Let $$p$$ and $$q$$ be roots of the equation $${x^2} - 2x + A = 0$$ and let $$r$$ and $$s$$ be the roots of the equation $${x^2} - 18x + B = 0.$$ If $$p < q < r < s$$ are in arithmetic progression, then $$A = \,..........$$ and $$B = \,..........$$

The real roots of the equation $$\,{\cos ^7}x + {\sin ^4}x = 1$$ in the interval $$\left( { - \pi ,\pi } \right)$$ are ...., ...., and ______.

For each natural number k, let $${C_k}$$ denote the circle with radius k centimetres and centre at the origin. On the circle $${C_k}$$, a-particle moves k centimetres in the counter-clockwise direction. After completing its motion on $${C_k}$$, the particle moves to $${C_{k + 1}}$$ in the radial direction. The motion of the patticle continues in the manner. The particle starts at (1, 0). If the particle crosses the positive direction of the x-axis for the first time on the circle $${C_n}$$ then n = ..............

Let C be any circle with centre $$\,\left( {0\, , \sqrt {2} } \right)$$. Prove that at the most two rational points can to there on C. (A rational point is a point both of whose coordinates are rational numbers.)