Paper II PYQs-2010

1.(a) Let $G=R-\{-1\}$ be the set of all real numbers omitting -1. Define the binary relation $\ast$ on $G$ by $a\ast b=a+b+ab$. Show $(G,\ast )$ is a group and it is abelian.

[12M]

1.(b) Show that a cyclic group of order 6 is isomorphic to the product of a cyclic group of order 2 and a cyclic group of order 3. Can you generalize this? Justify.

[12M]

1.(c) Discuss the convergence of the sequence $\left\{x_n\right\}$ where $X_n={\displaystyle \frac{\sin \left({\displaystyle \frac{n\pi }{2}}\right)}{8}}$.

[12M]

1.(d) Define $\left\{x_n\right\}$ by $x_1=5$ and $x_{n+1}=\sqrt{4+x_n}$ for $n>1$. Show that the sequence converges to $\left({\displaystyle \frac{1+\sqrt{17}}{2}}\right)$.

[12M]

1.(e) Show that $u(x,y)=2x-x^3+3x{y}^2$ is a harmonic function. Find a harmonic conjugate of $u(x,y)$. Hence, find the analytic function $f$ for which $u(x,y)$ is the real part.

[12M]

1.(f) Construct the dual of the primal problem.
Maximize $Z=2x_1+x_2+x_3$, subject to the constraints:
$x_1+x_2+x_3\ge 6$
$3x_1-2x_2+3x_3=3$
$-4x_1+3x_2-6x_3=3$
$x_1,x_2,x_3\ge 0$

[12M]

2.(a) Let $(R^{\ast },.)$ be the multiplicative group of non-zero reals and $(GL(n,R),X)$ be the multiplicative group of $n\times n$ non-singular real matrices. Show that the quotient group ${\displaystyle \frac{GL(n,R)}{SL(n,R)}}$ and $(R^{\ast },.)$ are isomorphic where $SL(n,R)=\{A\in GL(n,R)/\det A=1\}$. What is the center of $GL(n,R)$?

[15M]

2.(b) Let $C=\{f:I=[0,1]\to R/f$ is continuous $\}$. Show $C$ is a commutative ring with 1 under point wise addition and multiplication. Determine whether $C$s is an integral domain. Explain.

[15M]

2.(c) Define the function

$f(x)=\{\begin{array}{ll}{x}^2\sin {\displaystyle \frac{1}{x}},& \text{ if }x\ne 0\\ 0,& \text{ if }x=0\end{array}$.

Find $f^{\mathrm{\prime }}(x)$. Is $f^{\mathrm{\prime }}(x)$ continuous at $x=0$? Justify your answer.

[15M]

2.(d) Consider the series $\sum _{n=0}^{\mathrm{\infty }}{\displaystyle \frac{x^2}{{(1+x^2)}^2}}$. Find the values of $x$ for which it is convergent and also the sum function. Is the convergence uniform? Justify your answer.

[15M]

3.(a) Consider the polynomial ring $Q[x]$. Show $p(x)=x^3-2$ is irreducible over $Q$. Let $I$ be the ideal $Q[x]$ in generated by $p(x)$. Then show that ${\displaystyle \frac{Q[x]}{I}}$ is a field and that each element of it is of the form $a_0+a_{1}t+a_2{t}^2$ with ${\alpha }_0,a_1,a_2$ in $Q$ and $t=x+I$.

[15M]

3.(b) Show that the quotient ring ${\displaystyle \frac{Z[i]}{1+3i}}$ is isomorphic to the ring ${\displaystyle \frac{Z}{10Z}}$ where $Z[i]$ denotes the ring of Gaussian integers.

[15M]

3.(c) Let $f_n(x)=x^n$ on $-1<x\le 1$ for $n=1,2\dots \dots$. Find the limit function. Is the convergence uniform? Justify your answer.

[15M]

3.(d) Consider the series $\sum _{n=0}^{\mathrm{\infty }}{\displaystyle \frac{x^2}{(1+x^2{)}^n}}$.

Find the values of $x$ for which it is convergent and also the sum function. Is the convergence uniform? Justify your answer.

[15M]

4.(a)(i) Evaluate the line integral $\oint f(z)dz$ where $f(z)=z^2$, $c$ is the boundary of the triangle with vertices $A(0,0)$, $B(1,0)$, $C(1,2)$ in that order.

[7.5M]

4.(a)(ii) Find the image of the finite vertical strip $R$: $x=5$ to $x=5$, $-\pi \le \gamma \le \pi$ of $z-plane$ exponential function.

[7.5M]

4.(b) Find the Laurent series of the function $f(z)=\exp \left[{\displaystyle \frac{\lambda }{2}}(z-{\displaystyle \frac{1}{z}})\right]$ as $\sum _{n=-\mathrm{\infty }}^{\mathrm{\infty }}{C}_n{z}^n$ for 0, $|z|<\mathrm{\infty }$ $n=0$, $\pm 1$, $\pm 2$, $\dots$ with $\lambda$ a given complex number and taking the unit circle $C$ given by $z=e^{i\varphi }(-\pi \le \varphi \le \pi )$ as contour in this region.

