If the length of the chord of the circle,
x² + y² = r² (r > 0) along the line, y – 2x = 3 is r,
then r² is equal to :

If $$\alpha $$ and $$\beta $$ are the roots of the equation,
7x² – 3x – 2 = 0, then the value of
$${\alpha \over {1 - {\alpha ^2}}} + {\beta \over {1 - {\beta ^2}}}$$ is equal to :

The derivative of
$${\tan ^{ - 1}}\left( {{{\sqrt {1 + {x^2}} - 1} \over x}} \right)$$ with
respect to $${\tan ^{ - 1}}\left( {{{2x\sqrt {1 - {x^2}} } \over {1 - 2{x^2}}}} \right)$$ at x = $${1 \over 2}$$ is :

If x = 1 is a critical point of the function
f(x) = (3x² + ax – 2 – a)e^x , then :