Paper II PYQs-2013

1.(a) Show that the set of matrices $S=\{\left(\begin{array}{cc}a& -b\\ b& a\end{array}\right)a,b\in R\}$ is a field under the usual binary operations of matrix addition and matrix multiplication. What are the additive and multiplicative identities and what is the inverse of $\left(\begin{array}{cc}1& -1\\ 1& 1\end{array}\right)$? Consider the map $f:C\to S$ defined by $f(a+ib)=\left(\begin{array}{cc}a& -b\\ b& a\end{array}\right).$ Show that $f$ is an isomorphism. (Here $R$ is the set of real numbers and $C$ is the set of complex numbers).

[10M]

1.(b) Give an example of an infinite group in which every element has finite order.

[10M]

1.(c) Let $f(x)=\{\begin{array}{ll}{\displaystyle \frac{x^2}{2}}+4& \text{ if }x\ge 0\\ {\displaystyle \frac{-x^2}{2}}+2& \text{ if }x<0\end{array}$ Is $f$ Riemann integrable in the interval $[-1,2]$? Does there exist a function $g$ such that $g^{\mathrm{\prime }}(x)=f(x)?$ Justify your answer.

[10M]

1.(d) Prove that if $b{e}^{a+1}<1$, where $a$ and $b$ are positive and real, then the function $z^n{e}^{-a}-b{e}^z$ has $n$ zeros in the unit circle.

[10M]

1.(e) Maximize $z=2x_1+3x_2-5x_3$,

subject to $x_1+x_2+x_3=7$

and $2x_1-5x_2+x_3\ge 10$, $x_i\ge 0$

[10M]

2.(a) What are the orders of the following permutation in $S_{10}$?
$\left(\begin{array}{cccccccccc}1& 2& 3& 4& 5& 6& 7& 8& 9& 10\\ 1& 8& 7& 3& 10& 5& 4& 2& 6& 9\end{array}\right)$ and $(12345)(67)$.

[10M]

2.(b) What is the maximal possible order of an element in $S_{10}$? Why? Give an example of such a element. How many elements will there be in $S_{10}$ of that order?

[13M]

2.(c) Show that the series $\sum _1^{\mathrm{\infty }}{\displaystyle \frac{(-1{)}^{n-1}}{n+x^2}}$, is uniformly convergent but not absolutely for all real values of $x$.

[13M]

2.(d) Show that every open subset of $\$R$ is a countable union of disjoint open intervals.

[13M]

3.(a) Let $J=\{a+ib|a,b\in Z\}$ be the ring of Gaussian integers (subring of $C$). Which of the following is $J$ - Euclidean domain, principal ideal domain, and unique factorization domain? Justify your answer.

[15M]

3.(b) Let $R^C=$ ring of all real value continuous functions on $[0,1]$ under the operations $(f+g)x=f(x)+g(x)$, $(fg)x=f(x)g(x)$. Let $M=\{f\in R^C/f\left({\displaystyle \frac{1}{2}}\right)=0\}$. Is $M$ a maximal ideal of $R$? Justify your Answer.

[15M]

3.(c) Let $f(x,y)=y^2+4xy+3x^2+x^3+1$. At what points will $f(x,y)$ have a maximum or minimum?

[10M]

3.(d) Let $[x]$ denotes the integer part of the real number $x$, i.e., if $n\le x<n+1$ where $n$ is an integer, then $[x]=n$. Is the function $f(x)=[x{]}^2+3$ Riemann integrable in the interval $[-1,2]$ If not, explain why. If it is integrable, compute ${\int }_{-1}^2([x{]}^2+3)dx$.

[10M]

4.(a) Solve the minimum time assignment problem:
$\begin{array}{|ccccc|}\hline & I& II& III& IV\\ A& 3& 12& 5& 4\\ B& 7& 9& 8& 12\\ C& 5& 11& 10& 12\\ D& 6& 14& 4& 11\\ \hline\end{array}$

where $I$, $II$, $III$ and $IV$ are Machines;
$A$, $B$, $C$, $D$, $E$ are Jobs.

[15M]

4.(b) Using Cauchy’s residue theorem, evaluate the integral $I={\int }_0^{\pi }{\sin }^4\theta d\theta$.

[15M]

4.(c) Minimize $z=5x_1-4x_2+56x_3-8x_4$,

subject to the constraints

[30M]

5.(a) Form a partial differential equation by eliminating the arbitrary functions $f$ and $g$ from $z=yf(x)+xg(y)$.

