Applied Mathematics 4 - May 2012

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) Construct dual penalty of the following LPP:
Max z = 8x₁ + 3x₂
Subject to x₁ - 6x₂ ≥ 2
5x₁ + 7x₂ = -4; x₁, x₂ ≥ 0(5 marks) 1 (b) Find the orthogonal tracjectory of the family of curves given by:
2x - x³ + 3xy² = a(5 marks) 1 (c) Evaluate ∫{(1/z)dz}, z=C; where C is the upper half of | z |= 1.(5 marks) 1 (d) Show that every skew-hermitian matrix can be expressed in the form P+iQ where P is real skew-symmetric and Q is real Symmetric matrix.(5 marks) 2 (a) Determine the analytic function f(z) = - r³sin 3 Θ.(6 marks) 2 (b) Show that A = is derogatory.
(6 marks) 2 (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 + 4x₁ ≤ 15,
x₁, x₂, x₃ − 0(8 marks) 3 (a) Evaluate _-∞ ∫^∞{(x² - x + 2)/(x⁴ + 10x² + 9) dx}; using contour integration.(6 marks) 3 (b) Find the eigen values and eigen vectors of A³ where A =
(6 marks) 3 (c) Use penalty method (Big M) to solve Mini z = 4x + y subject to
3x + y = 3,
4x + 3y ≥ 6,
x + 2y ≤ 4,
x, y ≥ 0(8 marks) 4 (a) If w = f(z) then prove that
(dw/dz) = (cosθ - isinθ) (∂w/∂r)(6 marks) 4 (b) If A is orthogonal, find a,b,c. Also find A^-1.
(6 marks) 4 (c) Use dual simplex method to solve the LPP.
Minimize z = 6x₁ + 7x₂ + 3x₃ + 5x₄;
subject to 5x₁ + 6x₂ - 3x₃ + 4x₄ ≥ 12
x₂ + 5x₃ - 6x₄ ≥ 10
2x₁ + 5x₂ + x₃ + x₄ ≥ 8;
x₁, x₂, x₃, x₄ ≥ 0(8 marks) 5 (a) Prove that 3tan A = Atan 3, where:
(6 marks) 5 (b) Find the image of the line y - x + 1 = 0 under the transformation w = 1/z. Also find the image of the line y = x under the same transformation. Draw rough sketches.(6 marks) 5 (c) Solve the NLPP using the method of Lagrangian Multipliers.
Minimize z = x₁² + x₂² + x₃²;
subject to x₁ + x₂ + 3x₃ = 2, 5x₁ + 2x₂ + x₃ = 5,
x₁, x₂, x₃ ≥ 0(8 marks) 6 (a) Verify caley-hamiton theorem for A and hence find
A^-1
A³ - 5A².
where,
(6 marks) 6 (b) Find the bilinear transformation that maps the points ∞, i, 0 onto the points 0, i, ∞. Find the fixed points.(6 marks) 6 (c) State and prove Cauchy s integral theorem and hence evaluate
,
where,