Paper II PYQs-2019

1.(a) Let $G$ be a finite group, $H$ and $K$ subgroups of $G$ such that $K\subset H$. Show that $(G:K)=(G:H)(H:K)$.

[10M]

1.(b) Show that the function

is continuous and differentiable at $(1,-1)$.

[10M]

1.(c) Evaluate

[10M]

1.(d) Suppose $f(z)$ is analytic function on a domain $D\in C$ and satisfies the equation $Imf(z)=(Ref(z){)}^2$, $Z\in D$. Show that $f(z)$ is constant in $D$.

[10M]

1.(e) Use graphical method to solve the linear programming problem.
Maximize $Z=3x_1+2x_2$
subject to
$x_1-x_2\ge 1$,
$x_1+x_2\ge 3$ and
$x_1,x_2,x_3\ge 0$

[10M]

2.(a) If $G$ and $H$ are finite groups whose orders are relatively prime, then prove that there is only one homomorphism from $G$ to $H$, the trivial one.

[10M]

2.(b) Write down all quotient groups of the group $Z_{12}$.

[10M]

2.(c) Using differentials, find an approximate value of $f(4.1,4.9)$ where

[15M]

2.(d) Show that an isolated singular point $z_o$ of a function $f(z)$ is a pole of order $m$ if and only if $f(z)$ can be written in the form $f(z)={\displaystyle \frac{{\varphi }^{m-1}(z_o)}{(m-1)!}}$ if $m\ge 1$.

Moreover $Re{s}_{z=z_0}$ $f(z)={\displaystyle \frac{{\varphi }^{(m-1)}(z_0)}{(m-1)!}}$ if $m\ge 1$.

[15M]

3.(a) Discuss the uniform convergence of

$f_n(x)={\displaystyle \frac{nx}{1+n^2{x}^2}},\mathrm{\forall }x\in R(-\mathrm{\infty },\mathrm{\infty })$, $n=1,2,\mathrm{3..}$

[15M]

3.(b) Solve the linear programming problem using Simplex method.
Maximize $Z=x_1+2x_2-3x_3-2x_4$
subject to
$x_1+2x_2-3x_3+x_4=4$,
$x_1+2x_2+x_3+2x_4=4$ and
$x_1$, $x_2$, $x_3$, $x_4\ge 0$

[15M]

3.(c)(i) Evalute the integral ${\int }_{C}Re(z^2)dz$ from 0 to $2+4i$ along the curve $C$ where $C$ is a parabola $y=x^2$.

[10M]

3.(c)(ii) Let $a$ be an irreducible element of the Euclidean ring $R$, then prove that $R/(a)$ is a field.

[10M]

4.(a) Find the maximum value of $f(x,y,z)=x^2{y}^2{z}^2$ subject to the subsidiary condition $x^2+y^2+z^2=c^2$, $(x,y,z>0)$.

[15M]

4.(b) Obtain the first terms of the Laurent series expansion of the function $f(z)={\displaystyle \frac{1}{(e^x-1)}}$ about the point $z=0$ valid in the region $0<|z|<2\pi$.

[10M]

4.(c) Discuss the convergence of ${\int }_1^2{\displaystyle \frac{\sqrt{x}}{\ln x}}dx$.

[15M]

4.(d) Consider the following LPP

Maximize $Z=2x_1+4x_2+4x_3-3x_4$ subject to $x_1+x_2+x_3=4$, $x_1+4x_2+x_4=8$ and $x_1,x_2,x_3,x_4\ge 0$

Use the dual problem to verify that the basic solution $(x_1,x_2)$ is not optimal.

[10M]

5.(a) Form a partial differential equation of the family of surfaces given by the following expression:

$\psi (x^2+y^2+2z^2,y^2-2zx)=0$.

[10M]

5.(b) Apply Newton-Raphson method, to find a real root of transcendental equation $xlo{g}_{10}x=1.2$, correct to three decimal places.

