Paper II PYQs-2009

1.(a) If $R$ is the set of real numbers and $R_t$ is the set of positive real numbers, show that $R$ under addition $(R,+)$ and $R_{+}$ under multiplication $(R_{+},.)$ are isomorphic. Similarly if $Q$ is set of rational numbers and $Q_{+}$ is the set of positive rational numbers, are $(Q,+)$ and $(Q,.)$ isomorphic? Justify your answer.

[12M]

1.(b) Determine the number of homomorphisms from the additive group $Z_{15}$ to the additive group $Z_{10}$. ( $Z_n$ is the cyclic group of order $n$).

[4+8=12M]

1.(c) State Rolle’s theorem. Use it to prove that between two roots of $e^x\cos x=1$, there will be a root of $e^x\sin x=1$.

[12M]

1.(d) Let $f(x)=\{\begin{array}{ll}{\displaystyle \frac{|x|}{2}}+1,& \text{ if }x<1\\ {\displaystyle \frac{x}{2}}+1,& \text{ if }1\le x<2\\ -{\displaystyle \frac{|x|}{2}}+1,& \text{ if }2\le x\end{array}$.

What are the points of discontinuity of $f$, if any? What are the points where $f$ is not differentiable, if any? Justify your answers.

[12M]

1.(e) Let $f(z)={\displaystyle \frac{a_0+a_1\dots \dots +a_{n-1}{z}^{n-1}}{b_0+b_{1}z+\dots \dots \dots +b_n{z}^n}}$, $b_n\ne 0$. Assume that the zeros of the denominator are simple. Show that the sum of the residues of $f(z)$ at its poles is equal to ${\displaystyle \frac{a_n-1}{b_n}}$.

[12M]

1.(f) A paint factory produces both interior and exterior paint from materials $M_{1}$ and $M_{2}$ . The basic data is as follows:
$\begin{array}{|cccc|}\hline & \text{Exterior Paint}& \text{Interior Paint}& \text{Max. Daily Availability}\\ \text{ Raw Material }{M}_1& 6& 4& 24\\ \text{ Raw Material }{M}_2& 1& 2& 2\\ \text{ Profit per ton (Rs. 1000) }& 5& 4& \\ \hline\end{array}$

A market survey indicates that the daily demand interior paint cannot exceed that of exterior paint by more than 1 ton. The maximum daily demand of interior paint is 2 tons. The factory wants to determine the optimum product mix of interior and exterior paint that maximizes daily profits. Formulate the LP problem for this situation.

[12M]

2.(a) How many proper, non-zero ideals does the ring $Z_{12}$ have? Justify your answer. How many ideals does the ring $Z_{12}\oplus Z_{12}$ have? Why?

[2+3+4+6=15M]

2.(b) Show that the alternating group of four letters $A_4$ has no subgroup of order 6.

[15M]

2.(c) Show that the series ${\left({\displaystyle \frac{1}{3}}\right)}^2$+${\left({\displaystyle \frac{1.4}{3.6}}\right)}^2$+$\dots .+{\left({\displaystyle \frac{1.4.7\dots ..(3n-2)}{3.6.9\dots \dots \dots ..3n}}\right)}^2$ till infinity, converges.

[15M]

2.(d) Show that if $f:[a,b]\to R$ is a continuous function, then $f([a,b])=[c,d]$ for some real numbers $c$ and $d$, $c\le d$.

[15M]

3.(a) Show that $Z[X]$ is a unique factorization domain that is not a principal ideal domain ($Z$ is the ring of integers). Is it possible to give an example of principal ideal domain that is not a unique factorization domain? ($Z[X]$ is the ring of polynomials in the variable $X$ with integer.

[15M]

3.(b) How many elements does the quotient ring ${\displaystyle \frac{Z_5[X]}{X^2+1}}$ have? Is it an integral domain? Justify your answers.

[15M]

3.(c) Show that:

${\text{Lt}}_{x\to 1}\sum _{n=1}^{\mathrm{\infty }}{\displaystyle \frac{n^2{x}^2}{n^4+x^4}}$= $\sum _{n=1}^{\mathrm{\infty }}{\displaystyle \frac{n^2}{n^4+1}}$

Justify all steps of your answer by quoting the theorems you are using.

[15M]

3.(d) Show that a bounded infinite subset of $R$ must have a limit point.

[15M]

4.(a) If $\alpha$, $\beta$, $\gamma$ are real numbers such that ${\alpha }^2>{\beta }^2+{\gamma }^2$ show that: ${\int }_0^{2\pi }{\displaystyle \frac{d\theta }{\alpha +\beta \cos \theta +\gamma \sin \theta }}$=${\displaystyle \frac{2\pi }{\sqrt{{\alpha }^2-{\beta }^2-{\gamma }^2}}}$.

[30M]

4.(b) Maximize: $Z=3x_1+5x_2+4x_3$ subject to:

$2x_1+3x_2\le 8$,
$3x_1+2x_2+4x_3\le 15$,
$2x_2+5x_3\le 10$,
$x_i\ge 0$

[30M]

5.(a) Show that the differential equation of all cones which have their vertex at the origin is $px+qy=z$. Verify that this equation is satisfied by the surface $yz+zx+xy=0$.

