Match the following :

Column I	Column II
(A) In $ \mathbb{R}^2 $, if the magnitude of the projection vector of the vector $ \alpha \hat{i} + \beta \hat{j} $ on $ \sqrt{3}\hat{i} + \hat{j} $ is $ \sqrt{3} $ and if $ \alpha = 2 + \sqrt{3}\beta $, then possible value(s) of $ |\alpha| $ is (are)	$(P)\ 1$
(B) Let $ \alpha $ and $ b $ be real numbers such that the function $ f(x)= \begin{cases} -3\alpha x^2-2, & x<1 \\[4pt] bx+\alpha^2, & x\ge 1 \end{cases} $ is differentiable for all $ x \in \mathbb{R} $. Then possible value(s) of $ \alpha $ is (are)	$(Q)\ 2$
(C) Let $ \omega \ne 1 $ be a complex cube root of unity. If $ (3-3\omega+2\omega^2)^{4n+3} +(2+3\omega-3\omega^2)^{4n+3} +(-3+2\omega+3\omega^2)^{4n+3}=0, $ then possible value(s) of $ n $ is (are)	$(R)\ 3$
(D) Let the harmonic mean of two positive real numbers $ a $ and $ b $ be $ 4 $. If $ q $ is a positive real number such that $ a,\ 5,\ q,\ b $ is an arithmetic progression, then the value(s) of $ |q-a| $ is (are)	$(S)\ 4$
	$(T)\ 5$

match List $$I$$ with List $$II$$ and select the correct answer using the code given below the lists:

$$\,\,\,\,$$ $$\,\,\,\,$$ $$\,\,\,\,$$ List $$I$$
(P.)$$\,\,\,\,$$ Volume of parallelopiped determined by vectors $$\overrightarrow a ,\overrightarrow b $$ and $$\overrightarrow c $$ is $$2.$$ Then the volume of the parallelepiped determined by vectors $$2\left( {\overrightarrow a \times \overrightarrow b } \right),3\left( {\overrightarrow b \times \overrightarrow c } \right)$$ and $$\left( {\overrightarrow c \times \overrightarrow a } \right)$$ is
(Q.)$$\,\,\,\,$$ Volume of parallelopiped determined by vectors $$\overrightarrow a ,\overrightarrow b $$ and $$\overrightarrow c $$ is $$5.$$ Then the volume of the parallelepiped determined by vectors $$3\left( {\overrightarrow a + \overrightarrow b } \right),\left( {\overrightarrow b + \overrightarrow c } \right)$$ and $$2\left( {\overrightarrow c + \overrightarrow a } \right)$$ is
(R.)$$\,\,\,\,$$ Area of a triangle with adjacent sides determined by vectors $${\overrightarrow a }$$ and $${\overrightarrow b }$$ is $$20.$$ Then the area of the triangle with adjacent sides determined by vectors $$\left( {2\overrightarrow a + 3\overrightarrow b } \right)$$ and $$\left( {\overrightarrow a - \overrightarrow b } \right)$$ is
(S.)$$\,\,\,\,$$ Area of a parallelogram with adjacent sides determined by vectors $${\overrightarrow a }$$ and $${\overrightarrow b }$$ is $$30.$$ Then the area of the parallelogram with adjacent sides determined by vectors $$\left( {\overrightarrow a + \overrightarrow b } \right)$$ and $${\overrightarrow a }$$ is

$$\,\,\,\,$$ $$\,\,\,\,$$ $$\,\,\,\,$$ List $$II$$
(1.)$$\,\,\,\,$$ $$100$$
(2.)$$\,\,\,\,$$ $$30$$
(3.)$$\,\,\,\,$$ $$24$$
(4.)$$\,\,\,\,$$ $$60$$

Let $\overrightarrow{\mathrm{PR}}=3 \hat{i}+\hat{j}-2 \hat{k}$ and $ \overrightarrow{\mathrm{SQ}}=\hat{i}-3 \hat{j}-4 \hat{k}$ determine diagonals of a parallelogram $P Q R S$ and $\overrightarrow{\mathrm{PT}}=\hat{i}+2 \hat{j}+3 \hat{k}$ be another vector. Then the volume of the parallelopiped determined by the vectors $\overrightarrow{\mathrm{PT}}, \overrightarrow{\mathrm{PQ}}$ and $\overrightarrow{\mathrm{PS}}$ is :

If $$\overrightarrow a $$ and $$\overrightarrow b $$ are vectors such that $$\left| {\overrightarrow a + \overrightarrow b } \right| = \sqrt {29} $$ and $$\,\overrightarrow a \times \left( {2\widehat i + 3\widehat j + 4\widehat k} \right) = \left( {2\widehat i + 3\widehat j + 4\widehat k} \right) \times \widehat b,$$ then a possible value of $$\left( {\overrightarrow a + \overrightarrow b } \right).\left( { - 7\widehat i + 2\widehat j + 3\widehat k} \right)$$ is