The ratio of mass percent of C and H of an organic compound (C_XH_YO_Z) is 6 : 1. If one molecule of the above compound (C_XH_YO_Z) contains half as much oxygen as required to burn one molecule of compound C_XH_Y completely to CO₂ and H₂O. The empirical formula of compound C_XH_YO_Z is

From 6 different novels and 3 different dictionaries, 4 novels and 1 dictionary are to be selected and arranged in a row on a shelf so that the dictionary is always in the middle. The number of such arrangements is :

Let S = { $$x$$ $$ \in $$ R : $$x$$ $$ \ge $$ 0 and

$$2\left| {\sqrt x - 3} \right| + \sqrt x \left( {\sqrt x - 6} \right) + 6 = 0$$}. Then S

If $$\alpha ,\beta \in C$$ are the distinct roots of the equation
x² - x + 1 = 0, then $${\alpha ^{101}} + {\beta ^{107}}$$ is equal to :