2000

1) Use the mean value theforem to prove that ${\displaystyle \frac{2}{7}}<\log 1.4<{\displaystyle \frac{2}{5}}$.

[10M]

2) Let $f(x)=\{\begin{array}{l}0,x\text{ is irrational }\\ 1,x\text{ is rational }\end{array}$ show that $f$ is not Riemann-integrable on $[a,b]$

[10M]

3) Show that ${\displaystyle \frac{d^n}{d{x}^n}}\left({\displaystyle \frac{\log x}{x}}\right)=(-1{)}^n{\displaystyle \frac{n!}{x^{n+1}}}(\log x-1-{\displaystyle \frac{1}{2}}-{\displaystyle \frac{1}{3}}-\dots -{\displaystyle \frac{1}{n}})$

[10M]

4) Find constants $a$ and $b$ for which $F(a,b)={\int }_0^{\pi }{\{\sin x-(a{x}^2+bx)\}}^{2}dx$ is a minimum.

[10M]

1999

1) Determine the set of all points where the function $f(x)={\displaystyle \frac{x}{1+|x|}}$ is differentiable.

[10M]

2) Find three asymptotes of the curve $x^3+2x^{2}y-4x{y}^2-8y^3-4x+8y-10=0$. Also find the intercept of one asymptote between the other two.

[10M]

3) Find the dimensions of a right circular cone of minimum volume which can be circumscribed about a sphere of radius ${}^{\prime }{a}^{\prime }$.

[10M]

4) If $f$ is Riemann integrable over every interval of finite length and $f(x+y)=f(x)+f(y)$ for every pair of real numbers $x$ and $y$, show that $f(x)=cx$ where $c=f(1)$.

[10M]

5) Show that the area bounded by cissoid $x=asi{n}^{2}t,y=a{\displaystyle \frac{si{n}^{3}t}{cost}}$ and its asymptote is ${\displaystyle \frac{3\pi a^2}{4}}$.

[10M]

6) Show that $\int \int x^{m-1}{y}^{n-1}dxdy$ over the positive quadrant of the ellipse ${\displaystyle \frac{x^2}{a^2}}+{\displaystyle \frac{y^2}{b^2}}=1$ is ${\displaystyle \frac{a^m{b}^n}{4}}{\displaystyle \frac{\mathrm{\Gamma }({\displaystyle \frac{m}{2}})\mathrm{\Gamma }({\displaystyle \frac{n}{2}})}{\mathrm{\Gamma }({\displaystyle \frac{m}{2}}+{\displaystyle \frac{n}{2}}+1)}}$

[10M]

1998

1) Find the asymptotes of the curve $(2x-3y+1{)}^2(x+y)-8x+2y-9=0$ and show that they intersect the curve again in three points which lie on a straight line.

[10M]

2) A thin closed rectangular box is to have one edge $n$ times the length of another edge and the volume of the box is given to be $v$. Prove that the least surface $s$ is given by $n{s}^3=54(n+1{)}^2{v}^2$.

[10M]

3) If $x+y=1$, prove that ${\displaystyle \frac{d^n}{d{x}^n}}(x^n{y}^n)=n![y^n-({\displaystyle \frac{n}{1}}{)}^2{y}^{n-1}x+({\displaystyle \frac{n}{2}}{)}^2{y}^{n-2}{x}^2+...+(-1{)}^n{x}^n]$

[10M]

4) Show that ${\int }_0^{\mathrm{\infty }}{\displaystyle \frac{x^{p-1}}{(1+x{)}^{p+q}}}dx=B(p,q)$.

[10M]

5) Show that: $\iiint {\displaystyle \frac{dxdy}{\sqrt{(1-x^2-y^2-z^2)}}}={\displaystyle \frac{{\pi }^2}{8}}$, integral being extended over all positive values of $x,y,z$ for which the expression is real.

[10M]

6) The ellipse $b^2{x}^2+a^2{y}^2=a^2{b}^2$ is divided into two parts by the line $x={\displaystyle \frac{1}{2}}$a , the smaller part is rotated through four right angles about this line. Prove that the volume generated is

$\pi a^{2}b\{{\displaystyle \frac{3\sqrt{3}}{4}}-{\displaystyle \frac{\pi }{3}}\}$.

[10M]

1997

1) Suppose

Determine ${\displaystyle \frac{d}{dx}}\left(f^{-1}\right)$ at $\mathrm{x}=-1$ if it exists.

[10M]

2) Prove that the volume of the greatest parallelopiped that can be inscribed in the ellipsoid ${\displaystyle \frac{x^2}{a^2}}+{\displaystyle \frac{y^2}{b^2}}+{\displaystyle \frac{z^2}{c^2}}=1\text{ is }{\displaystyle \frac{8abc}{3\sqrt{3}}}$

[10M]

3) Show that the asymptotes of the curve $(x^2-y^2)(y^2-4x^2)+6x^3-5x^{2}y-3x{y}^2+z{y}^3-x^2+3xy-1=0$ cut the curve again in eight points which lie on a circle of radius 1 .

[10M]

4) An area bounded by a quadrant of a circle of radius a and the tangents at its extremities revolves about one of the tangents. Find the volume so generated.

