Let $f_1: \mathbb{R}\to \mathbb{R}, f_2\left(-\frac{\pi }{2},\frac{\pi }{2} \right)\to \mathbb{R}, f_3:\left(-1, e^{\frac{\pi }{2}}-2\right)\to \mathbb{R}$ and $f_4: \mathrm{ℝ}\to \mathrm{ℝ}$ be functions defined by
(i) $f_1\left(x\right)=\sin \left(\sqrt{1-e^{-x^2}}\right)$
(ii) $f_2\left(x\right)=\left\{\begin{array}{ll}\frac{\left|\sin x\right|}{{\tan }^{-1}\left(x\right)}& , x\ne 0\\ 1& , x=0\end{array}\right.$ , where the inverse trigonometric function ${\tan }^{-1}x$ assumes values in $\left(-\frac{\pi }{2},\frac{\pi }{2}\right)$
(iii) $f_3\left(x\right)=\left[\sin \left({\log }_{\text{e}}\left(x+2\right)\right)\right]$ , where, for $t\in R, \left[t\right]$ denotes the greatest integer less than or equal to $t$ ,
(iv) $f_4\left(x\right)=\left\{\begin{array}{cc}{x}^2\sin \left(\frac{1}{x}\right) & , x\ne 0\\ 0 & , x=0\end{array}\right.$

LIST-I	LIST-II
A. the function $f_1$ is	P. NOT continuous at $x=0$
B. The function $f_2$ is	Q. continuous at $x = 0$ and NOT differentiable at $x=0$
C. The function $f_3$ is	R. differentiable at $x=0$ and its derivative is NOT continuous at $x=0$
D. The function $f_4$ is	S. differentiable at $x=0$ and its derivative is continuous at $x=0$

The correct option is :

Let $P_1=I=\left[\begin{array}{ccc}1& 0& 0\\ 0& 1& 0\\ 0& 0& 1\end{array}\right], P_2=\left[\begin{array}{ccc}1& 0& 0\\ 0& 0& 1\\ 0& 1& 0\end{array}\right], P_3=\left[\begin{array}{ccc}0& 1& 0\\ 1& 0& 0\\ 0& 0& 1\end{array}\right],$$P_4=\left[\begin{array}{ccc}0& 1& 0\\ 0& 0& 1\\ 1& 0& 0\end{array}\right], P_5=\left[\begin{array}{ccc}0& 0& 1\\ 1& 0& 0\\ 0& 1& 0\end{array}\right],P_6=\left[\begin{array}{ccc}0& 0& 1\\ 0& 1& 0\\ 1& 0& 0\end{array}\right],$ and
$X=\sum _{k=1}^6{P}_K\left[\begin{array}{ccc}2& 1& 3\\ 1& 0& 2\\ 3& 2& 1\end{array}\right]P_K^T$
where $P_K^T$ denotes the transpose of the matrix $P_K.$ Then which of the following options is/are correct?

A vertical line passing through the point $\left(\text{h},\text{0}\right)$ intersects the ellipse $\frac{{\text{x}}^2}{4}+\frac{{\text{y}}^2}{3}=1$ at the points $\text{P}$ and $\text{Q}$. Let the tangents to the ellipse at $\text{P}$ and $\text{Q}$ meet at the point $\text{R}$. If $\Delta \left(\mathrm{h}\right)=$ area of triangle $\mathrm{PQR},{\Delta }_1=\underset{\frac{1}{2}\le \mathrm{h}\le 1}{\max }\Delta \left(\mathrm{h}\right)$ and ${\mathrm{\Delta }}_2=\underset{\frac{1}{2}\le \mathrm{h}\le 1}{\min }\Delta \left(\mathrm{h}\right)$, then $\frac{8}{\sqrt{5}}{\Delta }_1-8{\Delta }_2=$

Consider all possible permutations of the letters of the word ENDEANOEL. Match the Statements/Expressions in Column I with the Statements/Expressions in Column II.

