2008

1) Obtain the values of the constants a, $b$ and $c$ for which the function defined by

is continuous at $x=0$.

[10M]

2) If $u={\sin }^{-1}\left({\displaystyle \frac{x^3+y^3}{\sqrt{x}+\sqrt{y}}}\right),$ prove that
$x{\displaystyle \frac{\mathrm{\partial }u}{\mathrm{\partial }x}}+y{\displaystyle \frac{\mathrm{\partial }u}{\mathrm{\partial }y}}={\displaystyle \frac{5}{2}}\tan u$.

[10M]

3) Determine the value of
$\left({\int }_0^1{\displaystyle \frac{x^2}{{(1-x^4)}^{1/2}}}dx\right)\left({\int }_0^1{\displaystyle \frac{dx}{{(1+x^4)}^{1/2}}}\right)$

[10M]

4) Obtain the value of the double integral ${\iint }_D(x^2+y^2)dxdy$ where $D$ represents the region bounded by the straight line $y=x$ and the parabola $y^2=4x$.

[10M]

5) A wire of length $b$ is cut into two parts which are bent in the form of a square and a circle respectively. Find the minimum value of the sum of the areas so formed.

[10M]

6) Calculate the volume cut off from the sphere $x^2+y^2+z^2$ $=a^2$ by the right circular cylinder given by $x^2+y^2=b^2$.

[10M]

2007

1) If a function $f$ is such that its derivative $f$ is continuous on $[\mathrm{a},\mathrm{b}]$ and derivable on $]\mathrm{a},\mathrm{b}[$, then show that there exists a number $c$ between $a$ and $b$ such that $$ f(b)=f(a)+(b-a)f’(a)+\dfrac{1}{2}(b-a)^{2} f^{\prime \prime}(c) $$

[10M]

2) If

Show that both the partial derivatives exist at (0,0) but the function is not continuous thereat.

[10M]

3) Find the values of $a$ and $b$, so that

What are these conditions?

[10M]

4) Show that $f(xy,z2x)=0$ satisfies, under certain conditions, the equation

What are these conditions?

[10M]

5) Find the surface area generated by the revolution of the cardioids $r=a(\mathrm{I}+\cos \theta )$ about the initial line.

[10M]

6) The function $f$ is defined on $]0,1[$ by

Show that

Does not converge.

[10M]

2006

1) Show that

${\int }_0^{\mathrm{\infty }}{\displaystyle \frac{{\tan }^{-1}\alpha x{\tan }^{-1}\beta x}{x^2}}dx$ = ${\displaystyle \frac{\pi }{2}}{\displaystyle \frac{(\alpha +\beta {)}^{(\alpha +\beta )}}{{\alpha }^{\alpha }+{\beta }^{\beta }}}$, $\alpha$, $\beta >0$

[10M]

2) Show that $f_x(0,0)\ne f_y(a,0)$ where $f(x,y)=0$, if $xy=0$, $f(x,y)=x^2{\tan }^{-1}{\displaystyle \frac{y}{x}}-y^2{\tan }^{-1}{\displaystyle \frac{x}{y}}$, if $xy\ne 0$

[10M]

3) Find the volume under the spherical surface $x^2+y^2+z^2=a^2$ and over the lemniscate $r^2=a^2$ $\cos 2\theta$.

[10M]

4) Find the centre of gravity of the volume common to a cone of vertical angle $2\alpha$ and a sphere of radius $a$, the vertex of the cone being the centre of the sphere.

[10M]

5) Using Lagrange’s method of undetermined multipliers, find the stationary values of ${\mathrm{x}}^2+{\mathrm{y}}^2+$ $z^2$ subject to $a{x}^2+b{y}^2+c{z}^2=1$ and $lx+my+nz=0$. Interpret geometrically.

[10M]

6) Find the extreme values of $f(x,y)=2(x-y{)}^2-x^4-y^4$.

[10M]

2005

1) Let $f(x)=\{\begin{array}{c}{x}^2\sin {\displaystyle \frac{1}{x}}\text{ for }x\ne 0\\ 0.\text{ for }x=0\end{array}$
Show that $\mathrm{f}$ is differentiable at each point of reals but $\mathrm{f}(\mathrm{x})$ is not continuous at $\mathrm{x}=0$.

[10M]

2) Show that $f:R^2\to R$ defined by $f(x,y)2x^2-6xy+3y^2$ has a critical point at $(0,0)$ and that it is a saddle point.

[10M]

3.(i) Using Taylor’s theorem with remainder show that $x-{\displaystyle \frac{x^3}{6}}\le \sin x\le x-{\displaystyle \frac{x^3}{6}}+{\displaystyle \frac{x^5}{120}}\text{ for all }x\ge 0$

[5M]

3.(ii) Let $f:{\mathbf{R}}^2\to \mathbf{R}$ be defined by $\begin{array}{rl}f(x,y)& ={\displaystyle \frac{xy}{x^2-y^2}}\text{ if }x\ne \pm y\\ & =0\text{ if }x=\pm y\end{array}$

[5M]

4) Show that the curve given by $x^3-4x^{2}y+5x{y}^2-2y^3+3x^2-4xy+2y^2-3x+2y-1=0$ has only one asymptote given by $y={\displaystyle \frac{1}{2}}x+3$.

[10M]

5) Find the extremum values of $f(x,y)=2x^2-8xy+9y^2\text{ on }{x}^4+y^2-1=0$ using Lagrange multiplier method.

[10M]

6) A solid cuboid in $R^3$ given in spherical coordinates by $R=[0,a]$, $\theta =[0,\pi ]$, $\phi =0,\pi /4$ has a density function $\rho (\mathbf{R},\theta ,\phi )=4\mathbf{R}\sin {\displaystyle \frac{\theta }{2}}\cos \phi .$ Find the total mass of $\mathbf{C}$

[10M]