If a circle C, whose radius is 3, touches externally the circle,
$${x^2} + {y^2} + 2x - 4y - 4 = 0$$ at the point (2, 2), then the length of the intercept cut by this circle C, on the x-axis is equal to :

Let A = $$\left[ {\matrix{ 1 & 0 & 0 \cr 1 & 1 & 0 \cr 1 & 1 & 1 \cr } } \right]$$ and B = A²⁰. Then the sum of the elements of the first column of B is :

The least positive integer n for which $${\left( {{{1 + i\sqrt 3 } \over {1 - i\sqrt 3 }}} \right)^n} = 1,$$ is :

Let p, q and r be real numbers (p $$ \ne $$ q, r $$ \ne $$ 0), such that the roots of the equation $${1 \over {x + p}} + {1 \over {x + q}} = {1 \over r}$$ are equal in magnitude but opposite in sign, then the sum of squares of these roots is equal to :