Paper II PYQs-2010

1.(a) Let $G=\{\left[\begin{array}{ll}a& a\\ a& a\end{array}\right]\mid a\in \cdot R,a\ne 0\}$ Show that $G$ is a group under matrix multiplication.

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1.(b) Let $F$ be a field of order $32.$ Show that the only subfields of $F$ are $F$ itself and {0,1}

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1.(c) If $f:R\to R$ is such that $f(x+y)=f(x)f(y)$ for all $x,y$ in $R$ and $f(x)\ne 0$ for any $x$ in $R,$ show that $f^{\mathrm{\prime }}(x)=f(x)$ for all $x$ in $R$ given that $f^{\mathrm{\prime }}(0)=f(0)$ and the function is differentiable for all $x$ in $R$.

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1.(d) Determine the analytic function $f(z)=u+iv$ if $v=e^x(x\sin y+y\cos y)$

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1.(e) A captain of a cricket team has to allot four middle-order batting positions($IV,V,VI.VII$) to four batsmen($A,B,C,D$). The average number of runs scored by each batsman at these positions are as follows. Assign each batsman his batting position for maximum performance :

[10M]

2.(a) A rectangular box open at the top is to have a surface area of 12 square units. Find the dimensions of the box so that the volume is maximum.

[10M]

2.(b) Prove or disprove that $(R,+)$ and $(R^{+},\cdot )$ are isomorphic groups where $R^{+}$ denotes the set of all positive real numbers.

[10M]

2.(c) Using the method of contour integration, evaluate ${\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}{\displaystyle \frac{x^{2}dx}{{(x^2+1)}^2(x^2+2x+2)}}$

[10M]

3.(a) Show that zero and unity are only idempotents of $Z_n$ if $n=p^r,$ where $p$ is a prime.

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3.(b) Evaluate ${\iint }_R(x-y+1)dxdy$ where $R$ is the region inside the unit square $in$ which $x+y\ge {\displaystyle \frac{1}{2}}$

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3.(c) Solve the following linear programming problem by the simplex method: Maximize $Z=3x_1+4x_2+x_3$ subject to $\begin{array}{r}{x}_1+2x_2+7x_3\le 8\\ x_1+x_2-2x_3\le 6\\ x_1,x_2,x_3\ge 0\end{array}$

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4.(a) Let $R$ be a Euclidean domain with Euclidean valuation $d$. Let $n$ be an integer such that $d(1)+n\ge 0.$ Show that the function $d_n:R-\{0\}\to S,$ where $S$ is the set of all negative integers defined by $d_n(a)=d(a)+n$ for all $a\in R-\{0\}$ is a Euclidean valuation.

[10M]

4.(b) Obtain Laurent’s series expansion of the function $f(z)={\displaystyle \frac{1}{(z+1)(z+3)}}$ in the region $0<|z+1|<2$

[10M]

4.(c) $ABC$ Electricals manufactures and sells two models of lamps, $L_1$ and $L_2,$ the profit per unit being Rs$50$ and Rs$30$ respectively. The process involves two workers $W_1$ and $W_2,$ who are available for $40$ hours and $30$ hours per week, respectively. $W_1$ assembles each unit of $L_1$ in $30$ minutes and that of $L_2$ in $40$ minutes. $W_2$ paints each unit of $L_1$ in $30$ minutes and that of $L_2$ in $15$ minutes. Assuming that all lamps made can be sold, determine the weekly production figures that maximize the profit.

[10M]

5.(a) Find the general solution of $x(y^2+z)p+y(x^2+z)q=z(x^2-y^2)$

[10M]

5.(b) Solve $x{\log }_{10}x=1\cdot 2$ by regula falsi method.

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5.(c) Convert the following: (i) $(736\cdot 4{)}_8$ to decimal number (ii) $(41.6875{)}_{10}$ to binary number (iii) $(101101{)}_2$ to decimal number (iv) $(AF63{)}_{16}$ to decimal number (v) $(101111011111{)}_2$ to hexadecimal number

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5.(d) Show that the sum of the moments of inertia of an elliptic area about any two tangents at right angles is always the same.

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5.(e) A two-dimensional flow field is given by $\psi =xy$. Show that (i) the flow is irrotational; (ii) $\psi$ and $\varphi$ satisfy Laplace equation. Symbols $\psi$ and $\varphi$ convey the usual meaning.

[10M]

6.(a) Using Lagrange interpolation, obtain an approximate value of $\sin (0\cdot 15)$ and a bound on the truncation error for the given data: $\sin (0\cdot 1)=0\cdot 09983,\sin (0\cdot 2)=0\cdot 19867$

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6.(b) Draw a flow chart for finding the roots of the quadratic equation $a{x}^2+bx+c=0$.

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6.(c) Solve ${\displaystyle \frac{\mathrm{\partial }u}{\mathrm{\partial }t}}=4{\displaystyle \frac{{\mathrm{\partial }}^{2}u}{\mathrm{\partial }{x}^2}}$ given the conditions (i) $u(0,t)=u(\pi ,t)=0,nt>0$ (ii) $u(x,0)=\sin 2x,0<x<\pi$

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7.(a) Find the gereral solution of $(D-D^{\mathrm{\prime }}-1)(D-D^{\mathrm{\prime }}-2)z=e^{2x-y}+\sin (3x+2y)$

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7.(b) Show that $\varphi =(x-t)(y-t)$ represents the velocity potential of an incompressible two-dimensional fluid. Further show that the streamlines at time $t$ are the curves $(x-t{)}^2-(y-t{)}^2=\text{ constant }$

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7.(c) Find the interpolating polynomial for $(0,2)$,$(1,3)$,$(2,12)$ and $(5,147)$.

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8.(a) A mass $m_1,$ hanging at the end of a string, draws a mass $m_2$ along the surface of a smooth table. If the mass on the table be doubled, the tension of the string is increased by one-half. Show that $m_1:m_2=2:1$

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8.(b) Solve the initial value problem ${\displaystyle \frac{dy}{dx}}={\displaystyle \frac{y-x}{y+x}},y(0)=1$ for $x=0.1$ by Euler’s method.

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8.(c) Show that the vorticity vector $\overrightarrow{\mathrm{\Omega }}$ of an incompressible viscous fluid moving under no external forces satisfies the differential equation ${\displaystyle \frac{D\overrightarrow{\mathrm{\Omega }}}{Dt}}=\overrightarrow{\mathrm{\Omega }}\cdot \mathrm{\nabla }\overrightarrow{q}+v{\mathrm{\nabla }}^2\overrightarrow{\mathrm{\Omega }}$ where $v$ is the kinematic viscosity.

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