Columns 1, 2 and 3 contain conics, equations of tangents to the conics and points of contact, respectively.

Column 1	Column 2	Column 3
(I) $x^2+y^2=a^2$	(i) $my=m^{2}x+a$	(P) $\left(\frac{a}{m^2},\frac{2a}{m}\right)$
(II) $x^2+a^2{y}^2=a^2$	(ii) $y=mx+a \sqrt{m^2+1}$	(Q) $\left(\frac{-ma}{\sqrt{m^2+1}},\frac{a}{\sqrt{m^2+1}}\right)$
(III) $y^2=4ax$	(iii) $y=mx+ \sqrt{a^2{m}^2-1}$	(R) $\left(\frac{-a^{2}m}{\sqrt{a^2{m}^2+1}},\frac{1}{\sqrt{a^2{m}^2+1}}\right)$
(IV) $x^2-a^2{y}^2=a^2$	(iv) $y=mx+\sqrt{a^2{m}^2+1}$	(S) $\left(\frac{-a^{2}m}{\sqrt{a^2{m}^2-1}},\frac{-1}{\sqrt{a^2{m}^2-1}}\right)$

The tangent to a suitable conic (Column 1) at $\left(\sqrt{3},\frac{1}{2}\right)$ is found to be $\sqrt{3}x+2y=4$ , then which of the following options is the only Correct combination?

Consider the following frequency distribution :

Value	4	5	8	9	6	12	11
Frequency	5	\(f_1\)	\(f_2\)	2	1	1	3

Suppose that the sum of the frequencies is 19 and the median of this frequency distribution is 6. For the given frequency distribution, let \(\alpha\) denote the mean deviation about the mean, \(\beta\) denote the mean deviation about the median, and \(\sigma^2\) denote the variance.
Match each entry in List-I to the correct entry in List-II and choose the correct option.
\(\begin{array}{|l|l|l|l|} \hline & \text{LIST - I} & & \text{LIST - II} \\ \hline (P) & 7 f_1+9 f_2 \text{ is equal to} & (1) & 146 \\ \hline (Q) & 19 \alpha \text{ is equal to} & (2) & 47 \\ \hline (R) & 19 \beta \text{ is equal to} & (3) & 48 \\ \hline (S) & 19 \sigma^2 \text{ is equal to} & (4) & 145 \\ \hline & & (5) & 55 \\ \hline \end{array}\)

For any real number $x$, let $\left[x\right]$ denotes the largest integer less than or equal to $x$. If $I={\int }_0^{10}\left[\sqrt{\frac{10x}{x+1}}\right]dx$, then the value of $9I$ is ____.

Paragraph:
Consider the circle \(x^2+y^2=9\) and the parabola \(y^2=8 x\). They intersect at \(P\) and \(Q\) in the first and the fourth quadrants, respectively. Tangents to the circle at \(P\) and \(Q\) intersect the \(\mathrm{X}\)-axis at \(R\) and tangents to the parabola at \(P\) and \(Q\) intersect the \(\mathrm{X}\)-axis at \(S\).Question:
The ratio of the areas of the \(\triangle P Q S\) and \(\triangle P Q R\) is

Let $\text{z}=\frac{-1+\sqrt{3}\text{i}}{2}$ , where $\text{i}=\sqrt{-1}$ , and $\text{r},\text{s}\in \left\{1,2,3\right\}.$ Let $\text{P}=\left[\begin{array}{cc}{\left(-\mathrm{z}\right)}^{\text{r}}& {\text{z}}^{2\text{s}}\\ {\text{z}}^{2\text{s}}& {\text{z}}^{\text{r}}\end{array}\right]$ and $\text{I}$ be the identity matrix of order 2. Then the total number of ordered pairs $(\text{r},\text{s})$ for which ${\text{P}}^2=-\text{I}$ is

Let $T_1$ and $T_2$ be two distinct common tangents to the ellipse $E:\frac{x^2}{6}+\frac{y^2}{3}=1$ and the parabola $P:y^2=12x$. Suppose that the tangent $T_1$ touches $P$ and $E$ at the points $A_1$ and $A_2$, respectively and the tangent $T_2$ touches $P$ and $E$ at the points $A_4$ and $A_3$, respectively. Then which of the following statements is(are) true?

