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?

The letters of the word EQUATION are arranged in such a way that all vowels as well as consonants are together. How many such arrangements are there?