Let $$X = {({}^{10}{C_1})^2} + 2{({}^{10}{C_2})^2} + 3{({}^{10}{C_3})^2} + ... + 10{({}^{10}{C_{10}})^2}$$,

where $${}^{10}{C_r}$$, r $$ \in $${1, 2, ..., 10} denote binomial coefficients. Then, the value of $${1 \over {1430}}X$$ is ..........

Let $${E_1} = \left\{ {x \in R:x \ne 1\,and\,{x \over {x - 1}} > 0} \right\}$$ and


$${E_2} = \left\{ \matrix{ x \in {E_1}:{\sin ^{ - 1}}\left( {{{\log }_e}\left( {{x \over {x - 1}}} \right)} \right) \hfill \cr is\,a\,real\,number \hfill \cr} \right\}$$

(Here, the inverse trigonometric function $${\sin ^{ - 1}}$$ x assumes values in $$\left[ { - {\pi \over 2},{\pi \over 2}} \right]$$.).

Let f : E₁ $$ \to $$ R be the function defined by f(x) = $${{{\log }_e}\left( {{x \over {x - 1}}} \right)}$$ and g : E₂ $$ \to $$ R be the function defined by g(x) = $${\sin ^{ - 1}}\left( {{{\log }_e}\left( {{x \over {x - 1}}} \right)} \right)$$.

LIST-I	LIST-II
P. The range of $f$ is	1. $\left( -\infty, \frac{1}{1-e} \right] \cup \left[ \frac{e}{e-1}, \infty \right)$
Q. The range of $g$ contains	2. $(0, 1)$
R. The domain of $f$ contains	3. $\left[ -\frac{1}{2}, \frac{1}{2} \right]$
S. The domain of $g$ is	4. $(-\infty, 0) \cup (0, \infty)$
	5. $\left( -\infty, \frac{e}{e-1} \right)$
	6. $(-\infty, 0) \cup \left( \frac{1}{2}, \frac{e}{e-1} \right]$

The correct option is :

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