If $$\alpha $$ and $$\beta $$ be two roots of the equation
x² – 64x + 256 = 0. Then the value of
$${\left( {{{{\alpha ^3}} \over {{\beta ^5}}}} \right)^{1/8}} + {\left( {{{{\beta ^3}} \over {{\alpha ^5}}}} \right)^{1/8}}$$ is :

The position of a moving car at time t is
given by f(t) = at² + bt + c, t > 0, where a, b and c are real numbers greater than 1. Then the average speed of the car over the time interval [t₁ , t₂ ] is attained at the point :

If I₁ = $$\int\limits_0^1 {{{\left( {1 - {x^{50}}} \right)}^{100}}} dx$$ and
I₂ = $$\int\limits_0^1 {{{\left( {1 - {x^{50}}} \right)}^{101}}} dx$$ such
that I₂ = $$\alpha $$I₁ then $$\alpha $$ equals to :

If $$\overrightarrow a $$ and $$\overrightarrow b $$ are unit vectors, then the greatest value of

$$\sqrt 3 \left| {\overrightarrow a + \overrightarrow b } \right| + \left| {\overrightarrow a - \overrightarrow b } \right|$$ is_____.