a) State Rolle's theorem hence verify for $f(x)=x^2+2x$ defined in the interval [-2, 0].

b) Find the first six terms of the expansions of the function $e^x\log(1+y)$ in a Taylor series in the neighbourhood of the point (0, 0).

a) The temperature $u(x,y,z)$ at any point in space is $u=400xz^2$ find the highest temperature on surface of the sphere $x^2+y^2+z^2=1$.

b) If $u=x^2\tan^{-1}\frac{y}{x}-y^2\tan^{-1}\frac{x}{y}$ find the value of $\frac{\partial^2u}{\partial x \partial y}$.

a) Find shortest distance from the origin to the curve $x^2+4xy+6y^2=140$.

b) Find $\frac{du}{dt}$ if $u=x^2+y^2, x=a\cos t, y=b\sin t$.

a) Change the order of integration in $\int_{0}^{1}\int_{x^2}^{2-x}xy\ dy\ dx$ and hence evaluate.

b) Evaluate $\iint e^{2x+3y}dxdy$ over the triangle bounded by $x=0, y=0$ and $x+y=1$.

a) Test the convergence of the series $1+\frac{x}{2}+\frac{x^2}{5}+\frac{x^3}{10}+...+\frac{x^n}{n^2+1}+...$

b) Expand as a half range $f(x)=x\sin x$ series and cosine series for the interval $0

a) Expand $f(x)=x\sin x, 0

b) Find the $a_0$ and $a_n$ if the function $f(x)=x+x^2$ is expanded in Fourier series defined in (-1, 1).

a) i) If A is a skew symmetric matrix then show that $A^2$ is a symmetric matrix.

ii) Find eigen values of the matrix $\begin{pmatrix} 5 & 4 \\ 1 & 2 \end{pmatrix}$.

b) Find the inverse of $\begin{pmatrix} 0 & 1 & 2 \\ 1 & 2 & 3 \\ 3 & 1 & 1 \end{pmatrix}$ by using elementary row transformations.

a) Verify Cayley Hamilton theorem for the matrix A and hence find $A^{-1}$ for $\begin{pmatrix} 1 & 2 & -2 \\ -1 & 3 & 0 \\ 0 & -2 & 1 \end{pmatrix}$.

b) Test the consistency and hence, solve the following set of equations.

$x+2y-z=3$

$3x-y+2z=1$

$2x-2y+3z=2$

$x-y+z=-1$