The minimum number of times a fair coin needs to be tossed, so that the probability of getting at least two heads is at least $$0.96,$$ is

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)	$(S)\ 4$
	$(T)\ 5$

Let $$\Delta PQR$$ be a triangle. Let $$\vec a = \overrightarrow {QR} ,\vec b = \overrightarrow {RP} $$ and $$\overrightarrow c = \overrightarrow {PQ} .$$ If $$\left| {\overrightarrow a } \right| = 12,\,\,\left| {\overrightarrow b } \right| = 4\sqrt 3 ,\,\,\,\overrightarrow b .\overrightarrow c = 24,$$ then which of the following is (are) true?

Match the following :

	Column I		Column I
(A)	$\begin{array}{l}\text { In a triangle } \Delta X Y Z \text {, let } a, b \text { and } c \text { be the lengths of the sides } \\\text { opposite to the angles } X, Y \text { and } Z \text {, respectively. If } 2\left(a^2-b^2\right)=c^2 \\\text { and } \lambda=\frac{\sin (X-Y)}{\sin Z} \text {, then possible values of } n \text { for which } \cos (n \lambda) \\=0 \text { is (are) }\end{array}$	(P)	1
(B)	$\begin{array}{l}\text { In a triangle } \triangle X Y Z \text {, let } a, b \text { and } c \text { be the lengths of the sides } \\\text { opposite to the angles } X, Y \text { and } Z \text {, respectively. If } 1+\cos 2 X-2 \\\cos 2 Y=2 \sin X \sin Y \text {, then possible value(s) of } \frac{a}{b} \text { is (are) }\end{array}$	(Q)	2
(C)	$\begin{array}{l}\text { In } \mathbb{R}^2 \text {, let } \sqrt{3} \hat{i}+\hat{j}, \hat{i}+\sqrt{3} \hat{j} \text { and } \beta \hat{i}+(1-\beta) \hat{j} \text { be the position } \\\text { vectors of } X, Y \text { and } Z \text { with respect of the origin } \mathrm{O} \text {, respectively. If } \\\text { the distance of } \mathrm{Z} \text { from the bisector of the acute angle of } \overrightarrow{\mathrm{OX}} \text { with } \\\overrightarrow{\mathrm{OY}} \text { is } \frac{3}{\sqrt{2}} \text {, then possible value(s) of }|\beta| \text { is (are) }\end{array}$	(R)	3
(D)	$\begin{array}{l}\text { Suppose that } F(\alpha) \text { denotes the area of the region bounded by } \\x=0, x=2, y^2=4 x \text { and } y=|\alpha x-1|+|\alpha x-2|+\alpha x \text {, } \\\text { where, } \alpha \in\{0,1\} \text {. Then the value(s) of } F(\alpha)+\frac{8}{2} \sqrt{2} \text {, when } \alpha=0 \\\text { and } \alpha=1 \text {, is (are) }\end{array}$	(S)	5
		(T)	6