Paper II PYQs-2011

1.(a) Show that the set $G=\{f_1,f_2,f_3,f_4,f_5,f_6\}$ of six transformations on the set of Complex numbers defined by $f_1(z)=z$,
$f_2(z)=1-z$,
$f_3(z)={\displaystyle \frac{z}{(1-z)}}$,
$f_4(z)={\displaystyle \frac{1}{z}}$,
$f_5(z)={\displaystyle \frac{1}{(1-z)}}$,
$f_6(z)={\displaystyle \frac{(z-1)}{z}}$
is a non-abelian group of order 6 wrt composition of mappings.

[12M]

1.(b) Let $S=(0,1)$ and $f$ be defined by $f(x)={\displaystyle \frac{1}{x}}$ where $0<x\le 1$ (in $R$). Is $f$ uniformly continuous on $S$? Justify your answer.

[12M]

1.(c) If $f(z)=u+iv$ is an analytic function of $z=x+iy$ and $u-v={\displaystyle \frac{e^y-\cos x+\sin x}{\cosh y-\cos x}}$, find $f(z)$ subject to the condition, $f\left({\displaystyle \frac{\pi }{2}}\right)={\displaystyle \frac{3-i}{2}}$.

[12M]

1.(d) Solve by Simplex method. Maximize
$Z=5x_1+x_2$,
subject to constraints:
$3x_1+5x_2\le 15$
$5x_1+2x_2\le 10$
$x_1$, $x_2\ge 0$

[12M]

1.(e)(i) Prove that a group of Prime order is abelian.

[6M]

1.(e)(ii) How many generators are there of the cyclic group $(G,.)$ of order 8?

[6M]

2.(a) Give an example of a group $G$ in which every proper subgroup is cyclic but the group itself is not cyclic.

[15M]

2.(b) Let $f_n(x)=nx(1-x{)}^n$, $x\in [0,1]$. Examine the uniform convergence of $\{f_n(x)\}$ on $[0,1]$.

[15M]

2.(c) If the function $f(z)$ is analytic and one valued in $|z-a|<R$, prove that for $0<r<R$, $f^{\mathrm{\prime }}(a)={\displaystyle \frac{1}{\pi r}}{\int }_0^{2\pi }\mathrm{P}(\theta )e^{-i\theta }d\theta$, where $P(\theta )$ is the real part of $f(a+r{e}^{i\theta })$.

[15M]

2.(d) Find the shortest distance from the origin $(0,0)$ to the hyperbola $x^2+8xy+7y^2=225$.

[15M]

3.(a) Let $F$ be the set of all real valued continuous functions defined on the closed interval $[0,1]$. Prove that $(F,+,.)$ is a Commutative Ring with unity with respect to addition and multiplication of functions defined point wise as below:-
$(f+g)x=f(x)+g(x)$ and $(fg)x=f(x)g(x)$
$x\in [0,1]$ where $f,g\in F$.

[15M]

3.(b) Show that the series for which the sum of first $n$ terms, $f_n(x)={\displaystyle \frac{nx}{1+n^2{x}^2}}$, $0\le x\le 1$ cannot be differentiated term-by-term at $x=0$. What happens at $x\ne 0$?

[15M]

3.(c) Evaluate by Contour integration, ${\int }_0^1{\displaystyle \frac{dx}{{(x^2-x^3)}^{{\displaystyle \frac{1}{3}}}}}$.

[15M]

3.(d) Find the Laurent series for the function $f(z)={\displaystyle \frac{1}{1-z^2}}$ with centre $z=1$.

[15M]

4.(a) Let $a$ and $b$ be elements of a group, with $a^2=e$, $b^6=e$ and $ab=b^{4}a$. Find the order of $ab$, and express its inverse in each of the forms $a^m{b}^n$ and $b^m{a}^n$.

[20M]

4.(b) Show that if $S(x)=\sum _{n=1}^{\mathrm{\infty }}{\displaystyle \frac{1}{n^3+n^4{x}^2}}$, then its derivative $S^{\mathrm{\prime }}(x)=-2x\sum _{n=1}^{\mathrm{\infty }}{\displaystyle \frac{1}{n^2{(1+n{x}^2)}^2}}$, for all $x$.

[20M]

4.(c) Write down the dual of the following LP problem and hence solve it by graphical method.
Minimize $Z=6x_1+4x_2$,
subject to constraints:
$2x_1+x_2\ge 1$
$3x_1+4x_2\ge 1.5$
$x_1,x_2\ge 0$

[20M]

5.(a) Solve the PDE $(D^2-D^{\mathrm{\prime }2}+D+3D^{\mathrm{\prime }}-2)z$=$e^{(x-y)}-x^{2}y$.

