If the volume of tetrahedron, whose vertices are with position vectors $$\hat{i}-6 \hat{j}+10 \hat{k},-\hat{i}-3 \hat{j}+7 \hat{k}, 5 \hat{i}-\hat{j}+\lambda \hat{k}$$ and $$7 \hat{i}-4 \hat{j}+7 \hat{k}$$ is 11 cubic units, then value of $$\lambda$$ is

The argument of $$\frac{1+i \sqrt{3}}{\sqrt{3}+i}, i=\sqrt{-1}$$ is

If $$y=\tan ^{-1}\left(\frac{\log \left(\frac{\mathrm{e}}{x^2}\right)}{\log \left(e x^2\right)}\right)+\tan ^{-1}\left(\frac{4+2 \log x}{1-8 \log x}\right)$$, then $$\frac{\mathrm{d} y}{\mathrm{~d} x}$$ is

If the surface area of a spherical balloon of radius $$6 \mathrm{~cm}$$ is increasing at the rate $$2 \mathrm{~cm}^2 / \mathrm{sec}$$, then the rate of increase in its volume in $$\mathrm{cm}^3 / \mathrm{sec}$$ is