Column I	Column II
(A)The number of permutations containing the word ENDEA, is	(p) \(5 !\)
(B) The number of permutations in which the letter E occurs in the first and the last positions, is	(q) \(2 \times 5!\)
(C) The number of permutations in which none of the letters D, L, N occurs in the last five positions, is	(r) \(7 \times 5!\)
(D) The number of permutations in which the letters A, E, O occur only in odd positions, is	(s) \(21 \times 5!\)

Let \(z=\cos \theta+i \sin \theta\). Then, the value of \(\sum_{m=1}^{15} \operatorname{Im}\left(z^{2 m-1}\right)\) at \(\theta=2^{\circ}\) is

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

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Box \(1\) contains three cards bearing numbers \(1,2,3\); box \(2\) contains five cards bearing numbers \(1,2,3,4,5\); and box \(3\) contains seven cards bearing numbers \(1,2,3,4,5,6,7\). A card is drawn from each of the boxes. Let \(x_{i}\) be the number on the card drawn from the \(i^{t h}\) box, \(i=1,2,3\).

Question:
The probability that \(x_{1}, x_{2}, x_{3}\) are in an arithmetic progression, is

A block of mass \(2 \mathrm{~kg}\) is free to move along the \(x\)-axis. It is at rest and from \(t\) \(=0\) onwards it is subjected to a time-dependent force \(F(t)\) in the \(x\) direction. The force \(F(t)\) varies with \(t\) as shown in the figure. The kinetic energy of the block after \(4.5 \mathrm{~s}\) is

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When a particle is restricted to move along \(x\)-axis between \(x=0\) and \(x=a\), where \(a\) is of nanometer dimension, its energy can take only certain specific values. The allowed energies of the particle moving in such a restricted region, correspond to the formation of standing waves with nodes at its ends \(x=0\) and \(x=a\). The wavelength of this standing wave is related to the liner momentum \(p\) of the particle according to the de Broglie relation. The energy of the particle of mass \(m\) is related to its linear momentum as \(E=\frac{p^2}{2 m}\). Thus, the energy of the particle can be denoted by a quantum number \(n\) taking values \(1,2,3, \ldots(n=1\), called the ground state) corresponding to the number of loops in the standing wave.
Use the model described above to answer the following three questions for a particle moving in the line \(x=0\) to \(x=a\) [Take \(h=6.6 \times 10^{-34} \mathrm{Js}\) and \(e=1.6 \times 10^{-19} \mathrm{C}\) ]
Question:
The speed of the particle that can take discrete values is proportional to

The cubic unit cell structure of a compound containing cation $\mathrm{M}$ and anion $\mathrm{X}$ is shown below. When compared to the anion, the cation has smaller ionic radius. Choose the correct statement(s).

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Let \(F_{1}\left(x_{1}, 0\right)\) and \(F_{2}\left(x_{2}, 0\right)\), for \(x_{1}<0\) and \(x_{2}>0\), be the foci of the ellipse \(\frac{x^{2}}{9}+\frac{y^{2}}{8}=1 .\) Suppose a parabola having vertex at the origin and focus at \(F_{2}\) intersects the ellipse at point \(M\) in the first quadrant and at point \(N\) in the fourth quadrant.


Question:

The orthocentre of the triangle \(F_{1} M N\) is

In a scattering experiment, a particle of mass \(2 m\) collides with another particle of mass \(m\), which is initially at rest. Assuming the collision to be perfectly elastic, the maximum angular deviation \(\theta\) of the heavier particle, as shown in the figure, in radians is:

Let $\alpha , \beta$ and $\gamma$ be real numbers. Consider the following system of linear equations $x+2y+z=7$ $x+\alpha z=11$ $2x-3y+\beta z=\gamma$ Match each entry in List-I to the correct entries in List-II.

	List-I		List-II
$\left(\mathrm{P}\right)$	If $\beta =\frac{1}{2}\left(7\alpha -3\right)$ and $\gamma =28$ then the system has	$\left(1\right)$	a unique solution
$\left(\mathrm{Q}\right)$	If $\beta =\frac{1}{2}\left(7\alpha -3\right)$ and $\gamma \ne 28$, then the system has	$\left(2\right)$	no solution
$\left(\mathrm{R}\right)$	If $\beta \ne \frac{1}{2}\left(7\alpha -3\right)$ where $\alpha =1$ and $\gamma \ne 28$, then the system has	$\left(3\right)$	infinitely many solutions
$\left(\mathrm{S}\right)$	If $\beta \ne \frac{1}{2}\left(7\alpha -3\right)$ where $\alpha =1$ and $\gamma =28$, then the system has	$\left(4\right)$	$x=11, y=-2$ and $z=0$ as a solution
		$\left(5\right)$	$x=-15, y=4$ and $z=0$ as a solution

The correct option is: