a) Prove that $\frac{\pi}{3} > \frac{1}{\sqrt{53}}\cos^{-1}\frac{3}{5} > \frac{\pi}{3} - \frac{1}{8}$ using Lagrange's mean value theorem.

b) Find the minimum and maximum value of $f(x,y)=x^3+3xy^2-3x^2-3y^2+4$.

a) Find C of Cauchy's Mean value theorem on [a, b] for the function $f(x)=e^x$ and $g(x)=e^{-x}, (a,b>0)$.

b) Prove that $\Gamma(n)\Gamma(1-n)=\frac{\pi}{\sin n\pi}$.

a) By Changing the order of integration, evaluate $\int_{0}^{1}\int_{0}^{\sqrt{1-x^2}}y^2dydx$.

b) Find the area of a plane in the form of a quadrant of the ellipse $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$.

Let W be a subspace of a finite dimensional vector space V(F). Then $\dim(V/W) = \dim V - \dim W$.

Obtain the Fourier series to represent $f(x)=x\sin x, 0

a) Show that $T: V_2(R) \to V_3(R)$ is defined as $T(a, b) = (a-b, b-a, -a)$ is linear transformation.

b) Test the convergence of the series $\sum_{n=1}^{\infty}(\sqrt{n+1}-\sqrt{n-1})$.

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

b) Examine the consistency of the system of the following equations. If consistent, solve the equations.

$x+y+z=3$

$x+2y+3z=4$

$x+4y+9z=6$

Diagonalize the matrix $A = \begin{pmatrix} 1 & 1 & 0 \\ 6 & 2 & 0 \\ 1 & 0 & 3 \end{pmatrix}$.