## Applied Mathematics 4 - Dec 2013

### Information Technology (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 A^{8} -5A^{7} +7A^{6} -3A^{5} +A^{4}- 5A^{3}+ 8A^{2}- 2A+I

where,

(5 marks)
**1 (b)** Find the orthogonal trajectory of the family of curves x^{3}y-xy^{3}=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 = x_{1} + x_{2};

subject to 2x_{1} + x_{2} ≥ 2;

-x_{1} - x_{2} ≥ 1;

x_{1}, x_{2} ≥ 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 = 2x_{1} + 3x_{2}

subject to the constraints:

x_{1} + x_{2} ≥ 5,

x_{1} + 2x_{2} ≥ 5,

x_{1} , x_{2} ≥ 0
(8 marks)
**3 (a)** Use Lagrangian Multiplier method to optimize

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
(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 = 430x_{1} + 460x_{2} + 420x_{3}

subject to x_{1} + 3x_{2} + 4x_{3}≥ 3

2x_{1} + 4x_{2} ≥ 2

x_{1} + 2x_{2} ≥ 5

x_{1}, x_{2}, x_{3} ≥ 0
(8 marks)
**5 (a)** Consider the following problem

Maximize z = x_{1} + 3x_{2} + 4x_{3}

Subject to x_{1} + 2x_{2} + 3x_{3} = 4

2x_{1} + 3x_{2} + 5x_{3} = 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) / (z^{2} - 2z -3)] indicating regions of convergences.(6 marks)
**5 (c)** Verify caley-hamilton theorem for the matrix A and hence find A^{-1} and A^{4}

where,

(8 marks)
**6 (a)** If u = -r^{3}sin 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,><>(6 marks)
**6 (c)** Use simplex method to solve the LPP

Max z = 3x_{1} + 5x_{2} + 4x_{3} subject to the constraints:

2x_{1} + 3x_{2} ≤ 8,

2x_{2} + 5x_{3} ≤ 10

3x_{1} + 2x_{2} + 4x_{3} ≤ 15,

x_{1}, x_{2}, x_{3} ≥ 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 = 2x_{1} + 3x_{2} -x_{1}^{2} - 2x_{2}^{2}

subject to x_{1} + 3x_{2} ≤ 6

5x_{1} + 2x_{2} ≤ 10

x_{1},x_{2} ≥ 0.
(8 marks)