Paper II PYQs-2014

1.(a) Let $G$ be the set of all real $2\times 2\left(\begin{array}{cc}x& y\\ 0& z\end{array}\right)$, where $xz\ne 0$ matrices. Show that $G$ is group under matrix multiplication. Let $N$ denote the subset $\{\left[\begin{array}{cc}1& a\\ 0& 1\end{array}\right]:a\in R\}$. Is $N$ a normal subgroup of $G$? Justify your answer.

[10M]

1.(b) Test the convergence of the improper integral ${\int }_1^{\mathrm{\infty }}{\displaystyle \frac{dx}{x^2(1+e^{-x})}}$.

[10M]

1.(c) Prove that the function $f(z)=u+iv$, where $f(z)={\displaystyle \frac{x^3(1+i)-y^3(1-i)}{x^2+y^2}}$, $z\ne 0$; $f(0)=0$ 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 $f(z)={\displaystyle \frac{1}{z^2(z-1)}}$ about $z=0$ and $z=1$.

[10M]

1.(e) Solve graphically:

Maximize
$Z=6x_1+5x_2$,
subject to:
$2x_1+x_2\le 216$
$x_1+x_2\le 11$
$x_1+2x_2\ge 6$ $5x_1+6x_2\le 90$
$x_1$, $x_2\ge 0$

[10M]

2.(a) Show that $Z_7$ is a field. Then find $([5]+[6]{)}^{-1}$ and $(-[4]{)}^{-1}$ in $Z_7$.

[15M]

2.(b) Integrate ${\int }_1^{0}f(x)dx$, where $f(x)=\{\begin{array}{ll}2x\sin {\displaystyle \frac{1}{x}}-\cos {\displaystyle \frac{1}{x}},x& \in [0,1]\\ 0,& x=0\end{array}$

[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 $\{a+b\omega :{\omega }^3=1\}$, where $a$ and $b$ are real numbers, is a field with respect to usual addition and multiplication.

[15M]

3.(b) Obtain ${\displaystyle \frac{{\mathrm{\partial }}^{2}f(0,0)}{\mathrm{\partial }x\mathrm{\partial }y}}$ and ${\displaystyle \frac{{\mathrm{\partial }}^{2}f(0,0)}{\mathrm{\partial }y\mathrm{\partial }x}}$ for the function

Also, discuss the continuity ${\displaystyle \frac{{\mathrm{\partial }}^{2}f}{\mathrm{\partial }x\mathrm{\partial }y}}$ and ${\displaystyle \frac{{\mathrm{\partial }}^{2}f}{\mathrm{\partial }y\mathrm{\partial }x}}$ of $f$ at $(0,0)$.

[15M]

3.(c) Evaluate the integral ${\int }_0^{\pi }{\displaystyle \frac{d\theta }{{(1+{\displaystyle \frac{1}{2}}\cos \theta )}^2}}$ using residues.

[20M]

4.(a) Prove that the set $Q(\sqrt{5})=\{a+b\sqrt{5}:a,b\in Q\}$ is commutative ring with identity.

[15M]

4.(b) Find the minimum value of $x^2+y^2+z^2$ subject to the condition $xyz=a^3$ by the method of Lagrange multipliers.

[15M]

4.(c) Find all optimal solutions of the following linear programming problem by the simplex method.
Maximize
$Z=30x_1+24x_2$,
subject to:
$5x_1+4x_2\le 200$
$x_1\le 32$
$x_2\le 40$
$x_1{x}_2\ge 0$

[20M]

5.(a) Solve the partial differential equation $(2D^2-5D{D}^{\mathrm{\prime }}+2D^{\mathrm{\prime }2})z=24(y-x)$.

[10M]

5.(b) Apply Newton-Raphson method to determine a root of the equation $\cos x-x{e}^x=0$ correct up to four decimal places.

[10M]

5.(c) Use five subintervals to integrate ${\int }_0^1{\displaystyle \frac{dx}{1+x^2}}$ 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 ${\displaystyle \frac{{\mathrm{\partial }}^{2}z}{\mathrm{\partial }{x}^2}}=x^2{\displaystyle \frac{{\mathrm{\partial }}^{2}z}{\mathrm{\partial }{y}^2}}$ to canonical form.

[15M]

6.(b) Solve the system of equations $2x_1-x_2=7$
$-x_1+2x_2-x_3=1$
$-x_2+2x_3=1$
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 ${\displaystyle \frac{dy}{dx}}=\sqrt{x+y}$, $y(0.4)=0.41$. Take the step length $h=0.2$.

[20M]

7.(a) Find the deflection of a vibrating string (length $=\pi$, ends fixed, ${\displaystyle \frac{{\mathrm{\partial }}^{2}u}{\mathrm{\partial }{t}^2}}$=${\displaystyle \frac{{\mathrm{\partial }}^{2}u}{\mathrm{\partial }{x}^2}})$ corresponding to zero initial velocity and initial deflection. $f(x)=k(\sin x-\sin 2x)$.

[15M]

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

[15M]

7.(c) Given the velocity potential $\varphi ={\displaystyle \frac{1}{2}}\log \left[{\displaystyle \frac{(x+a{)}^2+y^2}{(x-a{)}^2+y^2}}\right]$, determine the streamlines.

[20M]

8.(a) Solve ${\displaystyle \frac{{\mathrm{\partial }}^{2}u}{\mathrm{\partial }{t}^2}}={\displaystyle \frac{{\mathrm{\partial }}^{2}u}{\mathrm{\partial }{x}^2}}$, $0<x<1$, $t>0$, given that: i) $u(x,0)=0$, $0\le x\le 1$
ii) ${\displaystyle \frac{\mathrm{\partial }u}{\mathrm{\partial }t}}(x,0)=x^2$, $0\le x\le 1$
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]