In a high school, a committee has to be formed from a group of 6 boys M₁, M₂, M₃, M₄, M₅, M₆ and 5 girls G₁, G₂, G₃, G₄, G₅.

(i) Let $$\alpha $$₁ be the total number of ways in which the committee can be formed such that the committee has 5 members, having exactly 3 boys and 2 girls.

(ii) Let $$\alpha $$₂ be the total number of ways in which the committee can be formed such that the committee has at least 2 members, and having an equal number of boys and girls.

i) Let $$\alpha $$₃ be the total number of ways in which the committee can be formed such that the committee has 5 members, at least 2 of them being girls.

(iv) Let $$\alpha $$₄ be the total number of ways in which the committee can be formed such that the committee has 4 members, having at least 2 girls such that both M₁ and G₁ are NOT in the committee together.

LIST-I	LIST-II
P. The value of $\alpha_1$ is	1. 136
Q. The value of $\alpha_2$ is	2. 189
R. The value of $\alpha_3$ is	3. 192
S. The value of $\alpha_4$ is	4. 200
	5. 381
	6. 461

The correct option is

Let $$H:{{{x^2}} \over {{a^2}}} - {{{y^2}} \over {{b^2}}} = 1$$, where a > b > 0, be a hyperbola in the XY-plane whose conjugate axis LM subtends an angle of 60$$^\circ $$ at one of its vertices N. Let the area of the $$\Delta $$LMN be $$4\sqrt 3 $$.

	List - I		List - II
P.	The length of the conjugate axis of H is	1.	8
Q.	The eccentricity of H is	2.	$${4 \over {\sqrt 3 }}$$
R.	The distance between the foci of H is	3.	$${2 \over {\sqrt 3 }}$$
S.	The length of the latus rectum of H is	4.	4

Let $${f_1}:R \to R,\,{f_2}:\left( { - {\pi \over 2},{\pi \over 2}} \right) \to R,\,{f_3}:( - 1,{e^{\pi /2}} - 2) \to R$$ and $${f_4}:R \to R$$ be functions defined by

(i) $${f_1}(x) = \sin (\sqrt {1 - {e^{ - {x^2}}}} )$$,

(ii) $${f_2}(x) = \left\{ \matrix{ {{|\sin x|} \over {\tan { - ^1}x}}if\,x \ne 0,\,where \hfill \cr 1\,if\,x = 0 \hfill \cr} \right.$$

the inverse trigonometric function tan^$$-$$1x assumes values in $$\left( { - {\pi \over 2},{\pi \over 2}} \right)$$,

(iii) $${f_3}(x) = [\sin ({\log _e}(x + 2))]$$, where for $$t \in R,\,[t]$$ denotes the greatest integer less than or equal to t,

(iv) $${f_4}(x) = \left\{ \matrix{ {x^2}\sin \left( {{1 \over x}} \right)\,if\,x \ne 0 \hfill \cr 0\,if\,x = 0 \hfill \cr} \right.$$

LIST-I	LIST-II
P. The function $$ f_1 $$ is	1. NOT continuous at $$ x = 0 $$
Q. The function $$ f_2 $$ is	2. continuous at $$ x = 0 $$ and NOT differentiable at $$ x = 0 $$
R. The function $$ f_3 $$ is	3. differentiable at $$ x = 0 $$ and its derivative is NOT continuous at $$ x = 0 $$
S. The function $$ f_4 $$ is	4. differentiable at $$ x = 0 $$ and its derivative is continuous at $$ x = 0 $$

A particle, of mass $${10^{ - 3}}$$ $$kg$$ and charge $$1.0$$ $$C,$$ is initially at rest. At time $$t=0,$$ the particle comes under the influence of an electric field $$\overrightarrow E \left( t \right) = {E_0}\sin \,\,$$ $$\omega t\widehat i,$$ where $${E_0} = 1.0\,N{C^{ - 1}}$$ and $$\omega = 10{}^3\,rad\,{s^{ - 1}}.$$ Consider the effect of only the electrical force on the particle. Then the maximum speed, in $$m{s^{ - 1}},$$ attained by the particle at subsequent times is _______________.