[12M]

5.(b)(i) Form the partial differential equation by elimination the arbitrary function $f$ given by: $f(x^2+y^2,z-xy)=0$.

[6M]

5.(b)(ii) Find the integral surface of: $x^{2}p+y^{2}p+z^2=0$ which passes through the curve: $xy=x+y$, $z=1$.

[6M]

5.(c)(i) The equation $x^2+ax+b=0$ has two real roots $\alpha$ and $\beta$. Show that the iterative method given by: $x_{k+1}=-{\displaystyle \frac{(a{x}_k+b)}{x_k}}$, $k=0,1,2\dots$ is convergent near $x=\alpha$, if $|\alpha |>|\beta |$.

[6M]

5.(c)(ii) Find the values of two valued Boolean variables $A$, $B$, $C$, $D$ by solving the following simultaneous equations:
$$\begin{gathered} \overline{A}+AB=0 \\ AB+AC \\ AB+A\overline{C}+CD=\overline{C}D \end{gathered}$$
where $\overline{x}$ represents the complement of $x$.

[6M]

5.(d)(i) Realize the following expressions by using NAND gates only:
$g=(\overline{a}+\overline{b}+c)\overline{d}(\overline{a}+e)f$, where $\overline{x}$ represents the complement of $x$.

[6M]

5.(d)(ii) Find the decimal equivalent of $(357.32{)}_8$.

[6M]

5.(e) The flat surface of a hemisphere of radius $r$ is cemented to one flat surface of a cylinder of the same radius and of the same material. If the length of the cylinder be $l$ and the total mass be $m$, show that the moment of inertia of the combination about the axis of the cylinder is given by: $m{r}^2{\displaystyle \frac{({\displaystyle \frac{l}{2}}+{\displaystyle \frac{4}{15}}r)}{(l+{\displaystyle \frac{2r}{3}})}}$.

[12M]

5.(f) Two sources, each of strength $m$ are placed at the point $(-a,0)$, $(a,0)$ and a sink of strength $2m$ is at the 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 a variable parameter. Show also that the fluid speed at any point is $\left(2m{a}^2\right)/\left(r_1{r}_2{r}_3\right)$, where $r_1{r}_2$ and $r_3$ are the distance of the points from the source and the sink.

[12M]

6.(a) Find the characteristics of: $y^{2}r-x^{2}t=0$ where $r$ and $t$ have their usual meanings.

[15M]

6.(b) Solve: $(D^2-D{D}^{\mathrm{\prime }}-2D^{\mathrm{\prime }2})z$=$(2x^2+xy-y^2)\sin xy-\cos xy$ where $D$ and $D^{\mathrm{\prime }}$ represent ${\displaystyle \frac{\mathrm{\partial }}{\mathrm{\partial }x}}$ and ${\displaystyle \frac{\mathrm{\partial }}{\mathrm{\partial }y}}$ and ${\displaystyle \frac{\mathrm{\partial }}{\mathrm{\partial }y}}$.

[15M]

6.(c) A tightly stretched string has its ends fixed at $x=0$ and $x=1$. At time $t=0$, the string is given a shape defined by $f(x)=\mu x(l-x)$, where $\mu$ is a constant, and then released. Find the displacement of any point $x$ of the string at time $t>0$.

[30M]

7.(a) Develop an algorithm for Regula-Falsi method to find a root of $f(x)=0$ starting with two initial iterates $x_0$ and $x_1$ to the root such that sign $\left(f\left(x_0\right)\right)\ne \mathrm{sign}\left(f\left(x_1\right)\right)$. Take $n$ as the maximum number of iterations allowed and epsilon be the prescribed error.

[30M]

7.(b) Using Lagrange interpolation formula, calculate the value of $f(3)$ from the following table of values of $x$ and $f(x)$:
$\begin{array}{|ccccccc|}\hline x& 0& 1& 2& 4& 5& 6\\ f(x)& 1& 14& 15& 5& 6& 19\\ \hline\end{array}$

[15M]

7.(c) Find the value of $y(1.2)$ using Runge-Kutta fourth order method with step size $h=0.2$ from the initial value problem: $y^{\mathrm{\prime }}=xy$, $y(1)=2$.

[15M]

8.(a) A perfectly rough sphere of mass $m$ and radius $b$, rests on the lowest point of a fixed spherical cavity a radius $a$. To the highest point of the movable sphere is attached a particle of mass $m^{\mathrm{\prime }}$ and the system is disturbed. Show that the oscillations are the same as of a simple pendulum of length $(a-b){\displaystyle \frac{4m^{\mathrm{\prime }}+{\displaystyle \frac{7}{5}}m}{m+m^{\mathrm{\prime }}(2-{\displaystyle \frac{a}{b}})}}$.

[30M]

8.(b) An infinite mass of fluid is acted on by a force ${\displaystyle \frac{\mu }{r^{3/2}}}$ per unit mass directed to the origin. If initially the fluid is at rest and there is a cavity in the form of the sphere $r=C$ in it, show the cavity will be filled up after an interval of time ${\left({\displaystyle \frac{2}{5\mu }}\right)}^{{\displaystyle \frac{1}{2}}}\cdot C^{{\displaystyle \frac{5}{4}}}$.

[30M]