a) Prove that a rectangular solid of maximum volume within a sphere is a cube.

b) If $u=\tan^{-1}\frac{x^3+y^3}{x-y}$ then prove that $x\frac{\partial u}{\partial x}+y\frac{\partial u}{\partial y}=\sin 2u$.

a) Show that the surface area of solid generated by revolution of the loop of curve $x=t^2, y=t-t^3/3$ about the x-axis is $3\pi$.

b) Change the order of integration in the following integral and then evaluate $\int_{0}^{\infty}\int_{x}^{\infty}\frac{e^{-y}}{y}dydx$.

a) Find the Fourier Series for $f(x)=x+x^2$ in $(-\pi,\pi)$.

b) Find the half range sine series for $f(x)=x(\pi-x)$ in $(0,\pi)$. Hence Deduce that $1-\frac{1}{3^3}+\frac{1}{5^3}-\cdots=\frac{\pi^3}{32}$.

a) Let $W_1$ and $W_2$ be subspaces of a vector space V and assume that $W_1 \cap W_2 = \{0\}$. Let $w_1 \in W_1$ and $w_2 \in W_2$ be such that $w_1 \neq 0$ and $w_2 \neq 0$. Prove that $\{w_1, w_2\}$ is linearly independent.

b) i) Prove that $W_1 \cap W_2$ is a subspace of V.

ii) Give an example to show that $W_1 \cup W_2$ need not be a subspace of V.

iii) Is $W_1 \cup W_2$ a subspace of V?

a) Find Eigen Value and Eigen vectors of $\begin{pmatrix} 2 & -2 & 2 \\ 1 & 1 & 1 \\ 1 & 3 & -1 \end{pmatrix}$.

b) Diagonalizable the matrix. $\begin{pmatrix} 3 & -1 & 1 \\ -1 & 5 & -1 \\ 1 & -1 & 3 \end{pmatrix}$.

a) Reduce the matrix $\begin{pmatrix} 3 & 2 & -1 \\ 4 & 2 & 6 \\ 7 & 4 & 5 \end{pmatrix}$ to the normal form, hence find its rank.

b) Find the first 3 terms in the Maclaurin series for i) $\sin^{-1}x$ ii) $xe^{-x}$.

a) Investigate for what values of $\lambda$ and $\mu$ the simultaneous equation

$x+y+z=6$

$x+2y+3z=10$

$x+2y+\lambda z=\mu$

Have i) no solution ii) a unique solution iii) an infinite number of solutions.

b) Find the Taylor series for the function $x^4+x-2$ centered at a=1.

a) Evaluate $\int_{0}^{\infty}\int_{x}^{\infty}\frac{e^{-y}}{y}dydx$.

b) If $x^x y^y z^z = C$ then show that $\frac{\partial^2 z}{\partial x \partial y}=-(x\log ex)^{-1}$.