## Applied Mathematics 4 - May 2013

### Computer Engineering (Semester 4)

TOTAL MARKS: 80

TOTAL TIME: 3 HOURS
(1) Question 1 is compulsory.

(2) Attempt any **three** from the remaining questions.

(3) Assume data if required.

(4) Figures to the right indicate full marks.
**1 (a)** Check if the following function is harmonic.

f(γ,θ) = [γ + (a^{2}/γ)] cosθ(5 marks)
**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 marks)
**1 (c)** Prove that the eigen values of an orthogonal matrix are +1 or -1.(5 marks)
**1 (d)** Construct the dual of the following LPP.

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 marks)
**2 (a)** Evaluate:

where C is a circle |z|=1.(6 marks)
**2 (b)** Diagonalise the hermitian matrix

(6 marks)
**2 (c)** Use simplex method to solve the LPP

Maximize z = 1000x_{1} + 4000x_{2} + 5000x_{3} subject to the constraints:

x_{1} + 2x_{2} + 3x_{3}≤ 14

x_{1} + 2x_{1} ≤ 14

x_{1}, x_{2}, x_{3} ≥ 0(8 marks)
**3 (a)** Evaluate
using contour integration:

(6 marks)
**3 (b)** State caley-hamilton theorem. Use it to find A^{-1} and A^{4}

where,

(6 marks)
**3 (c)** Use penalty method to

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} ≥ 8

x_{1}, x_{2},x_{3} ≥ 0(8 marks)
**4 (a)** Find A^{100}

where,

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

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

Minimize z = 3x_{1} + 2x_{2} + x_{3} + 4x_{4};

subject to 2x_{1} + 4x_{2} + 5x_{3} + x_{4} ≥ 10

3x_{1} - x_{2} + 7x_{3} - 2x_{4} ≥ 2

5x_{1} + 2x_{2} + x_{3} + 6x_{4} ≥ 15

x_{1}, x_{2}, x_{3}, x_{4} ≥ 0(8 marks)
**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 marks)
**5 (b)** Is A derogatory?

(6 marks)
**5 (c)** Evaluate:

where 0 < b < a.(8 marks)
**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.(6 marks)
**6 (b)** Find the image of region bounded by x=1, y=1, x+y=1 under the transformation w=z^{2}.(6 marks)
**6 (c)** Use Lagrangian Multiplier method to optimize (8)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(8 marks)
**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 (6 marks)
**7 (b)** Find the analytic function f(z) whose imaginary part is

e^{-x} {2xy cos y + (y^{2} - x^{2}) sin y}. (6 marks)
**7 (c)** Solve the following N.L.P.P. using Kuhn-Tucker conditions.

Optimize z = 2x_{1} + 3x_{2} - (x_{1}^{2} + x_{2}^{2} + x_{3}^{2})

subject to x_{1} + x_{2} ≤ 1

2x_{1} + 3x_{2} ≤ 6

x_{1} , x_{2} ≥ 0.(8 marks)