2000

1) Given that the terms of a sequence $\left\{a_n\right\}$ are such that each $a_k$,$k\le 3$, is the arithmetic mean of its two immediately preceding terms. Show that the sequence converges. Also find the limit of the sequence.

[12M]

2) Determine the values of $x$ for which the infinite product $\prod _{n=0}^{\mathrm{\infty }}(1+{\displaystyle \frac{1}{x^{2^n}}})$ converges absolutely. Find its value whenever it converges.

[12M]

3) Suppose $f$ is twice differentiable real valued function in $(0,\mathrm{\infty })$ and $M_0,M_1$ and $M_2$ the least upper bounds of $|f(x)|,|f^{\mathrm{\prime }}(x)|$ and $|f^{\mathrm{\prime }\mathrm{\prime }}(x)|$ respectively in $(0,\mathrm{\infty })$. Prove for each $x>0,h>0$ that $f^{\mathrm{\prime }}(x)={\displaystyle \frac{1}{2h}}[f(x+2h)-f(x)]-h{f}^{\mathrm{\prime }}(u)$ for some $\mathbf{u}\in (\mathbf{x},\mathbf{x}+2\mathbf{h}).$ Hence show that ${\mathrm{M}}_1^2\le 4{\mathrm{M}}_0{\mathrm{M}}_2$

[15M]

4) Evaluate ${\iint }_S(x^{3}dydz+x^{2}ydzdx+x^{2}zdxdy)$ by transforming into triple integral where $S$ is the closed surface formed by the cylinder $x^2+y^2=a^2,0\le z\le b$ and the circular discs $x^2+y^2\le a^2$, $z=0$ and $x^2+y^2\le a^2,z=b$

[15M]

5) Suppose $\mathrm{f}(\zeta )$ is continuous on a circle $\mathrm{C}$. Show that ${\int }_C{\displaystyle \frac{f(\zeta )d\zeta }{(\zeta -z)}}$ as $z$ varies inside of $\mathrm{C},$ is differentiable under the integral sign. Find the derivative. Hence or otherwise, derive an integral representation for $f^{\mathrm{\prime }}(z)$ if $f(z)$ is analytic on and inside of $C$.

[15M]

1999

1) A sequence $\left\{{\mathrm{S}}_n\right\}$ is defined by the recursion formula $S_{n+1}=\sqrt{3S_n};S_1=1.$ Does this sequence converge? If so, find limit ${\mathrm{S}}_{\mathrm{n}}$.

2) Find the shortest distance from the origin to the hyperbola $x^2+8xy+7y^2=225$, $z=0$.

[20M]

3) Show that the double $\int {\int }_R{\displaystyle \frac{x-y}{(x+y{)}^3}}dxdy$ does not exist over $R=[0,1;0,1]$.

[20M]

4) Show that the double integral ${\iint }_R\frac{x-y}{(x+y{)}^3}dxdy$ does not exist over $R=[0,1;0,1]$.

1998

1) Show that the function $f(x,y)=2x^4-3x^{2}y+y^2$ has (0,0) as the only critical point but the function neither has a minima nor a maxima at $(0,0)$.

2) Test the convergence of the integral ${\int }_0^{\mathrm{\infty }}{e}^{-ax}{\displaystyle \frac{sinx}{x}}dx$, $a\ge 0$.

3) Test the series $\sum _{n=1}^{\mathrm{\infty }}{\displaystyle \frac{x}{(n+x^2{)}^2}}$ for uniform convergence.

4) Let $f(x)=x$ and $g(x)=x^2$. Does ${\int }_0^{1}fdg$ exist? If it exist, then find its value.s

1997

1) Show that a non-empty set P in ${\mathrm{R}}^n$ each of whose points is a limit-point is uncountable.

[10M]

2) Show that ${\iiint }_{D}xydxdydz={\displaystyle \frac{a^2{b}^2{c}^2}{6}}$ where domain $D$ is given by ${\displaystyle \frac{x^2}{a^2}}+{\displaystyle \frac{y^2}{b^2}}+{\displaystyle \frac{z^2}{c^2}}\le 1$.

[10M]

3) If $u={\sin }^{-1}\left[{(x^2+y^2)}^{1/5}\right],$ prove that $x^2{\displaystyle \frac{{\mathrm{\partial }}^{2}u}{\mathrm{\partial }{x}^2}}+2xy{\displaystyle \frac{{\mathrm{\partial }}^{2}u}{\mathrm{\partial }y\cdot \mathrm{\partial }x}}+y^2{\displaystyle \frac{{\mathrm{\partial }}^{2}x}{\mathrm{\partial }{y}^2}}={\displaystyle \frac{2}{25}}\tan u(2{\tan }^{2}u-3)$

[10M]

1996

1) A function $\mathrm{f}$ is defined in the interval $(\mathrm{a},\mathrm{b})$ as follows: $\begin{array}{l}f(x)=q^{-2},\text{ when }x=p{q}^{-1}\\ f(x)=q^{-3},\text{ when }x={\left(p{q}^{-1}\right)}^{1/2}\end{array}$ where $\mathrm{p},\mathrm{q}$ are relatively prime integers; $f(\mathrm{x})=0,$ for all other values of $\mathrm{x}$. Is $\mathrm{f}$ Riemann integrable ? Justify your answer.

[15M]

2) Test for uniform convergence, the series $\sum _{n=1}^{\mathrm{\infty }}{\displaystyle \frac{2^n{x}^{(2^n-1)}}{1+x^{2n}}}$

[15M]

3) Evaluate ${\int }_0^{\pi /2}{\int }_0^{\pi /2}\sin x{\sin }^{-1}(\sin x\sin y)dxdy$

[15M]

1995

1) Let $f$ be a continuous real function on $R$ such that $f$ maps open interval onto open intervals. Prove that $f$ is monotonic.

[15M]

2) Suppose $f$ maps an open ball $U\subset R^n$ into $R^m$ and $f$ is differentiable on $U$. Suppose there exists a real number $M>0$ such that $|f^{\prime }(x)|\le M$ for all $x\in U$. Prove that $|f(b)-f(a)|\le M|b-a|$ for all $a,b\in U$

[15M]

3) Find and classify the extreme values of the function $f(x,y)=x^2+y^2+x+y+xy$

[15M]

4) Suppose $\alpha$ is a real number not equal to $n\pi$ for any integer $n$. Prove that ${\int }_0^{\mathrm{\infty }}{\int }_0^{\mathrm{\infty }}{e}^{-(x^2+2xy\cos \alpha +y^2)}dxdy={\displaystyle \frac{\alpha }{2\sin \alpha }}$

[15M]