Let \(E^c\) denotes the complement of an event \(E\). Let \(E, F, G\) be pairwise independent events with \(P(G)>0\) and \(P(E \cap F \cap G)=0\). Then, \(P\left(E^c \cap F^c \mid G\right)\) equals

For $x\in \mathrm{ℝ}$, let $y\left(x\right)$ be a solution of the differential equation $\left(x^2-5\right)\frac{dy}{dx}-2xy=-2x{\left(x^2-5\right)}^2$ such that $y\left(2\right)=7$. Then the maximum value of the function $y\left(x\right)$ is

Let $E_1=${$x\in \mathrm{ℝ}: x\ne 1$ and $\frac{x}{x-1}>0$} and $E_2=${$x\in E_1: {\sin }^{-1}\left({\log }_e\left(\frac{x}{x-1}\right)\right)$ is a real number} (Here the inverse trigonometric function ${\sin }^{-1}\left(x\right)$ assumes values in $\left[-\frac{\mathrm{\pi }}{2}. \frac{\mathrm{\pi }}{2}\right]$.)
Let $f:E_1\to \mathrm{ℝ}$ be the function defined by $f\left(x\right)={\log }_e\left(\frac{x}{x-1}\right)$
And $g:E_2\to \mathbb{R}$ be the function defined by $g\left(x\right)={\sin }^{-1}\left({\log }_e\left(\frac{x}{x-1}\right)\right)$ .

LIST-I	LIST-II
A. The range of $f$ is	P. $\left(-\infty ,\frac{1}{1-e}\right]\cup \left[\frac{e}{e-1}, \infty \right)$
B. The range of $g$ contains	Q. (0, 1)
C. The domain of $f$ contains	R. $\left[-\frac{1}{2},\frac{1}{2}\right]$
D. The domain of $g$ is	S. $\left(-\infty ,0\right)\cup \left(0, \infty \right)$
	T. $\left(-\infty ,\frac{e}{e-1}\right]$
	U. $\left(-\infty ,0\right)\cup \left(\frac{1}{2},\frac{e}{e-1}\right]$

The correct option is:

Let P be a matrix of order $3\times 3$ such that all the entries in P are from the set $\left\{-1, \mathrm{0,} 1\right\}$ . Then, the maximum possible value of the determinant of P is _______.

If $y=y\left(x\right)$ satisfies the differential equation $8\sqrt{x}\left(\sqrt{9+\sqrt{x}}\right)dy={\left(\sqrt{4+\sqrt{9+\sqrt{x}}}\right)}^{-1}dx$ and $x>0, y\left(0\right)=\sqrt{7},$ then $y\left(256\right)=$

Let \(a_0, a_1, \ldots, a_{23}\) be real numbers such that
\(\left(1+\frac{2}{5} x\right)^{23}=\sum_{i=0}^{23} a_i x^i\)
for every real number \(x\). let \(a_r\) be the largest among the numbers \(a_j\) for \(0 \leq j \leq 23\).
The the value of \(r\) is ______

Match the following

Column I	Column II
(A) \(CH _3- CHBr - CD _3\) on treatment with alc. KOH gives \(CH _2= CH - CD _3\) as a major product.	(P) E1 reaction
(B) \(Ph - CHBr - CH _3\) reacts faster than \(Ph - CHBr - CD _3\)	(Q) E2 reaction
(C) \(Ph - CH _2- CH _2 Br\) on treatment with \(C _2 H _5 OD / C _2 H _5 O ^{-}\)gives \(Ph - CD = CH _2\) as the major product	(R) El cb reaction
(D) \(PhCH _2 CH _2 Br\) and \(PhCH _2 CH _2 Br\) react with same rate	(S) First order reaction