1988

1) Show that the Cauchy product of $\sum _{n=0}^{\mathrm{\infty }}{\displaystyle \frac{(-1{)}^n}{\sqrt{n+1}}}$ with itself diverges.

2) Prove that the sequence $\{S_n(x)\}$ where $S_n(x)=nx{e}^{-n{x}^2}$ is not uniformly convergent in any interval $(0,\mathrm{k}),\mathrm{k}>0$.

3) Evaluate
$\iint {(1-{\displaystyle \frac{x^2}{a^2}}-{\displaystyle \frac{y^2}{b^2}})}^{1/2}dxdy$ over the area of the ellipse ${\displaystyle \frac{x^2}{a^2}}+{\displaystyle \frac{y^2}{b^2}}=1$.

4) Discuss the convergence of the improper integral ${\int }_0^x{t}^{x-1}{e}^{-t}dt$.

5) Show that a local extreme value of f given by $f(\overline{x})=x_1^k+\dots .+x_n^k$, $\overline{x}=(x_1,x_2,\dots ,x_n)$
subject to the condition $x_1+x_2+\dots \dots +x_n=a,$ is $a^k{n}^{1-k}$.

6) The function $f:R^2\to R^1$ is given by
$f(x,y)=\{\begin{array}{cc}{\displaystyle \frac{x^{2}y}{x^2+y^2}}& \text{ when }(x,y)\ne (0,0)\\ 0& \text{ when }(x,y)=(0,0)\end{array}$ Prove that at (0,0) $\mathrm{f}$ is continuous and possesses all directional derivatives but is not differentiable.

1987

1) Let

2) If $f$ is continuous on a closed and finite interval $[a,b]$ then show that $f$ is uniformly continuous on $[\mathrm{a},\mathrm{b}]$.

3) Test for convergence ${\int }_0^{-x^2}dx$.

4) If $\mathrm{f}$ is monotonic and $\mathrm{g}$ is bounded and real valued function Riemann integrable over $[\mathrm{a},\mathrm{b}]$, then Prove that there exists a $c\in [a,b]$ such that

${\int }_0^{b}f(x)g(x)dx$=$f(a){\int }_0^{c}g(x)dx+f(b){\int }_c^{b}g(x)dx$

5) Test for uniform convergence of the sequence $\{S_n(x)\}$ where $S_n(x)=nx(1-x{)}^n$ when $0^{\mathrm{\prime }\mathrm{\prime }}{x}^{\mathrm{\prime }\mathrm{\prime }\mathrm{\prime }}1$.

6) Find the maximum and minimum value of $f(x)=xy$ subject to the condition that $x^2+y^2+xy=a^2$.

1986

1) If $f(x)=x^2$ for all $x\in R,$ then show that $f$ is uniformly continuous on every closed and finite interval, but is not uniformly continuous on $\mathrm{R}$.

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

3) If fand $\mathrm{g}$ are differentiable on $[\mathrm{a},\mathrm{b}]$ and $f^{\mathrm{\prime }},{\mathrm{g}}^{\mathrm{\prime }}$ are Riemann integrable over $[\mathrm{a},\mathrm{b}]$, then show that ${\int }_0^{b}f(x)dg(x)+{\int }_a^{b}g(x)df(x)=f(b)g(b)-f(a)g(a)$.

4) If $\mathrm{f}$ is monotonic and $\mathrm{g}$ is Riemann integrable over $[\mathrm{a},\mathrm{b}],$ then show that there exists a $C\in [a,b]$ such that

${\int }_0^{b}f(x)g(x)dx$=$f(a)\int g(x)dx+f(b){\int }_c^{b}g(x)dx$

5) Find the maximum and minimum values of $f(x,y)=7x^2+8xy+y^2$ where $x$, $y$ are connected by the relation $x^2+y^2=1$.

1985

1) Examine the convergence of the integral ${\int }_0^{\mathrm{\infty }}{\displaystyle \frac{\sin x^{\mathrm{\infty }}}{x^n}}dx$.

2) State and prove the second mean value theorem for Riemann integrals.

3) Show that for the function $f(x,y)=\{\begin{array}{cc}{\displaystyle \frac{x^2{y}^2}{x^2+y^2}}& ;(x,y)\ne (0,0)\\ 0& ;x=0,y=0\end{array}$
(i) $f_x$ is not differentiable at $(0,0)$
(ii) $f_{yx}$ is not continuous at $(0,0)$
(iii) $f_{xy}(0,0)=f_{yx}(0,0)$

1984

1) If $f$ is monotonic on $[a,b]$ and if $\alpha$ is continuous on $[\mathrm{a},\mathrm{b}],$ then prove that ${\int }_a^{b}fdx$ exists.

2) If $f$ and ${\alpha }^{\mathrm{\prime }}$ are integrable in the sense of Riemann on $[\mathrm{a},\mathrm{b}],$ then prove that $\int fd\alpha ={\int }_0^{b}f(x){\alpha }^{\mathrm{\prime }}(x)dx$.

3) Show that the maximum and minimum values of the function $u=x^2+y^2+xy$, where $a{x}^2+b{y}^2=ab(a>b>0)$ are given by $4(u-a)(u-b)=ab$.

4) Discuss the continuity and differentiability at (0,0) of the function $f(x,y)=x^2{\tan }^{-1}(y/x)-y^2{\tan }^{-1}(x/y)$; $f(0,0)=0$.

Also examine if $fxy$ and $fyx$ are equal at $(0,0)$.

1983

1) If $f(x)\ge 0$ and $f$ decreases monotonically for $x\ge 1,$ then prove that ${\int }_1^{\mathrm{\infty }}f(x)dx$ converges iff $\sum _{n=1}^{\mathrm{\infty }}f(n)$ converges.

2) Examine the convergence of the integral ${\int }_0^1(x^p+x^{-p})\log (1+x){\displaystyle \frac{dx}{x}}$.

3) Obtain a set of sufficient conditions such that for a function $f(x,y)$, ${\displaystyle \frac{{\mathrm{\partial }}^{2}f}{\mathrm{\partial }x\mathrm{\partial }y}}$=${\displaystyle \frac{{\mathrm{\partial }}^{2}f}{\mathrm{\partial }y\mathrm{\partial }x}}$.

4) Find the maximum and minimum values of $x^2+y^2+z^2$ subject to the conditions $x+y+z=1$; $xyz+1=0$.