[10M]

5) Show how the change of order in the integral ${\int }_0^{\mathrm{\infty }}{\int }_0^{\mathrm{\infty }}{e}^{-xy}\sin xdxdy$ leads to the evaluation of ${\int }_0^{\mathrm{\infty }}{\displaystyle \frac{\sin x}{x}}dx,$ Hence evaluate it.

[10M]

6) Show that in $\sqrt{n}\sqrt{n+{\displaystyle \frac{1}{2}}}={\displaystyle \frac{\sqrt{\pi }}{2^{2n-1}}}$ where $\mathrm{n}>0$ and $\sqrt{n}$ denotes gamma function.

[10M]

1996

1) Find the asymptotes of the curve $4(x^4+y^4)-17x^2{y}^2-4x(4y^2-x^2)+2(x^2-2)=0$ and show that they pass through the points of intersection of the curve with the ellipse $x^2+4y^2=4$

[15M]

2) Show that any continuous function defined for all real $x$ and satisfying the equation $f(x)=f$ $(2x+1)$ for all $x$ must be a constant function.

[15M]

3) Show that the maximum and minimum of the radii vectors of the sections of the surface ${(x^2+y^2+z^2)}^2={\displaystyle \frac{x^2}{a^2}}+{\displaystyle \frac{y^2}{b^2}}+{\displaystyle \frac{z^2}{c^2}}$ by the plane $\lambda x+\mu y+vz=0$ are given by the equation ${\displaystyle \frac{a^2{\lambda }^2}{1-a^2{r}^2}}+{\displaystyle \frac{b^2{\mu }^2}{1-b^2{r}^2}}+{\displaystyle \frac{c^2{v}^2}{1-c^2{r}^2}}=0$

[15M]

4) If $u=f({\displaystyle \frac{x}{y}},{\displaystyle \frac{y}{z}},{\displaystyle \frac{z}{x}}),$ prove that $x{\displaystyle \frac{\mathrm{\partial }u}{\mathrm{\partial }x}}+y{\displaystyle \frac{\mathrm{\partial }u}{\mathrm{\partial }y}}+z{\displaystyle \frac{\mathrm{\partial }u}{\mathrm{\partial }z}}=0$

[12M]

5) Evaluate ${\int }_0^{\mathrm{\infty }}{\int }_0^{\mathrm{\infty }}{\displaystyle \frac{e^{-y}}{y}}dxdy$

[12M]

6) The area cut off from the parabola $y^2=4ax$ by the chord joining the vertex to an end of the latus rectum is rotated through four right angles about the chord. Find the volume of the solid so formed.

[12M]

1995

1) If $\mathrm{g}$ is the inverse of $\mathrm{f}$ and $f^{\mathrm{\prime }}(x)={\displaystyle \frac{1}{1+x^3}}$, prove that $g^{\mathrm{\prime }}(x)=1+[g(x){]}^3$.

[20M]

2) Taking the nth derivative of ${\left({\mathrm{x}}^n\right)}^2$ in two different ways, show that $1+{\displaystyle \frac{n^2}{1^2}}+{\displaystyle \frac{n^2(n-1{)}^2}{1^2\cdot 2^2}}+{\displaystyle \frac{n^2(n-1{)}^2(n-2{)}^2}{1^2\cdot 2^2\cdot 3^2}}+\dots \mathrm{to}(n+1)$ terms $={\displaystyle \frac{(2n)!}{(n!{)}^2}}$

[20M]

3) Let $f(x,y),$ which possesses continuous partial derivatives of second order, be a homogeneous function of $x$ and $y$ of degree $n$. Prove that ${\mathrm{x}}^2{\mathrm{f}}_{\mathrm{x}\mathrm{x}}+2\mathrm{x}\mathrm{y}{\mathrm{f}}_{\mathrm{x}\mathrm{y}}+{\mathrm{y}}^2{\mathrm{f}}_{yy}=\mathrm{n}(\mathrm{n}-1)\mathrm{f}$

[20M]

4) Find the area bounded by the curve ${({\displaystyle \frac{x^2}{4}}+{\displaystyle \frac{y^2}{9}})}^2={\displaystyle \frac{x^2}{4}}-{\displaystyle \frac{y^2}{9}}$

[20M]

5) Let $f(x),x\ge 1$ be such that the area bounded by the curve $y=f(x)$ and the lines $x=1,x=b$ is equal to $\sqrt{1+b^2}-\sqrt{2}$ for all $\mathrm{b}\ge 1$, Does $f$ attain its minimum $?$ If so what is its value?

[20M]

6) Show that $\mathrm{\Gamma }\left({\displaystyle \frac{1}{n}}\right)\mathrm{\Gamma }\left({\displaystyle \frac{2}{n}}\right)\mathrm{\Gamma }\left({\displaystyle \frac{3}{n}}\right)\dots .\mathrm{\Gamma }\left({\displaystyle \frac{n-1}{n}}\right)={\displaystyle \frac{(2\pi )}{\sqrt{n}}}{\displaystyle \frac{n-1}{2}}$

[20M]