Paper II PYQs-2013

1.(a) Evaluate: $\underset{x\to 0}{lim}\left({\displaystyle \frac{e^{ax}-e^{bx}+\tan x}{x}}\right)$

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1.(b) Prove that if every element of a group $(G,0)$ be its own inverse, then it is an abelian group.

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1.(c) Construct an analytic function $\begin{array}{l}f(z)=u(x,y)+iv(x,y),\text{ where }\\ v(x,y)=6xy-5x+3\end{array}$ Express the result as a function of $z$.

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1.(d)Find the optimal assignment cost from the following cost matrix : | |$A$|$B$|$C$|$D$| |—–|—|—|—|—| |$I$ | 4 | 5 | 4 | 3 | |$II$ | 3 | 2 | 2 | 6 | |$III$| 4 | 5 | 3 | 5 | |$IV$ | 2 | 4 | 2 | 6 |

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2.(a) Show that any finite integral domain is a field.

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2.(b) Every field is an integral domain. Prove it.

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2.(c) Solve the following Salesman problem : | | $A$ | $B$ | $C$ | $D$ | |—|——–|——–|——–|——–| |$A$|$\mathrm{\infty }$| 12 | 10 | 15 | |$B$| 16 |$\mathrm{\infty }$| 11 | 13 | |$C$| 17 | 18 |$\mathrm{\infty }$| 20 | |$D$| 13 | 11 | 18 |$\mathrm{\infty }$|

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3.(a) Show that the function $f(x)=x^2$ is uniformly continuous in $(0,1)$ but not in $R$.

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3.(b) Prove that: (i) the intersection of two ideals is an ideal. (ii) a field has no proper ideals.

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4.(a) Find the area of the region between the x-axis and $y=(x-1{)}^3$ from $x=0$ to $x=2$.

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4.(b) Find Laurent series about the indicated singularity. Name the singularity and give the region of convergence. ${\displaystyle \frac{z-\sin z}{z^3}};z=0$

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4.(c) $x_1=4,x_2=1,x_3=3$ is a feasible solution of the system of equations $\begin{array}{l}2x_1-3x_2+x_3=8\\ x_1+2x_2+3x_3=15\end{array}$ Reduce the feasible solution to two different basic feasible solutions.

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5.(a) Use Newton - Raphson method and derive the iteration scheme $x_{n+1}={\displaystyle \frac{1}{2}}(x_n+{\displaystyle \frac{N}{x_n}})$ to calculate an approximate value of the square root of a number $N$. Show that the formula $\sqrt{N}\approx {\displaystyle \frac{A+B}{4}}+{\displaystyle \frac{N}{A+B}}$, where $AB=N,$ can easily be obtained if the above scheme is applied two times. Assume $A=1$ as an initial guess value and use the formula twice to calculate the value of $\sqrt{2}$ [For $2^{\text{nd }}$ iteration, one may take $A=$ result of the $1^{\text{st }}$ iteration].

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5.(b) Eliminate the arbitrary function $f$ from the given equation $f(x^2+y^2+z^2,x+y+z)=0$

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5.(c) Derive the Hamiltonian and equation of motion for a simple pendulum.

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6.(a) Solve the PDE: $x{u}_x+y{u}_y+z{u}_z=xyz$

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6.(b) Convert $(0\cdot 231{)}_5,(104\cdot 231{)}_5$ and $(247{)}_7$ to base $10$.

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6.(c) Rewrite the hyperbolic equation $x^2{u}_{xx}-y^2{u}_{yy}=0(x>0,y>0)$ in canonical form.

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7.(a) Find the values of $a$ and $b$ in the $2-D$ velocity field $\overrightarrow{v}=(3y^2-a{x}^2)\hat{i}+$ bxy $\hat{j}$ so that the flow becomes incompressible and irrotational. Find the stream function of the flow.

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7.(b) Write an algorithm to find the inverse of a given non-singular diagonally dominant square matrix using Gauss - Jordan method.

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7.(c) Find the solution of the equation ${\left({\displaystyle \frac{\mathrm{\partial }u}{\mathrm{\partial }x}}\right)}^2+{\left({\displaystyle \frac{\mathrm{\partial }u}{\mathrm{\partial }y}}\right)}^2=1$ that passes through the circle $x^2+y^2=1,u=1$

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8.(a) Solve the following heat equation, using the method of separation of variables: ${\displaystyle \frac{\mathrm{\partial }u}{\mathrm{\partial }t}}={\displaystyle \frac{{\mathrm{\partial }}^{2}u}{\mathrm{\partial }{x}^2}},0<x<1,t>0$ subject to the conditions $u=0$ at $x=0$ and $x=1,$ for $t>0$ $u=4x(1-x),$ at $t=0$ for $0\le x\le 1$

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8.(b) Use the Classical Fourth-order Runge-Kutta method with $h=0\cdot 2$ to calculate a solution at $x=0\cdot 4$ for the initial value problem ${\displaystyle \frac{du}{dx}}=4-x^2+u,u(0)=0$ on the interval $[0,0\cdot 4]$

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8.(c) Draw a flow chart for testing whether a given real number is a prime or not.

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