Paper II PYQs-2012

1.(a) How many elements of order 2 are there in the group of order 16 generated by $a$ and $b$ such that the order of $a$ is 8, the order of $b$ is 2 and $ba{b}^{-1}=a^{-1}$.

[12M]

1.(b) Let
$f_n(x)=\{\begin{array}{cl}0,& \text{ if }x<{\displaystyle \frac{1}{n+1}}\\ \sin {\displaystyle \frac{\pi }{x}},& \text{ if }x<{\displaystyle \frac{1}{n+1}}\le x\le {\displaystyle \frac{1}{n}}\\ 0,& \text{ if }x>{\displaystyle \frac{1}{n}}\end{array}$
Show that $f_n(x)$ converges to a continuous function but not uniformly.

[12M]

1.(c) Show that the function defined by $f(z)=\{\begin{array}{c}{\displaystyle \frac{x^3{y}^5(x+iy)}{x^6+y^{10}}},z\ne 0\\ 0,z=0\end{array}$ is not analytic at the origin though it satisfies Cauchy Riemann-Equations at the origin.

[12M]

1.(d) For each hour per day that Ashok studies mathematics, it yields him 10 marks and for eaach hour that he studies physics, it yields him 5 marks. He can study at most 14 hours a day and he must get at least 40 marks in each. Determine graphically how many hours a day he should study mathematics and physics each, in order to maximize his marks?

[12M]

1.(e) Show that the series $\sum _{n=1}^{\mathrm{\infty }}{\left({\displaystyle \frac{\pi }{\pi +1}}\right)}^n{n}^6$ is convergent.

[12M]

2.(a) How many conjugacy classes does the permutation group $S_5$ of permutation 5 numbers have? Write down one element in each class (preferably in terms of cycles).

[15M]

2.(b) Let $f(x,y)=\{\begin{array}{ll}{\displaystyle \frac{(x+y{)}^2}{x^2+y^2}},& \text{ if }(x,y)\ne (0,0)\\ 1,& \text{ if }(x,y)=(0,0)\end{array}$.
Show that ${\displaystyle \frac{\mathrm{\partial }f}{\mathrm{\partial }x}}$ and ${\displaystyle \frac{\mathrm{\partial }f}{\mathrm{\partial }y}}$ exist at $(0,0)$ though $f(x,y)$ is not continuous at $(0,0)$.

[15M]

2.(c) Use Cauchy integral formula to evaluate ${\int }_c{\displaystyle \frac{e^{3z}}{(z+1{)}^4}}dz$, where $c$ is the circle $|z=2$.

[15M]

2.(d) Find the minimum distance of the line given by the planes $3x+4y+5z=7$ and $x-z=9$ and from the origin, by the method of Lagrange’s multipliers.

[15M]

3.(a) Is the ideal generated by 2 and $X$ in the polynomial ring $Z[X]$ of polynomials in a single variable $X$ with coefficients in the ring of integers $Z$, a principal ideal? Justify your answer.

[15M]

3.(b) Let $f(x)$ be differentiable on $[0,1]$ such that $f(1)=0$ and ${\int }_0^1{f}^2(x)dx=1$. Prove that ${\int }_0^{1}xf(x)f^{\mathrm{\prime }}(x)dx=-{\displaystyle \frac{1}{2}}$.

[15M]

3.(c) Expand the function $f(z)={\displaystyle \frac{1}{(z+1)(z+3)}}$ in Laurent series valid for:
i) $1<|z|<3$
ii) $|z|>3$
iii) $0<|z+1|<2$
iv) $|z|<1$

[15M]

3.(d) Evaluate by contour integration $I={\int }_0^{2\pi }{\displaystyle \frac{d\theta }{1-2a\cos \theta +a^2}}$, $a^2<1$.

[15M]

4.(a) Describe the maximal ideals in the ring of Gaussian integers $Z[i]=\{a+bi|a,b\in Z\}$.