[10M]

5.(c) A uniform rod $OA$, of length $2a$, free to turn about its end $O$, revolves with angular velocity $\omega$ about the vertical $OZ$ through $O$, and is inclined at a constant angle $\alpha$ to $OZ$; find the value of $\alpha$.

[10M]

5.(d) Using Runge-Kutta method of fourth order, solve ${\displaystyle \frac{dy}{dx}}={\displaystyle \frac{y^2-x^2}{y^2+x^2}}$ with $y(0)=1$ at $x=0.2$. Use four decimal places for calculation and step length 0.2.

[10M]

5.(e) Draw a flow chart and write a basic algorithm (in FORTRAN/C/C++) for evaluating $y={\int }_0^6{\displaystyle \frac{dx}{1+x^2}}$ using Trapezoidal rule.

[10M]

6.(a) Solve the first order quasilinear partial differential equation by the method of characteristics:

$x{\displaystyle \frac{\mathrm{\partial }u}{\mathrm{\partial }x}}+(u-x-y){\displaystyle \frac{\mathrm{\partial }u}{\mathrm{\partial }y}}=x+2y$ in $x>0,-\mathrm{\infty }<u<\mathrm{\infty }$ with $u=1+y$ on $x=1$.

[15M]

6.(b) Find the equivalent numbers given in a specified number to the system mentioned against them: (i) Integer 524 in binary system.
(ii) 101010110101.101101011 to octal system.
(iii) Decimal number 5280 to hexadecimal system
(iv) Find the unknown number $(1101.101{)}_8$ to $(?{)}_{10}$.

[15M]

6.(c) A circular cylinder of radius $a$ and radius of gyration $k$ rolls without slipping inside a fixed hollow cylinder of radius $b$. Show Show that the plane through axes moves in a circular pendulum of length $(b-a)(1+{\displaystyle \frac{k^2}{a^2}})$.

[20M]

7.(a) Using Hamilton’s equation, find the acceleration for a sphere rolling down a rough inclined plane, if $x$ be a distance of the point of contact of the sphere from a fixed point of the plane.

[15M]

7.(b) Apply Gauss-Seidel iteration method to solve the following system of equation:
$2x+y-2z=17$
$3x+20y-z=-18$
$2x-3y+20z=25$, correct to three decimal places.

[15M]

7.(c) Reduce the following second order partial differential equation to canonical from and find the general solution:

${\displaystyle \frac{{\mathrm{\partial }}^{2}u}{\mathrm{\partial }{x}^2}}-2x{\displaystyle \frac{{\mathrm{\partial }}^{2}u}{\mathrm{\partial }x\mathrm{\partial }y}}+x^2{\displaystyle \frac{{\mathrm{\partial }}^{2}u}{\mathrm{\partial }{y}^2}}$=${\displaystyle \frac{\mathrm{\partial }u}{\mathrm{\partial }y}}+12x$

[20M]

8.(a) Given the Boolean expression

(i) Draw the logical diagram for the expression.
(ii) Minimize the expression.
(iii) Draw the logical diagram for the reduced expression.

[15M]

8.(b) A sphere of radius $R$, whose centre is at rest, vibrates radially in an infinte incompressible fluid of density $\rho$, which is at rest at infinity. If the pressure at infinity is $\mathrm{\Pi }$, show that the pressure at the surface of the sphere at time $t$ is

$\mathrm{\Pi }+{\displaystyle \frac{1}{2}}\rho \{{\displaystyle \frac{d^2{R}^2}{d{t}^2}}+({\displaystyle \frac{dR}{dt}}{)}^2\}$.

[15M]

8.(c) Two sources, each of strength $m$, are placed at the point at the points $(-a,0)$, $(a,0)$ and a sink of strength $2m$ at origin. Show that the stream lines are the curves $(x^2+y^2{)}^2$=$a^2(x^2-y^2+\lambda xy)$, where $\lambda$ is variable parameter.

Show also that the fluid speed at any point is $(2m{a}^2)/(r_1{r}_2{r}_3)$, where $r_1$, $r_2$ and $r_3$ are the distances of the points from the sources and the sink, respectively.

[20M]