Let x >1, y >1, z >1 be in GP. Then $\rm \frac{1}{1+ln x}, \frac{1}{1+lny}, \frac{1}{1+\ln z}$ are

If $\rm \omega=-\frac{1}{2}+i\frac{\sqrt3}{2}$ then what is $\rm \begin{vmatrix}1+\omega &1+\omega^2&\omega+\omega^2\\\ 1&\omega\ & \omega^2\\\ \frac{1}{\omega}&\frac{1}{\omega^2}&1\end{vmatrix}$ equal to? 

If the sum of the first n terms of a series is n(2n+1), then what is the nth term?

In how many ways can the letters of the word INDIA be permutated such that in each combination, vowels should occupy odd positions?