Let a, b $$ \in $$ R, a $$ \ne $$ 0 be such that the equation, ax² – 2bx + 5 = 0 has a repeated root $$\alpha $$, which is also a root of the equation, x² – 2bx – 10 = 0. If $$\beta $$ is the other root of this equation, then $$\alpha $$² + $$\beta $$² is equal to :

The number of real roots of the equation,
e^4x + e^3x – 4e^2x + e^x + 1 = 0 is :

Let S be the set of all real roots of the equation,
3^x(3^x – 1) + 2 = |3^x – 1| + |3^x – 2|. Then S :

Let $$\alpha = {{ - 1 + i\sqrt 3 } \over 2}$$.
If $$a = \left( {1 + \alpha } \right)\sum\limits_{k = 0}^{100} {{\alpha ^{2k}}} $$ and
$$b = \sum\limits_{k = 0}^{100} {{\alpha ^{3k}}} $$, then a and b are the roots of the quadratic equation :