[20M]

4.(b) Give an example of a function $f(x)$, that is not Riemann integrable but $|f(x)|$ is Riemann integrable. Justify your answer.

[20M]

4.(c) By the method of Vogel, determine an initial basic feasible solution for the following transportation problem. Products $P_1$, $P_2$, $P_3$ and $P_4$ have to be sent of destinations $D_1$, $D_2$ and $D_3$. The cost of sending product $P_i$ to destinations $D_j$ is $C_{ij}$, where the matrix:
$\left[C_{ij}\right]=\left[\begin{array}{cccc}10& 0& 15& 5\\ 7& 3& 6& 15\\ 0& 11& 9& 13\end{array}\right]$
The total requirements of destinations $D_1$, $D_2$ and $D_3$ are given by 45, 45, 95 respectively and the availability of the products $P_1$, $P_2$, $P_3$ and $P_4$ are respectively 25, 35, 55 and 70.

[12M]

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

[12M]

5.(b) Use Newton-Raphson method to find the real root of the equation $3x=\cos x+1$ correct to four decimal places.

[12M]

5.(c) Provide a computer algorithm to solve an ordinary differential equation ${\displaystyle \frac{dy}{dx}}=f(x,y)$ in the interval $[a,b]$ for $n$ number of discrete points, where the initial value is $y(a)=\alpha$, using Euler’s method.

[12M]

5.(d) Obtain the equations governing the motion of a spherical pendulum.

[12M]

5.(e) A rigid sphere of radius $a$ is placed in a stream of fluid whose velocity in the undisturbed state is $V$. Determine the velocity of the fluid at any point of the disturbed stream.

[12M]

6.(a) Solve the partial differential equation $px+qy=3z$.

[20M]

6.(b) A string of length $l$ is fixed at its ends. The string from the mid-point is pulled up to a height $k$ and then released from rest. Find the deflection $y(x,t)$ of the vibrating string.

[20M]

6.(c) Solve the following system of simultaneous equations, using Gauss-Seidel iterative method:

$3x+20y-z=-18$
$20x+y-2z=17$
$2x-3y+20z=25$

[20M]

7.(a) Find ${\displaystyle \frac{dy}{dx}}$ at $x=0.1$ from the following data:

[20M]

7.(b) The edge $r=\alpha$ of a circular plate is kept at temperature $f(\theta )$. The plate is insulated so that there is no loss of heat from either surface. Find the temperature distribution in steady state.

[20M]

7.(c) In a certain examination, a candidate has to appear for one major & two major subjects. The rules for declaration of results are marks for major are denoted by $M_1$ and for minors by $M_2$ and $M_3$. If the candidate obtains 75% and above marks in each of the three subjects, the candidate is declared to have passed the examination in first class with distinction. If the candidate obtains 60% and above marks in each of the three subjects, the candidate is declared to have passed the examination in first class. If the candidate obtains 50% or above in major, 40% or above in each of the two minors and an average of 50% or above in all the three subjects put together, the candidate is declared to have passed the examination in second class. All those candidates, who have obtained 50% and above in major and 40% or above in minor, are declared to have passed the examination. If the candidate obtains less than 50% in major or less than 40% in anyone of the two minors, the candidate is declared to have failed in the examinations. Draw a flow chart to declare the results for the above.

[20M]

8.(a) A pendulum consists of a rod of length $2a$ and mass $m$; to one end of spherical bob of radius ${\displaystyle \frac{a}{3}}$ and mass $15m$ is attached. Find the moment of inertia of the pendulum:

(i) About an axis through the other end of the rod and at right angle to the rod.

(ii) About a parallel axis through the centre of mass of the pendulum. [Given: the centre of mass of the pendulum is ${\displaystyle \frac{a}{12}}$ above the centre of the sphere].

[30M]

8.(b) Show that $\varphi =xf(r)$ is a possible form for the velocity potential for an incompressible fluid motion. If the fluid velocity $\dot{q}\to 0$ as $r\to \mathrm{\infty }$, find the surfaces of constant speed.

[30M]