Applied Mathematics 4 - Dec 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) 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 marks) 1 (b) Find the orthogonal trajectory of the family of curves x³y-xy³=c. (5 marks) 1 (c) Evaluate:

where C is a circle |z|=1.(5 marks) 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 marks) 2 (a) Find the eigen values and eigen vectors of the matrix.
(6 marks) 2 (b) Find the imaginary part of the analytic function whose real part is -
e^2x(xcos 2y - sin 2y). Also verify that it is harmonic.(6 marks) 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 marks) 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 marks) 3 (b) Evaluate $$\displaystyle\int\limits_c\dfrac{z^2}{(z-1)^2(z-2)}dz$$(6 marks) 3 (c) Show that A is derogatory.
(8 marks) 4 (a) Show that A is diagonaisable. Also find the transforming and diagonal matrix.
(6 marks) 4 (b) Show that f(z)= √(|xy|) is not analytic at the origin although Cauchy-Reimann. Equations are satisfied at that point.(6 marks) 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 marks) 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 marks) 5 (b) Obtain Taylor^'s and Laurent^'s expansion of f(z)= [(z-1) / (z² - 2z -3)] indicating regions of convergences.(6 marks) 5 (c) Verify caley-hamilton theorem for the matrix A and hence find A^-1 and A⁴
where,
(8 marks) 6 (a) If u = -r³sin 3θ,find the analytic function f(z) whose real part is u. (6 marks) 6 (b) Prove that 3 tan A=A tan 3.
<where,><br></where,></span>(6 marks) 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 marks) 7 (a) Find the bilinear transformation that maps the points ∞, i, 0 onto the points 0, i, ∞.(6 marks) 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 marks) 7 (c) Using Kuhn-Tucker conditions:
Minimize z = 2x₁ + 3x₂ -x₁² - 2x₂²
subject to x₁ + 3x₂ ≤ 6
5x₁ + 2x₂ ≤ 10
x₁,x₂ ≥ 0.(8 marks)