Consider the family of all circles whose centers lie on the straight line $\mathrm{y}=\mathrm{x}.$ If this family of circles is represented by the differential equation $\mathrm{P}\mathrm{y}"\mathrm{ }+\mathrm{ }\mathrm{Q}\mathrm{y}\mathrm{\prime }\mathrm{ }+\mathrm{ }1\mathrm{ }=\mathrm{ }0,\mathrm{ }$ where $\mathrm{P},\mathrm{ }\mathrm{Q}$ are functions of $\mathrm{x},\mathrm{ }\mathrm{y}\mathrm{ }\mathrm{a}\mathrm{n}\mathrm{d}\mathrm{ }\mathrm{y}\mathrm{\prime }$ $\left(\mathrm{h}\mathrm{e}\mathrm{r}\mathrm{e}\mathrm{ }{\mathrm{y}}^{\mathrm{\prime }}=\mathrm{ }\frac{\mathrm{d}\mathrm{y}}{\mathrm{d}\mathrm{x}},\mathrm{ }\mathrm{y}"\mathrm{ }=\mathrm{ }\frac{{\mathrm{d}}^2\mathrm{y}}{\mathrm{d}{\mathrm{x}}^2}\right)$ , then which of the following statements is (are) true ?

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Football teams \(T_{1}\) and \(T_{2}\) have to play two games against each other. It is assumed that the outcomes of the two games are independent. The probabilities of \(T_{1}\) winning, drawing and losing a game against \(T_{2}\) are \(\frac{1}{2}, \frac{1}{6}\) and \(\frac{1}{3}\), respectively. Each team gets \(3\) points for a win, \(1\) point for a draw and \(0\) point for a loss in a game. Let \(X\) and \(Y\) denote the total points scored by teams \(T_{1}\) and \(T_{2}\), respectively, after two games.


Question:

\(P(X>Y)\) is

Let $f:\left[0, 1\right]\to \left[0, 1\right]$ be the function defined by $f\left(x\right)=\frac{x^3}{3}-x^2+\frac{5}{9}x+\frac{17}{36}$. Consider the square region $S=\left[0, 1\right]\times \left[0, 1\right]$. Let $G=\left\{\left(x, y\right)\in S:y>f\left(x\right)\right\}$ be called the green region and $R=\left\{\left(x, y\right)\in S:y<f\left(x\right)\right\}$ be called the red region. Let $L_h=\left\{\left(x, h\right)\in S:x\in \left[0, 1\right]\right\}$ be the horizontal line drawn at a height $h\in \left[0, 1\right]$. Then which of the following statements is(are) true?

For every twice differentiable function $f:\mathbb{R}\to \left[-2, 2\right] \mathrm{w}\mathrm{i}\mathrm{t}\mathrm{h} {\left(f\left(0\right)\right)}^2+{\left(f^{\prime }\left(0\right)\right)}^2=85$ , which of the following statement(s) is (are) TRUE?

Let \(p(x)\) be a polynomial of degree 4 having extremum at \(x=1,2\) and \(\lim _{x \rightarrow 0}\left[1+\frac{p(x)}{x^2}\right]=2\). Then, the value of \(p(2)\) is

Let \(f:[-1,2] \rightarrow[0, \infty)\) be a continuous function such that \(f(x)=f(1-x)\) for all \(x \in[-1,2]\). Let \(R_1=\int_{-1}^2 x f(x) d x\) and \(R_2\) be the area of the region bounded by \(y=f(x), x=-1, x=2\) and the \(X\)-axis. Then,

If $M=\left(\begin{array}{cc}\frac{5}{2}& \frac{3}{2}\\ -\frac{3}{2}& -\frac{1}{2}\end{array}\right)$, then which of the following matrices is equal to $M^{2022}$ ?

For $x\in \mathrm{ℝ}$, let ${\tan }^{-1}\left(x\right)\in \left(-\frac{\pi }{2},\frac{\pi }{2}\right)$. Then the minimum value of the function $f:\mathrm{ℝ}\to \mathrm{ℝ}$ defined by $f\left(x\right)={\int }_0^{x{\tan }^{-1}x}\frac{e^{\left(t-\cos t\right)}}{1+t^{2023}}dt$ is

Consider the system of equations \(x-2 y+3 z=-1, x-3 y+4 z=1\) and \(-x+y-2 z=k\)
Statement 1 The system of equations has no solution for \(k \neq 3\).
Statement 2 The determinant \(\left|\begin{array}{ccc}1 & 3 & -1 \\ -1 & -2 & k \\ 1 & 4 & 1\end{array}\right| \neq 0\), for \(k \neq 3\).

Consider the lines \(L_{1}\) and \(L_{2}\) defined by
\[
L_{1}: x \sqrt{2}+y-1=0 \text { and } L_{2}: x \sqrt{2}-y+1=0
\]
For a fixed constant \(\lambda\), let \(C\) be the locus of a point \(P\) such that the product of the distance of \(P\) from \(L_{1}\) and the distance of \(P\) from \(L_{2}\) is \(\lambda^{2}\). The line \(y=2 x+1\) meets \(C\) at two points \(R\) and \(S\), where the distance between \(R\) and \(S\) is \(\sqrt{270}\).
Let the perpendicular bisector of \(R S\) meet \(C\) at two distinct points \(R^{\prime}\) and \(S^{\prime}\). Let \(D\) be the square of the distance between \(R^{\prime}\) and \(S^{\prime}\).
The value of ${\lambda }^2$ is

In an insulated vessel, \(0.05 \mathrm{~kg}\) steam at \(373 \mathrm{~K}\) and \(0.45 \mathrm{~kg}\) of ice at \(253 \mathrm{~K}\) are mixed. Find the final temperature of the mixture (in kelvin).
\( \text { Given, }L_{\text {fusion }} =80 \mathrm{cal} / \mathrm{g}=336 \mathrm{~J} / \mathrm{g},\) \( L_{\text {vaporization }}=540 \mathrm{cal} / \mathrm{g}=2268 \mathrm{~J} / \mathrm{g}, \)
\( S_{\text {ice }} =2100 \mathrm{~J} / \mathrm{kg}, K=0.5 \mathrm{cal} / \mathrm{gK} \text { and } S_{\text {water }}=\) \(4200 \mathrm{~J} / \mathrm{kg}, K=1 \mathrm{cal} / \mathrm{gK} .\)

Let \(f_{1}:(0, \infty) \rightarrow \mathbb{R}\) and \(f_{2}:(0, \infty) \rightarrow \mathbb{R}\) be defined by

\[

f_{1}(x)=\int_{0}^{x} \prod_{j=1}^{21}(t-j)^{j} d t, \quad x>0

\]

and

\(

f_{2}(x)=98(x-1)^{50}-600(x-1)^{49}+2450, \quad x>0

\)

where, for any positive integer \(n\) and real numbers \(a_{1}, a_{2}, \ldots, a_{n}, \prod_{i=1}^{n} a_{i}\) denotes the product of \(a_{1}, a_{2}, \ldots, a_{n} .\) Let \(m_{i}\) and \(n_{i}\), respectively, denote the number of points of local minima and the number of points of local maxima of function \(f_{i}, i=1,2\), in the interval \((0, \infty)\).

The value of $2m_1+3n_1+m_1{n}_1$ is _____.

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