[10M]

5.(b) Reduce the equation $y{\displaystyle \frac{{\mathrm{\partial }}^{2}z}{\mathrm{\partial }{x}^2}}+(x+y){\displaystyle \frac{{\mathrm{\partial }}^{2}z}{\mathrm{\partial }x\mathrm{\partial }y}}+x{\displaystyle \frac{{\mathrm{\partial }}^{2}z}{\mathrm{\partial }{y}^2}}=0$ to its canonical from when $x\ne y$.

[10M]

5.(c) In an examination, the number of students who obtained marks between certain limits were given in the following table:

Using Newton forward interpolation formula, find the number of students whose marks lie between 45 and 50.

[10M]

5.(d) Prove that the necessary and sufficient conditions that the vortex lines may be at right angles to the stream lines are $u$, $v$, $m=\mu ({\displaystyle \frac{\mathrm{\partial }\varphi }{\mathrm{\partial }x}},{\displaystyle \frac{\mathrm{\partial }\varphi }{\mathrm{\partial }y}},{\displaystyle \frac{\mathrm{\partial }\varphi }{\mathrm{\partial }z}})$ where $\mu$ and $\varphi$ are functions of $x$, $y$, $z$, $t$.

[10M]

5.(e) Four solid spheres $A$, $B$, $C$ and $D$, each of mass $m$ and radius $a$, are placed with their centers on the four corners of a square of side $b$. Calculate the moment of inertia of the system about a diagonal of the square.

[10M]

6.(a) Solve $(D^2+D{D}^{\mathrm{\prime }}-6D^{\mathrm{\prime }2})z=x^2\sin (x+y)$ where $D$ and $D^{\mathrm{\prime }}$ denote ${\displaystyle \frac{\mathrm{\partial }}{\mathrm{\partial }x}}$ and ${\displaystyle \frac{\mathrm{\partial }}{\mathrm{\partial }y}}$.

[15M]

6.(b) Find the surface which intersects the surfaces of the system $z(x+y)=C(3z+1)$, ($C$ being a constant) orthogonally and which passes through the circle $x^2+y^2=1$, $z=1$.

[15M]

6.(c) A tightly stretched string with fixed end points $x=0$ and $x=l$ is initially at rest in equilibrium position. If it is set vibrating by giving each point a velocity $\lambda x(l-x)$, find the displacement of the string at any distance $x$ from one end at any time $t$.

[20M]

7.(a) Develop an algorithm for Newton-Raphson method to solve $f(x)=0$ starting with initial iterate $x_0$, $n$ be the number of iterations allowed, epsilon be the prescribed relative error and delta be the prescribed lower bound for $f^{\mathrm{\prime }}(x)$.

[20M]

7.(b) Use Euler’s method with step size $h=0.15$ to compute the approximate value of $y(0.6)$, correct up to five decimal places from the initial value problem, where $y^{\mathrm{\prime }}=x(y+x)-1$, $y(0)=2$.

[15M]

7.(c) The velocity of a train which starts from rest is given in the following table. The time is in minutes and velocity is in km/hour.

Estimate approximately the total distance run in 30 minutes by using composite Simpson’s ${\displaystyle \frac{1}{3}}$rd rule.

[15M]

8.(a) Two equal rods $AB$ and $BC$ each of length $l$ smoothly jointed at $B$, are suspended from $A$ and oscillate in a vertical plane through $A$. Show that the periods of normal oscillation are ${\displaystyle \frac{2\pi }{n}}$, where $n^2=(3\pm {\displaystyle \frac{6}{\sqrt{7}}}){\displaystyle \frac{g}{l}}$.

[15M]

8.(b) If fluid fills the region of space on the positive side of the $x-axis$, which is a right boundary and if there be a sources $m$ at the point $(0,a)$ and an equal sink at $(0,b)$ and if the pressure on the negative side be the same as the pressure at infinity, show that the resultant pressure on the boundary is ${\displaystyle \frac{\pi \rho m^2(a-b{)}^2}{\{2ab(a+b)\}}}$ where $\rho$ is the density of the fluid.

[15M]

8.(c) If $n$ rectilinear vortices of the same strength $K$ are symmetrically arranged as generators of a circle cylinder of radius $a$ in an infinite liquid, prove that the vortices will move round the cylinder uniformly in time ${\displaystyle \frac{8{\pi }^2{a}^2}{(n-1)\mathrm{K}}}$. Find the velocity at any point of the liquid.

[20M]