$${5 \over 4}{\cos ^2}\,2x + {\cos ^4}\,x + {\sin ^4}\,x + {\cos ^6}\,x + {\sin ^6}\,x\, = \,2$$
in the interval $$\left[ {0,\,2\pi } \right]$$ is
$$\,\,\,\,$$ $$\,\,\,\,$$ $$\,\,\,\,$$ 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 $$f\left( x \right) = \left\{ {\matrix{
{ - 3a{x^2} - 2,} & {x < 1} \cr
{bx + {a^2},} & {x \ge 1} \cr
} } \right.$$ if differentiable for all $$x \in R$$. Then possible value of $$a$$ is (are)
(C)$$\,\,\,\,$$ Let $$\omega \ne 1$$ 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)$$\,\,\,\,$$ $$5$$