a) State Lagrange'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 $e^x\cos y$ in a Taylor series in the neighbourhood of the point (0, 0).

a) Estimate the extreme values of the function $x^3+y^3-63(x+y)+12xy$.

b) If $u=\left(\frac{y-x}{xy}\right)\left(\frac{z-x}{xz}\right)$ find the value of $x^2u_x+y^2u_y+z^2u_z$.

a) Show that the rectangular solid of maximum volume that can be inscribed in a given sphere is a cube.

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) i) Find the value of $\Gamma(\frac{3}{2})$.

ii) Evaluate $\int_{0}^{1}x^3(1-\sqrt{x})^2dx$.

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

a) Show that $\beta(l,m)=\frac{\Gamma(l)\Gamma(m)}{\Gamma(l+m)}$.

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

a) Transform the following matrix into normal form and hence find its rank $\begin{pmatrix} 5 & 3 & 14 & 4 \\ 0 & 1 & 2 & 1 \\ 1 & -1 & 2 & 0 \end{pmatrix}$.

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

a) Find the eigen values and eigen vectors of matrix $\begin{pmatrix} 2 & -2 & 2 \\ 1 & 1 & 1 \\ 1 & 3 & -1 \end{pmatrix}$.

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

$x+2y-z=3, 3x-y+2z=1, 2x-2y+3z=2, x-y+z=-1$.