}subject tox

_{1}+ 2x_{2}≤ 10x

_{1}+ 3 x_{2}≤ 9x

_{1}, x_{1}≥ 0.
8 M

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