A six faced die is a biased one. It is thrice more likely to show an odd numbers than show an even number. It is thrown twice. The probability that the sum of the numbers in two throws is even, is

The sum of all the solution of the equation $$\cos \theta \cos \left( {{\pi \over 3} + \theta } \right)\cos \left( {{\pi \over 3} - \theta } \right) = {1 \over 4},\theta \in [0,6\pi ]$$

Let $$\alpha$$ be the solution of $${16^{{{\sin }^2}\theta }} + {16^{{{\cos }^2}\theta }} = 10$$ in $$\left( {0,{\pi \over 4}} \right)$$. If the shadow of a vertical pole is $${1 \over {\sqrt 3 }}$$ of its height, then the altitude of the sun is

For each parabola y = x² + px + q, meeting coordinate axes at 3-distinct points, if circles are drawn through these points, then the family of circles must pass through