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 + (a²/γ)] cosθ(5 marks) 1 (b) Integrate function f(z) = x² + iy from A (1,1) to B (2,4) along the curve x=t, y=t²(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 = x₁ + 3x₂ - 2x₃ + 5x₄;
subject to 3x₁ - x₂ + x₃ -4x₄ = 6;
5x₁ + 3x₂ - x₃ - 2x₄ = 4;
x₁, x₃ ≥ 0; x₃, x₄ 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 = 1000x₁ + 4000x₂ + 5000x₃ subject to the constraints:
x₁ + 2x₂ + 3x₃≤ 14
x₁ + 2x₁ ≤ 14
x₁, x₂, x₃ ≥ 0(8 marks) 3 (a) Evaluate using contour integration:
(6 marks) 3 (b) State caley-hamilton theorem. Use it to find A^-1 and A⁴
where,
(6 marks) 3 (c) Use penalty method to
Minimize z = x₁ + 2x₂ + x₃
subject to x₁ + (x₂/2) + (x₃/2) ≤ 1
(3/2) x₁ + 2x₂ + x₃ ≥ 8
x₁, x₂,x₃ ≥ 0(8 marks) 4 (a) Find A¹⁰⁰
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 = 3x₁ + 2x₂ + x₃ + 4x₄;
subject to 2x₁ + 4x₂ + 5x₃ + x₄ ≥ 10
3x₁ - x₂ + 7x₃ - 2x₄ ≥ 2
5x₁ + 2x₂ + x₃ + 6x₄ ≥ 15
x₁, x₂, x₃, x₄ ≥ 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=z².(6 marks) 6 (c) Use Lagrangian Multiplier method to optimize (8)z=2x₁² + x₂² + 3x₃² + 10x₁ + 8x₂ + 6x₃ - 100;
subject to x₁ + x₂ + x₃ = 20
x₁, x₂, x₃ ≥ 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 + (y² - x²) sin y}. (6 marks) 7 (c) Solve the following N.L.P.P. using Kuhn-Tucker conditions.
Optimize z = 2x₁ + 3x₂ - (x₁² + x₂² + x₃²)
subject to x₁ + x₂ ≤ 1
2x₁ + 3x₂ ≤ 6
x₁ , x₂ ≥ 0.(8 marks)