1 (a)
Find the characteristic equation of the matrix A given below and hence, find the matrix represented by A8 -5A7 +7A6 -3A5 +A4- 5A3+ 8A2- 2A+I
where,
where,
5 M
1 (b)
Find the orthogonal trajectory of the family of curves x3y-xy3=c.
5 M
1 (c)
Evaluate:
![](data:image/png;base64,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)
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 = x1 + x2;
subject to 2x1 + x2 ≥ 2;
-x1 - x2 ≥ 1;
x1, x2 ≥ 0.
Minimise z = x1 + x2;
subject to 2x1 + x2 ≥ 2;
-x1 - x2 ≥ 1;
x1, x2 ≥ 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 -
e2x(xcos 2y - sin 2y). Also verify that it is harmonic.
e2x(xcos 2y - sin 2y). Also verify that it is harmonic.
6 M
2 (c)
Use penalty method to solve the following LPP
Minimize z = 2x1 + 3x2
subject to the constraints:
x1 + x2 ≥ 5,
x1 + 2x2 ≥ 5,
x1 , x2 ≥ 0
Minimize z = 2x1 + 3x2
subject to the constraints:
x1 + x2 ≥ 5,
x1 + 2x2 ≥ 5,
x1 , x2 ≥ 0
8 M
3 (a)
Use Lagrangian Multiplier method to optimize
z=2x12 + x22 + 3x32 + 10x1 + 8x2 + 6x3 - 100;
subject to x1 + x2 + x3 = 20,
x1, x2, x3 ≥ 0
z=2x12 + x22 + 3x32 + 10x1 + 8x2 + 6x3 - 100;
subject to x1 + x2 + x3 = 20,
x1, x2, x3 ≥ 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 = 430x1 + 460x2 + 420x3
subject to x1 + 3x2 + 4x3≥ 3
2x1 + 4x2 ≥ 2
x1 + 2x2 ≥ 5
x1, x2, x3 ≥ 0
Minimize z = 430x1 + 460x2 + 420x3
subject to x1 + 3x2 + 4x3≥ 3
2x1 + 4x2 ≥ 2
x1 + 2x2 ≥ 5
x1, x2, x3 ≥ 0
8 M
5 (a)
Consider the following problem
Maximize z = x1 + 3x2 + 4x3
Subject to x1 + 2x2 + 3x3 = 4
2x1 + 3x2 + 5x3 = 7
Determine:-
(i) All basic solutions.
(ii) All feasible basic solutions.
(iii) Optimal feasible basic solution.
Maximize z = x1 + 3x2 + 4x3
Subject to x1 + 2x2 + 3x3 = 4
2x1 + 3x2 + 5x3 = 7
Determine:-
(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) / (z2 - 2z -3)] indicating regions of convergences.
6 M
5 (c)
Verify caley-hamilton theorem for the matrix A and hence find A-1 and A4
where,
where,
8 M
6 (a)
If u = -r3sin 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 = 3x1 + 5x2 + 4x3 subject to the constraints:
2x1 + 3x2 ≤ 8,
2x2 + 5x3 ≤ 10
3x1 + 2x2 + 4x3 ≤ 15,
x1, x2, x3 ≥ 0
Max z = 3x1 + 5x2 + 4x3 subject to the constraints:
2x1 + 3x2 ≤ 8,
2x2 + 5x3 ≤ 10
3x1 + 2x2 + 4x3 ≤ 15,
x1, x2, x3 ≥ 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 s series which represents the function
f(z) = 2/[(z-1)(z-2)]
When (i) |z| < 1
(ii) 1 < |z|< 2
(iii) |z|> 2
f(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 = 2x1 + 3x2 -x12 - 2x22
subject to x1 + 3x2 ≤ 6
5x1 + 2x2 ≤ 10
x1,x2 ≥ 0.
Minimize z = 2x1 + 3x2 -x12 - 2x22
subject to x1 + 3x2 ≤ 6
5x1 + 2x2 ≤ 10
x1,x2 ≥ 0.
8 M
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