Let n be the number of ways in which 5 boys and 5 girls can stand in a queue in such a way that all the girls stand consecutively in the queue. Let m be the number of ways in which 5 boys and 5 girls can stand in queue in such a way that exactly four girls stand consecutively in the queue. Then the value of $\frac{\mathrm{m}}{\mathrm{n}}$ is

Let ${\tan }^{-1}\left(x\right)\in \left(-\frac{\pi }{2}, \frac{\pi }{2}\right)$, for $x\in \mathrm{ℝ}$. Then the number of real solutions of the equation $\sqrt{1+\cos \left(2x\right)}=\sqrt{2}{\tan }^{-1}\left(\tan x\right)$ in the set $\left(-\frac{3\pi }{2}, -\frac{\pi }{2}\right)\cup \left(-\frac{\pi }{2}, \frac{\pi }{2}\right)\cup \left(\frac{\pi }{2}, \frac{3\pi }{2}\right)$ is equal to

Let $f :\left(-\frac{\pi }{2},\frac{\pi }{2}\right)\to R$ be given by $f\left(x\right)={\left(\log \left(\sec x+\tan x\right)\right)}^3.$ Then

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Let \(S=S_{1} \cap S_{2} \cap S_{3}\), where

\(S_{1}=\{z \in \mathbb{C}:|z|<4\}, \quad S_{2}=\left\{z \in \mathbb{C}: \operatorname{Im}\left[\frac{z-1+\sqrt{3} i}{1-\sqrt{3} i}\right]>0\right\}\) and \(S_{3}=\{z \in \mathbb{C}: \operatorname{Re} z>0\}\).


Question:

Area of \(S=\)

Let \(\mathbb{R}\) denote the set of all real numbers. Then the area of the region \(\left\{(x, y) \in \mathbb{R} \times \mathbb{R}: x>0, y>\frac{1}{x}, 5 x-4 y-1>0,4 x+4 y-17 <0\right\}\) is

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Read the following passage and answer the questions.
Question:
The value of \(|U|\) is

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Let \(P Q\) be a focal chord of the parabola \(y^{2}=4 a x\). The tangents to the parabola at \(P\) and \(Q\) meet at a point lying on the line \(y=2 x+a, a>0\).


Question:

If chord \(P Q\) subtends an angle \(\theta\) at the vertex of \(y^{2}=4 a x\), then \(\tan \theta=\)

Let \(z_k=\cos \left(\frac{2 k \pi}{10}\right)+i \sin \left(\frac{2 k \pi}{10}\right) ; k=1,2, \ldots, 9\)

List - I	List - II
(A) For each \(z_k\) there exists a \(z_j\) such \(z_k \cdot z_j=1\)	(P) True
(B) There exists a \(k \in\{1,2, \ldots, 9\}\) such that \(z_1 \cdot z=z_k\) has no solution z in the set of complex numbers	(Q) False
(C) \(\frac{\left|1-z_1\right|\left|1-z_2\right|\ldots\left|1-z_9\right|}{10}\) equals	(R) 1
(D) \(1-\sum_{k=1}^9 \cos \left(\frac{2 k \pi}{10}\right)\) equals	(S) 2

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A special metal \(S\) conducts electricity without any resistance. A closed wire loop, made of \(S\), does not allow any change in flux through itself by inducing a suitable current to generate a compensating flux. The induced current in the loop cannot decay due to its zero resistance. This current gives rise to a magnetic moment which in turn repels the source of magnetic field or flux. Consider such a loop, of radius \(a\), with its center at the origin. A magnetic dipole of moment \(m\) is brought along the axis of this loop from infinity to a point at distance \(r(\gg a)\) from the center of the loop with its north pole always facing the loop, as shown in the figure below.

The magnitude of magnetic field of a dipole \(m\), at a point on its axis at distance \(r\), is \(\frac{\mu_{0}}{2 \pi} \frac{m}{r^{3}}\), where \(\mu_{0}\) is the permeability of free space. The magnitude of the force between two magnetic dipoles with moments, \(m_{1}\) and \(m_{2}\), separated by a distance \(r\) on the common axis, with their north poles facing each other, is \(\frac{k m_{1} m_{2}}{r^{4}}\), where \(k\) is a constant of appropriate dimensions. The direction of this force is along the line joining the two dipoles.

Question :
When the dipole m is placed at a distance r from the centre of the loop (as shown in the figure), the current induced in the loop will be proportional to:

If \(f(x)=\min \left\{1, x^2, x^3\right\}\), then

For a real number $\alpha ,$ if the system $\left[\begin{array}{ccc}1& \alpha & {\alpha }^2\\ \alpha & 1& \alpha \\ {\alpha }^2& \alpha & 1\end{array}\right] \left[\begin{array}{c}x\\ y\\ z\end{array}\right]=\left[\begin{array}{c}1\\ -1\\ 1\end{array}\right]$ of linear equations, has infinitely many solutions, then $1+\alpha +{\alpha }^2=$

Consider the planes \(3 x-6 y-2 z=15\) and \(2 x+y-2 z=5\).
Statement I The parametric equations of the line of intersection of the given planes are \(x=3+14 t, y=1+2 t\) and \(z=15 t\).
Statement II The vectors \(14 \hat{\mathbf{i}}+2 \hat{\mathbf{j}}+15 \hat{\mathbf{k}}\) is parallel to the line of intersection of the given planes.

The specific heat capacity of a substance is temperature dependent and is given by the formula \(C=k T\), where \(k\) is a constant of suitable dimensions in SI units, and \(T\) is the absolute temperature. If the heat required to raise the temperature of \(1 \mathrm{~kg}\) of the substance from \(-73^{\circ} \mathrm{C}\) to \(27^{\circ} \mathrm{C}\) is \(n k\), the value of \(n\) is _____
[Given: \(0 \mathrm{~K}=-273^{\circ} \mathrm{C}\).]