[15M]

4.(c) Determine an optimal transportation programme so that the transportation cost of 340 tons of a certain type of material from three factories to five warehouses $W_1$, $W_2$, $W_3$, $W_4$, $W_5$ is minimized. The five warehouses must receive 40 tons, 50 tons, 70 tons, 70 tons and 90 tons respectively. The availability of the material at $F_1$, $F_2$, $F_3$ is 100 tons, 120 tons, 120 tons respectively. The transportation costs per ton factories to warehouses are given in the table below:
$\begin{array}{|ccccc|}\hline W_1& W_2& W_3& W_4& W_5\\ F_1& 4& 1& 2& 6& 9\\ F_2& 6& 4& 3& 5& 7\\ F_3& 5& 2& 6& 4& 8\\ \hline\end{array}$

Use Vogel’s approximation method to obtain the initial basic feasible solution.

[30M]

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

[12M]

5.(b) Find the surface satisfying the PDE $(D^2-2D{D}^{\mathrm{\prime }}+D^{\mathrm{\prime }2})Z=0$ and the conditions that $bZ=y^2$ when $x=0$ and $aZ=x^2$ when $y=0$.

[12M]

5.(c) Find the positive root of the equation $10x{e}^{-x^2}-1=0$ correct up to 6 decimal places by using Newton-Raphson method. Carry out computations only for three iterations.

[12M]

5.(d)(i) Suppose a computer spends 60 per cent of its time handling a particular type of computation when running a given program and its manufacturers make a change that improves its performance on that type of computation by a factor of 10. If the program takes 100 sec to execute, what will its execution time be after the change?

[6M]

5.(d)(ii) If $A\oplus B=A{B}^{\mathrm{\prime }}+A^{\mathrm{\prime }}B$, find the value of $x\oplus y\oplus z$.

[6M]

5.(e) A uniform lamina is bounded by a parabolic are of latus rectum 4$a$ and a double ordinate at a distance $b$ from the vertex. If $b={\displaystyle \frac{a}{3}}(7+4\sqrt{7})$, show that two of the principal axis at the end of a latus rectum are the tangent and normal there.

[12M]

5.(f) In an incompressible fluid the vorticity at every point is constant in magnitude and direction, show that the components of velocity $u$, $v$, $w$ are solutions of Laplace’s equation.

[12M]

6.(a) Solve the following partial differential equation, $zp+yq=x$, $x_0(s)=s$, $y_0(s)=1$, $z_0(s)=2s$, by the method of characteristics.

[20M]

6.(b) Reduce the following $2^{nd}$ order partial differential equation into canonical form and find find its general solution.
$x{u}_{xx}+2x^2{u}_{xy}-u_x=0$.

[20M]

6.(c) Solve the following heat equation
$\begin{array}{ll}{u}_t-u_{xx}=0,& 0<x<2,t>0\\ u(0,t)=u(2,t)=0& t>0\\ u(x,0)=x(2-x),& 0\le x\le 2\end{array}$.

[20M]

7.(a) Given the system of equations:

$2x+3y=1$
$2x+4y+z=2$
$2x+6z+Aw=4$
$4z+Bw=C$

State the solvability and uniqueness conditions for the system. Give the solution when it exists.

[20M]

7.(b) Find the value of the integral ${\int }_1^{\mathrm{\infty }}{\log }_{10}xdx$ by using Simpson’s ${\displaystyle \frac{1}{3}}$rd rule, correct up to 4 decimal places. Take 8 subintervals in your computation.

[20M]

7.(c)(i) Find the hexadecimal equivalent of the decimal number $(587632{)}_{10}$.

[5M]

7.(c)(ii) For the given set of data points $(x_1,f(x_1))$, $(x_2,f(x_2))$, $\cdots$ $(x_n,f(x_n))$, write an algorithm to find the value of $f(x)$ by using Lagrange’s interpolation formula.

[10M]

7.(c)(iii) Using Boolean algebra, simplify the following expressions:

(a) $a+a^{\mathrm{\prime }}b+a^{\mathrm{\prime }}{b}^{\mathrm{\prime }}c+a^{\mathrm{\prime }}{b}^{\mathrm{\prime }}{c}^{\mathrm{\prime }}d+\dots$

(b) $x^{\mathrm{\prime }}{y}^{\mathrm{\prime }}z+yz+xz$, where $x^{\prime }$ represents the complement of $x$.

[5M]

8.(a) A sphere of radius $a$ land mass $m$ rolles down a rough plane inclined at an angle $\alpha$ to the horizontal. If $x$ be the distance of the point of contact of the sphere from a fixed point on the plane, find the acceleration by using Hamilton’s equation.

[30M]

8.(b) When a pair of equal and opposite rectilinear vortices are situated in a long circle cylinder at equal distance from its axis, show that the path of each vortex is given by the equation $(r^2{\sin }^2\theta -b^2){(r^2-a^2)}^2=4a^2{b}^2{r}^2{\sin }^2\theta$, $\theta$ being measured from the line through the centre perpendicular to the joint of the vortices.

[30M]