Let f be a differentiable function from

R to R such that $$\left| {f\left( x \right) - f\left( y \right)} \right| \le 2{\left| {x - y} \right|^{{3 \over 2}}},$$   

for all  $$x,y \in $$ R.

If   $$f\left( 0 \right) = 1$$  

then   $$\int\limits_0^1 {{f^2}} \left( x \right)dx$$  is equal to :

If   $$f\left( x \right) = \int {{{5{x^8} + 7{x^6}} \over {{{\left( {{x^2} + 1 + 2{x^7}} \right)}^2}}}} \,dx,\,\left( {x \ge 0} \right),$$

$$f\left( 0 \right) = 0,$$    then the value of $$f(1)$$ is :

The number of all possible positive integral values of $$\alpha $$  for which the roots of the quadratic equation, 6x² $$-$$ 11x + $$\alpha $$ = 0 are rational numbers is :

Let  $$\overrightarrow a = \widehat i + \widehat j + \sqrt 2 \widehat k,$$   $$\overrightarrow b = {b_1}\widehat i + {b_2}\widehat j + \sqrt 2 \widehat k$$,    $$\overrightarrow c = 5\widehat i + \widehat j + \sqrt 2 \widehat k$$   be three vectors such that the projection vector of $$\overrightarrow b $$ on $$\overrightarrow a $$ is $$\overrightarrow a $$.
If   $$\overrightarrow a + \overrightarrow b $$   is perpendicular to $$\overrightarrow c $$ , then $$\left| {\overrightarrow b } \right|$$ is equal to :