Let $$A = \{ (x,y) \in R \times R|2{x^2} + 2{y^2} - 2x - 2y = 1\} $$, $$B = \{ (x,y) \in R \times R|4{x^2} + 4{y^2} - 16y + 7 = 0\} $$ and $$C = \{ (x,y) \in R \times R|{x^2} + {y^2} - 4x - 2y + 5 \le {r^2}\} $$.

Then the minimum value of |r| such that $$A \cup B \subseteq C$$ is equal to

For real numbers $$\alpha$$ and $$\beta$$, consider the following system of linear equations :

x + y $$-$$ z = 2, x + 2y + $$\alpha$$z = 1, 2x $$-$$ y + z = $$\beta$$. If the system has infinite solutions, then $$\alpha$$ + $$\beta$$ is equal to ______________.

Let $$\overrightarrow a = \widehat i + \widehat j + \widehat k,\overrightarrow b $$ and $$\overrightarrow c = \widehat j - \widehat k$$ be three vectors such that $$\overrightarrow a \times \overrightarrow b = \overrightarrow c $$ and $$\overrightarrow a \,.\,\overrightarrow b = 1$$. If the length of projection vector of the vector $$\overrightarrow b $$ on the vector $$\overrightarrow a \times \overrightarrow c $$ is l, then the value of 3l² is equal to _____________.

If $${\log _3}2,{\log _3}({2^x} - 5),{\log _3}\left( {{2^x} - {7 \over 2}} \right)$$ are in an arithmetic progression, then the value of x is equal to _____________.