Match the following : $\,\,\,\,\,\,\,\,\,\,\,\,$ Column $I$ (A) $\,\,\,\,$ In ${R^2},$ If the magnitude of the projection vector of the vector $\alpha \widehat i + \beta \widehat j$ on $\sqrt 3 \widehat i + \widehat j$ and If $\alpha = 2 + \sqrt 3 \beta ,$ then possible value of $$\left| \alpha \right|$$ is/are (B) $\,\,\,\,$ Let $a$ and $b$ be real numbers such that the function $$f\left( x \right) = \left\{ {\begin{matrix} { - 3a{x^2} - 2,} & {x \lt 1} \\ {bx + {a^2},} & {x \ge 1} \\ \end{matrix} } \right.$$ if differentiable for all $x \in R$. Then possible value of $a$ is (are) (C) $\,\,\,\,$ Let $\omega e 1$ be a complex cube root of unity. If $${\left( {3 - 3\omega + 2{\omega ^2}} \right)^{4n + 3}} + {\left( {2 + 3\omega - 3{\omega ^2}} \right)^{4n + 3}} + {\left( { - 3 + 2\omega + 3{\omega ^2}} \right)^{4n + 3}} = 0,$$ then possible value (s) of $n$ is (are) (D) $\,\,\,\,$ Let the harmonic mean of two positive real numbers $a$ and $b$ be $4.$ If $q$ is a positive real nimber such that $a, 5, q, b$ is an arithmetic progression, then the value(s) of $$\left| {q - a} \right|$$ is (are) $\,\,\,\,\,\,\,\,\,\,\,\,$ Column $II$ (p) $\,\,\,\,1$ (q) $\,\,\,\,2$ (r) $\,\,\,\,3$ (s) $\,\,\,\,4$ (t) $\,\,\,\,5$