The number of values of $$\alpha$$ for which the system of equations :

x + y + z = $$\alpha$$

$$\alpha$$x + 2$$\alpha$$y + 3z = $$-$$1

x + 3$$\alpha$$y + 5z = 4

is inconsistent, is

If the sum of the squares of the reciprocals of the roots $$\alpha$$ and $$\beta$$ of

the equation 3x² + $$\lambda$$x $$-$$ 1 = 0 is 15, then 6($$\alpha$$³ + $$\beta$$³)² is equal to :

The set of all values of k for which

$${({\tan ^{ - 1}}x)^3} + {({\cot ^{ - 1}}x)^3} = k{\pi ^3},\,x \in R$$, is the interval :

For the function

$$f(x) = 4{\log _e}(x - 1) - 2{x^2} + 4x + 5,\,x > 1$$, which one of the following is NOT correct?