Let $$S$$ be the set of all non-zero real numbers $$\alpha $$ such that the quadratic equation $$\alpha {x^2} - x + \alpha = 0$$ has two distinct real roots $${x_1}$$ and $${x_2}$$ satisfying the inequality $$\left| {{x_1} - {x_2}} \right| < 1.$$ Which of the following intervals is (are) $$a$$ subset(s) os $$S$$?

For any integer k, let $${a_k} = \cos \left( {{{k\pi } \over 7}} \right) + i\,\,\sin \left( {{{k\pi } \over 7}} \right)$$, where $$i = \sqrt { - 1} \,$$. The value of the expression $${{\sum\limits_{k = 1}^{12} {\left| {{\alpha _{k + 1}} - {a_k}} \right|} } \over {\sum\limits_{k = 1}^3 {\left| {{\alpha _{4k - 1}} - {\alpha _{4k - 2}}} \right|} }}$$ is

Suppose that $$\overrightarrow p ,\overrightarrow q $$ and $$\overrightarrow r $$ are three non-coplanar vectors in $${R^3}$$. Let the components of a vector $$\overrightarrow s $$ along $$\overrightarrow p ,$$ $$\overrightarrow q $$ and $$\overrightarrow r $$ be $$4, 3$$ and $$5,$$ respectively. If the components of this vector $$\overrightarrow s $$ along $$\left( { - \overrightarrow p + \overrightarrow q + \overrightarrow r } \right),\left( {\overrightarrow p - \overrightarrow q + \overrightarrow r } \right)$$ and $$\left( { - \overrightarrow p - \overrightarrow q + \overrightarrow r } \right)$$ are $$x, y$$ and $$z,$$ respectively, then the value of $$2x+y+z$$ is

Let $${n_1}$$ and $${n_2}$$ be the number of red and black balls, respectively, in box $${\rm I}$$. Let $${n_3}$$ and $${n_4}$$ be the number of red and black balls, respectively, in box $${\rm I}{\rm I}.$$

A ball is drawn at random from box $${\rm I}$$ and transferred to box $${\rm I}$$$${\rm I}.$$ If the probability of drawing a red ball from box $${\rm I},$$ after this transfer, is $${1 \over 3},$$ then the correct option(s) with the possible values of $${n_1}$$ and $${n_2}$$ is(are)