[12M]

5.(b) Solve the PDE $(x+2z){\displaystyle \frac{\mathrm{\partial }z}{\mathrm{\partial }x}}+(4zx-y){\displaystyle \frac{\mathrm{\partial }z}{\mathrm{\partial }y}}=2x^2+y$.

[12M]

5.(c) Calculate ${\int }_2^{10}{\displaystyle \frac{dx}{1+x}}$ (up to 3 places of decimal) by dividing the range into 8 equal parts by Simpson’s ${\displaystyle \frac{1}{3}}$rd rule.

[12M]

5.(d)(i) Compute $(3205{)}_{10}$ to the base 8.

[6M]

5.(d)(ii) Let $A$ be an arbitrary but fixed Boolean algebra with operations $\wedge$, $\vee$ and ${}^{\prime }$, and the zero and the unit element denoted by 0 and 1 respectively. Let $x$, $y$, $z\dots$ be elements of $A$. If $x,y\in A$ be such that $x\wedge y=0$ and $x\vee y=1$, then prove that $y=x^{\mathrm{\prime }}$.

[6M]

5.(e) Let $a$ be the radius of the base of a right circular cone of height $h$ and mass $M$ . Find the moment of inertia of that right circular cone about a line through the vertex perpendicular to the axis.

[12M]

6.(a) Find the surface satisfying ${\displaystyle \frac{{\mathrm{\partial }}^{2}z}{\mathrm{\partial }{x}^2}}=6x+2$ and touching $z=x^3+y^3$ along its section by the plane $x+y+1=0$.

[20M]

6.(b) Solve ${\displaystyle \frac{{\mathrm{\partial }}^{2}u}{\mathrm{\partial }{x}^2}}+{\displaystyle \frac{{\mathrm{\partial }}^{2}u}{\mathrm{\partial }{y}^2}}=0,0\le x\le a,0\le y\le b$ satisfying the boundary conditions $u(0,y)=0$, $u(x,0)=0$, $u(x,b)=0$, ${\displaystyle \frac{\mathrm{\partial }u}{\mathrm{\partial }x}}(a,y)=T{\sin }^3{\displaystyle \frac{\pi y}{a}}$.

[20M]

6.(c) Obtain temperature distribution $y(x,t)$ in a uniform bar of unit length whose one end is kept at ${10}^0$ and the other end is insulated. Also it is given that $y(x,0)=1-x$, $0<x<1$.

[20M]

7.(a) A solid of revolution is formed by rotating about the $x-axis$, the area between the $x-axis$, the line $x=0$ and $x=1$ and a curve through the points with the following co-ordinates:
$\begin{array}{|cccccc|}\hline x& 0.00& 0.25& 0.50& 0.75& 1\\ y& 1& 0.9896& 0.9589& 0.9089& 0.8415\\ \hline\end{array}$

Find the volume of the solid.

[20M]

7.(b) Find the logic circuit that represents the following Boolean function. Find also an equivalent simpler circuit:
$\begin{array}{|cccc|}\hline x& y& z& f(x,y,z)\\ 1& 1& 1& 1\\ 1& 1& 0& 0\\ 1& 0& 1& 0\\ 1& 0& 1& 0\\ 0& 1& 1& 1\\ 0& 1& 0& 0\\ 0& 0& 1& 0\\ 0& 0& 0& 0\\ \hline\end{array}$

[20M]

7.(c) Draw a flow chart for Lagrange’s interpolation formula.

[20M]

8.(a) The ends of a heavy rod of length $2a$ are rigidly attached to two light rings which can respectively slide on the thin smooth fixed horizontal and vertical wires $O$. The rod starts at an angle $\alpha$ to the horizon with an angular velocity $\sqrt{[3g(1-\sin \alpha )/2a]}$ and moves downwards. Show that it will strike the horizontal wire at the end of time $-2\sqrt{a/(3g)}\log [\tan ({\displaystyle \frac{\pi }{8}}-{\displaystyle \frac{\alpha }{4}})\cot {\displaystyle \frac{\pi }{8}}]$.

[30M]

8.(b) An infinite row of the equidistance rectilinear vortices are at distance $a$ apart. The vortices are of the same numerical strength $K$ but they are alternately of opposite signs. Find the Complex function that determines the velocity potential and the stream function.

[30M]