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MU Computer Engineering (Semester 4)
Applied Mathematics 4
December 2013
Total marks: --
Total time: --
INSTRUCTIONS
(1) Assume appropriate data and state your reasons
(2) Marks are given to the right of every question
(3) Draw neat diagrams wherever necessary

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,
5 M
1 (b) Find the orthogonal trajectory of the family of curves x3y-xy3=c.
5 M
1 (c) Evaluate:

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.
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.
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
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
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
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.
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,
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
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
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.
8 M

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