Let $$S$$ be the set of all solutions of the equation $$\cos ^{-1}(2 x)-2 \cos ^{-1}\left(\sqrt{1-x^{2}}\right)=\pi, x \in\left[-\frac{1}{2}, \frac{1}{2}\right]$$. Then $$\sum_\limits{x \in S} 2 \sin ^{-1}\left(x^{2}-1\right)$$ is equal to :

Let $$R$$ be a relation on $$\mathbb{R}$$, given by $$R=\{(a, b): 3 a-3 b+\sqrt{7}$$ is an irrational number $$\}$$. Then $$R$$ is

The shortest distance between the lines

$${{x - 5} \over 1} = {{y - 2} \over 2} = {{z - 4} \over { - 3}}$$ and

$${{x + 3} \over 1} = {{y + 5} \over 4} = {{z - 1} \over { - 5}}$$ is :

Let $$S$$ denote the set of all real values of $$\lambda$$ such that the system of equations

$$\lambda x+y+z=1$$

$$x+\lambda y+z=1$$

$$x+y+\lambda z=1$$

is inconsistent, then $$\sum_\limits{\lambda \in S}\left(|\lambda|^{2}+|\lambda|\right)$$ is equal to