Paper II PYQs-2014

1.(a) Let G be the set of all real 2×2(xy0z), where xz≠0 matrices. Show that G is group under matrix multiplication. Let N denote the subset {[1a01]:a∈R}. Is N a normal subgroup of G? Justify your answer.

[10M]

1.(b) Test the convergence of the improper integral ∫∞1dxx2(1+e−x).

[10M]

1.(c) Prove that the function f(z)=u+iv, where f(z)=x3(1+i)−y3(1−i)x2+y2, z≠0; f(0)=0 satisfies Cauchy-Riemann equations at the origin, but the derivative of f at z=0 does not exist.

[10M]

1.(d) Expand in Laurent series the function f(z)=1z2(z−1) about z=0 and z=1.

[10M]

1.(e) Solve graphically:

Maximize
Z=6x1+5x2,
subject to:
2x1+x2≤216
x1+x2≤11
x1+2x2≥6 5x1+6x2≤90
x1, x2≥0

[10M]

2.(a) Show that Z7 is a field. Then find ([5]+[6])−1 and (−[4])−1 in Z7.

[15M]

2.(b) Integrate ∫01f(x)dx, where f(x)={2xsin1x−cos1x,x∈[0,1]0,x=0

[15M]

2.(c) Find the initial basic feasible solution to the following transportation problem by Vogel’s approximation method. Also, find its optimal solution and the minimum transportation cost.

[20M]

3.(a) Show that the set {a+bω:ω3=1}, where a and b are real numbers, is a field with respect to usual addition and multiplication.

[15M]

3.(b) Obtain ∂2f(0,0)∂x∂y and ∂2f(0,0)∂y∂x for the function

Also, discuss the continuity ∂2f∂x∂y and ∂2f∂y∂x of f at (0,0).

[15M]

3.(c) Evaluate the integral ∫π0dθ(1+12cosθ)2 using residues.

[20M]

4.(a) Prove that the set Q(√5)={a+b√5:a,b∈Q} is commutative ring with identity.

[15M]

4.(b) Find the minimum value of x2+y2+z2 subject to the condition xyz=a3 by the method of Lagrange multipliers.

[15M]

4.(c) Find all optimal solutions of the following linear programming problem by the simplex method.
Maximize
Z=30x1+24x2,
subject to:
5x1+4x2≤200
x1≤32
x2≤40
x1x2≥0

[20M]

5.(a) Solve the partial differential equation (2D2−5DD′+2D′2)z=24(y−x).

[10M]

5.(b) Apply Newton-Raphson method to determine a root of the equation cosx−xex=0 correct up to four decimal places.

[10M]

5.(c) Use five subintervals to integrate ∫10dx1+x2 using trapezoidal rule.

[10M]

5.(d) Use only AND and OR logic gates to construct a logic circuit for the Boolean expression z=xy+uv.

[10M]

5.(e) Find the equation of motion of a compound pendulum using Hamilton’s equations.

[10M]

6.(a) Reduce the equation ∂2z∂x2=x2∂2z∂y2 to canonical form.

[15M]

6.(b) Solve the system of equations 2x1−x2=7
−x1+2x2−x3=1
−x2+2x3=1
using Gauss-Seidel iteration method (perform three iterations).

[15M]

6.(c) Use Runge-Kutta formula of fourth order to find the value of y at x=0.8, where dydx=√x+y, y(0.4)=0.41. Take the step length h=0.2.

[20M]

7.(a) Find the deflection of a vibrating string (length =π, ends fixed, ∂2u∂t2=∂2u∂x2) corresponding to zero initial velocity and initial deflection. f(x)=k(sinx−sin2x).

[15M]

7.(b) Draw a flowchart for Simpson’s one-third rule.

[15M]

7.(c) Given the velocity potential ϕ=12log[(x+a)2+y2(x−a)2+y2], determine the streamlines.

[20M]

8.(a) Solve ∂2u∂t2=∂2u∂x2, 0<x<1, t>0, given that: i) u(x,0)=0, 0≤x≤1
ii) ∂u∂t(x,0)=x2, 0≤x≤1
iii) u(0,t)=u(1,t)=0, for all t

[15M]

8.(b) For any Boolean variables x and y, show that x+xy=x.

[15M]

8.(c) Find Navier-Stokes equation for steady laminar flow of a viscous incompressible fluid between two infinite parallel plates.

[20M]