The value of $\frac{15}{\sqrt{10} + \sqrt{20} + \sqrt{40} - \sqrt{5} - \sqrt{80}}$ is

Ethanol when reacted with $PCl_5$ gives $A$,$POCl_3$ and $HCl$. $A$ reacts with silver nitrite to form $B$ (major product) and $AgCl$. $A$ and $B$ respectively are

If $\alpha, \beta, \gamma$ are the roots of the equation $x^3-6x^2+11x-6=0$ and if $a=\alpha^2+\beta^2+\gamma^2$,$b=\alpha\beta+\beta\gamma+\gamma\alpha$ and $c=(\alpha+\beta)(\beta+\gamma)(\gamma+\alpha)$,then the correct inequality among the following is

$\int \frac{x-1}{(x+1) \sqrt{x(x^2+x+1)}} dx = $

Let $\overline{a}=\hat{i}+\hat{j}+\hat{k}$,$\overline{b}=\hat{i}-\hat{j}+\hat{k}$ and $\overline{c}=\hat{i}-\hat{j}-\hat{k}$ be three vectors. $A$ vector $\overline{v}$ in the plane of $\overline{a}$ and $\overline{b}$,whose projection on $\overline{c}$ is $\frac{1}{\sqrt{3}}$,is given by

If $f(x) = x^2 + 2bx + 2c^2$ and $g(x) = -x^2 - 2cx + b^2$ such that $\min f(x) > \max g(x)$,then the relation between $b$ and $c$ is: