Paper II PYQs-2009

1.(a) Prove that a non-empty subset $H$ of $a$ group G is normal subgroup of $G\iff$ for all $x,y\in H,g\in G,(gx)(gy{)}^{-1}\in H$

[10M]

1.(b) Show that the function $f(x]={\displaystyle \frac{1}{x}}$ is not uniformly continuous [0,1].

[10M]

1.(c) Show that under the transformation $w={\displaystyle \frac{z-i}{z+i}}$ real axis in the $E$ plane is mapped into the circle $|w|=1$ What portion of the Z-plane corresponds to the interior of the circle?

[10M]

1.(d) If $G$ is a finite Abelian group, then show that $O(a,b)$ is a divisor of $1.c\cdot m$. of $O(a),O(b)$

[10M]

1.(e) Evaluate ${\int }_C{\displaystyle \frac{2z+1}{z^2+z}}dz$ by Cauchy’s integral formula, where $C$ is $|z|={\displaystyle \frac{1}{2}}$

[10M]

2.(a) Find the dimensions of the largest rectangular parallelopiped that has three faces in the coordinate planes and one vertex in the plane ${\displaystyle \frac{x}{a}}+{\displaystyle \frac{y}{b}}+{\displaystyle \frac{z}{c}}=1$

[10M]

2.(b) Determine the analytic function $w=u+iv,$ if $u={\displaystyle \frac{2\sin 2x}{e^{2y}+e^{-2y}-2\cos 2x}}$

[10M]

2.(c) Find the multiplicative inverse of the element $\left[\begin{array}{ll}2& 5\\ 1& 3\end{array}\right]$ of the ring, $M$ of all matrices of order two over the integers.

[10M]

3.(a) Evaluate $\iint xy(x+y)dxdy$ over the area between $y=x^2$ and $y=x$

[10M]

3.(b) Evaluate by contour integration ${\int }_0^{2\pi }{\displaystyle \frac{d\theta }{1-2a\sin \theta +a^2}},0<a<1$

[10M]

3.(c) Write the dual of the following LPP and hence, solve it by graphical method : Minimize $Z=6x_1+4x_2$ constraints $\begin{array}{rl}2x_1+x_2& \ge 1\\ 3x_1+4x_2& \ge 1.5\\ x_1,x_2& \ge 0\end{array}$

[10M]

4.(a) Show that $d(a)<d(ab),$ where $a,b$ be two non-zero elements of a Euclidean domain $R$ and $b$ is not a unit in $R$

[10M]

4.(b) Show that a field is an integral domain and a non-zero finite integral domain is a field.

[10M]

4.(c) Solve by simplex method, the following LPP : Maximize $Z=5x_1+3x_2$ constraints $\begin{array}{rl}3x_1+5x_2& \le 15\\ 5x_1+2x_2& \le 10\\ x_1,x_2& \ge 0\end{array}$

[10M]

5.(a) Find complete and singular integrals of $(p^2+q^2)y=qz$.

[10M]

5.(b) Obtain the iterative scheme for finding $p$th root of a function of single variable using Newton-Raphson method. Hence, find $\sqrt[7]{277}\overline{234}$ correct to four decimal places.

[10M]

5.(c) Convert the following binary numbers to the base indicated: (i) $(10111011001\cdot 101110{)}_2$ to octal (ii) $(10111011001\cdot 10111000{)}_2$ to hexadecimal (iii) $(0\cdot 101{)}_2$ to decimal

[10M]

5.(d) A cannon of mass $M$, resting on a rough horizontal plane of coefficient of friction $\mu$ is fired with such a charge that the relative velocity of the ball and cannon at the moment when it leaves the cannon is $u.$ Show that the cannon will recoil a distance ${\left({\displaystyle \frac{mu}{M+m}}\right)}^2{\displaystyle \frac{1}{2\mu g}}$ along the plane, $m$ being the mass of the ball.

[10M]

5.(e) If the velocity of an incompressible fluid at the point $(x,y,z)$ is given by $({\displaystyle \frac{3xz}{r^5}},{\displaystyle \frac{3yz}{r^5}},{\displaystyle \frac{3z^2-r^2}{r^5}})$ where $r^2=x^2+y^2+z^2,$ prove that the liquid motion is possible and that the velocity potential is $\cos \theta /r^2$

[10M]

6.(a) A rod of length $l$ with insulated sides, is initially at a uniform temperature $u_0$. Its ends are suddenly cooled to $0^{\circ }C$ and are kept at that temperature. Find the temperature distribution in the rod at any time $t$.

[10M]

6.(b) Convert the following to the base indicated: (i) $(266\cdot 375$ ) ${}_{10}$ to base 8 (ii) $(341\cdot 24{)}_5$ to base 10 (iii) $(43\cdot 3125{)}_{10}$ to base 2

[10M]

6.(c) Draw the circuit diagram for $\overline{F}=A\overline{B}C+\overline{C}B$ using NAND to NAND logic long.

[10M]

6.(d) Using Runge-Kutta method, solve $y^{\mathrm{\prime }\mathrm{\prime }}=x{y}^{{}^{\prime }2}-y^2$ for $x=0.2.$ Initial conditions are at $x=0,y=1$ and $y^{\mathrm{\prime }}=0.$ Use four decimal places for computations.

[10M]

7.(a) Prove that the equation of motion of a homageneous inviscid liquid moving under conservative forces may be written as ${\displaystyle \frac{\mathrm{\partial }\overrightarrow{q}}{\mathrm{\partial }t}}-\overrightarrow{q}\times \mathrm{curl}\overrightarrow{q}=-\mathrm{grad}[{\displaystyle \frac{p}{\rho }}+{\displaystyle \frac{1}{2}}{q}^2+\overrightarrow{\mathrm{\Omega }}]$

[10M]

7.(b) Find the general solution of $\begin{array}{rl}\{D^2-D{D}^{\mathrm{\prime }}& -2D^{\mathrm{\prime }2}+2D+2D^{\mathrm{\prime }}\}z\\ & =e^{2x+3y}+xy+\sin (2x+y)\end{array}$

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7.(c) From the following data $\begin{array}{cccccc}x& :& 1& 8& 27& 64\\ y& :& 1& 2& 3& 4\end{array}$ calculate $y(20),$ using Lagrangian interpolation decimal points for computations.

[10M]

8.(a) A homogeneous sphere of radius $a$, rotating with angular velocity about hoizontal diameter, is gently placed on a table whose coefficient of friction is $\mu$. Show that there will be slipping at the point of contact for a time ${\displaystyle \frac{2\omega a}{7\mu g}}$ and that then the sphere will roll with angular velocity ${\displaystyle \frac{2\omega }{7}}$.

[10M]

8.(b) Derive composite ${\displaystyle \frac{1}{3}}rd$ Simpson’s rule. Hence, evaluate ${\int }_0^{0.6}{e}^{-x^2}dx$ by taking seven ordinates. Tabulate the integrand for these ordinates to four decimal places.

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8.(c) Show that for an incompressible steady flow with constant viscosity, the velocity components $\begin{array}{l}u(y)=y{\displaystyle \frac{U}{h}}+{\displaystyle \frac{h^2}{2\mu }}(-{\displaystyle \frac{dp}{dx}}){\displaystyle \frac{y}{h}}(1-{\displaystyle \frac{y}{h}})\\ v=0,w=0\end{array}$ satisfy the equations of motion, when the body force is neglected. $h,U,{\displaystyle \frac{dp}{dx}}$ are constants and $p=p(x).$

[10M]