1994

1) Examine the

(i) absolute convergence
(ii) uniform convergence

of the series $(1-x)+x(1-x)+x^2(1-x)+\dots$ in $[-c,1]$, where $0<c<1$.

[10M]

2) Prove that $S(x)=\sum {\displaystyle \frac{1}{n^p+n^q{x}^2}},p>1$ is uniformly convergent for all values of $x$ and can be differentiated term by term if $q<3p-2$

[10M]

3) Let the function $f$ be defined on $[0,1]$ by the condition $f(x)=2rx$ when ${\displaystyle \frac{1}{r+1}}<x<{\displaystyle \frac{1}{r}},r>0$. Show that $f$ is Riemann integrable in $[0,1]$ and ${\int }_0^{1}f(x)dx={\displaystyle \frac{{\pi }^2}{6}}$.

[10M]

4) By means of the substitution $x+y+z=u,y+z=uv,z=uvw$, evaluate $\iiint (x+y+z{)}^{n}xyzdxdydz$ taken over the volume bounded by $x=0,y=0,z=0,x+y+z=1$.

[10M]

1993

1) Show that $x=0$ is a point of non-uniform convergence of the series whose $nth$ term is $n^{2}x{e}^{n^2{x}^2}-(n-1{)}^{2}x{e}^{-(n-1)}2x^2$.

[10M]

2) Find all the maxima and minima of $f(x,y)=x^3+y^3-63(x+y)+12xy$

[10M]

3) Examine for Riemann integrability over [0,2] of the function defined in [0,2] by $f(x)=\{\begin{array}{r}x+x^2,\text{ for rational values of }x\\ x^2+x^3,\text{ for irrational values of }x\end{array}$

[10M]

4) Prove that ${\int }_0^{\mathrm{\infty }}{\displaystyle \frac{\sin x}{x}}dx$ converges and conditionally converges.

[10M]

5) Evaluate $\iiint {\displaystyle \frac{dxdydz}{x+y+z+1}}$ over the volume bounded by the coordinate planes and the plane $x+y+z=1$.

[10M]

1992

1) Examine $f(x,y,z)=2xyz-4zx-2yz+x^2+y^2+z^2-2x-4y-4z$ for extreme values.

[10M]

2) If $U_n={\displaystyle \frac{1+nx}{n{e}^{nx}}}-{\displaystyle \frac{1+(n+1)x}{(n+1)e^{(n+1)x}}},0<x<1$ Prove that ${\displaystyle \frac{d}{dx}}\mathrm{\Sigma }{U}_n=\mathrm{\Sigma }{\displaystyle \frac{d}{dx}}{u}_n$ Is the series uniformly convergent in $[0,1]$ ? Justify your claim.

[10M]

3) Find the upper and lower Riemann integral for the function defined in the interval [0,1] as follows: $f(x)=\{\begin{array}{ll}\sqrt{1-x^2}& \text{when x rational }\\ 1-x& \text{ when x is irrational }\end{array}$ and show that $f(x)$ is not Riemann integrable in [0,1]

[10M]

4) Discuss the convergence or divergence of ${\int }_0^{\mathrm{\infty }}{\displaystyle \frac{x^{\beta }dx}{1+x\alpha {\sin }^{2}x}},\alpha >\beta >0$

[10M]

5) Evaluate $\iint {\displaystyle \frac{\sqrt{(a^2{b}^2-b^2{x}^2-a^2{y}^2)}}{\sqrt{(a^2{b}^2+b^2{x}^2+a^2{y}^2)}}}dxdy$ over the positive quadrant of the ellipse ${\displaystyle \frac{x^2}{a^2}}+{\displaystyle \frac{y^2}{b^2}}=1$

[10M]

1991

1) Examine whether the function $y=\tan x$ is uniformly continuous in the open interval $(0,{\displaystyle \frac{\pi }{2}})$.

2) Evaluate ${\int }_0^{\mathrm{\infty }}\frac{\log (1+a^2{x}^2)}{1+b^2{x}^2}dx$.

3) If the rectangular axes $(x,y)$ are rotated through an angle $\alpha$ about the origin and the new co-ordinates are $(\overline{x},\overline{y})$ show that for any function $u$, ${\displaystyle \frac{{\mathrm{\partial }}^{2}u}{\mathrm{\partial }{x}^2}}+{\displaystyle \frac{{\mathrm{\partial }}^{2}u}{\mathrm{\partial }{y}^2}}={\displaystyle \frac{{\mathrm{\partial }}^{2}u}{\mathrm{\partial }{\overline{x}}^2}}+{\displaystyle \frac{{\mathrm{\partial }}^{2}u}{\mathrm{\partial }{\overline{y}}^2}}$.

4) A rectangle is inscribed in the ellipse ${\displaystyle \frac{x^2}{a^2}}+{\displaystyle \frac{y^2}{b^2}}=1$. What is the maximum possible area of the rectangle?

1990

1) Discuss the convergence of ${\int }_0^{\pi /2}\log \sin xdx$ and evaluate it, if it is convergent.

2) Find the point on the parabola $y^2=2x$, $z=0$, which is nearest the plane $z=x+2y+8$. Show that this minimum distance is $\sqrt{6}$.

3) Show that $\iiint {(a^2{b}^2{c}^2-b^2{c}^2{x}^2-c^2{a}^2{y}^2-a^2{b}^2{z}^2)}^{1/2}dxdydz$ over the volume bounded by ${\displaystyle \frac{x^2}{a^2}}+{\displaystyle \frac{y^2}{b^2}}+{\displaystyle \frac{z^2}{c^2}}=1$ is equal to ${\displaystyle \frac{{\ne }^2{a}^2{b}^2{c}^2}{4}}$.

TBC

1989

1) The function $f:[a,\mathrm{\infty })\to \mathbb{R}$ is continuous and $f(x)\to l$ as $x\to \mathrm{\infty },$ prove that $f$ is uniformly continuous on $[a,\mathrm{\infty })$.

2) Find the values of $x$ for which the series $\sum _{n=1}^{\mathrm{\infty }}{\displaystyle \frac{x^a}{1+n^2{x}^2}},x\ge 0,\alpha \ge 0,$ converges uniformly on (i) $[0,1]$ and
(ii) $[0,\mathrm{\infty })$

3) Discuss the existence of the improper integral ${\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}{e}^{-x^2/2}dx$.

4) Show that the value of $\iint (1-x-y{)}^2{x}^{1/2}{y}^{1/2}dxdy$ taken over the interior of the triangle whose vertices are the origin and the points $(0,1)$ and $(1,0)$ is ${\displaystyle \frac{\ne }{410}}$.

TBC