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

The value of \(\frac{(5050) \int_0^1\left(1-x^{50}\right)^{100} d x}{\int_0^1\left(1-x^{50}\right)^{101} d x}\)

For each positive integer $n$, let $y_n={\frac{1}{n}\left((n+1)(n+2)\mathrm{...}(n+n)\right)}^{\frac{1}{n}}$. If $\underset{n\to \infty }{\lim }{y}_n=L,$ then the value of $\left[L\right]$ (where $\left[x\right]$ is the greatest integer less than or equal to $x$) is ____

Internal bisector of \(\angle A\) of \(\triangle A B C\) meets side \(B C\) at \(D\). A line drawn through \(D\) perpendicular to \(A D\) intersects the side \(A C\) at \(E\) and side \(A B\) at \(F\). If \(a, b\) and \(c\) represent sides of \(\triangle A B C\), then

The total number of real solutions of the equation \(\theta=\tan ^{-1}(2 \tan \theta)-\frac{1}{2} \sin ^{-1}\left(\frac{6 \tan \theta}{9+\tan ^2 \theta}\right)\) is (Here, the inverse trigonometric functions \(\sin ^{-1} x\) and \(\tan ^{-1} x\) assume values in \(\left[-\frac{\pi}{2}, \frac{\pi}{2}\right]\) and \(\left(-\frac{\pi}{2}, \frac{\pi}{2}\right)\), respectively.)

Consider the following lists:

	List-$\mathrm{I}$		List-$\mathrm{II}$
$\left(\mathrm{I}\right)$	$\left\{x\in \left[-\frac{2\pi }{3},\frac{2\pi }{3}\right]:\cos x+\sin x=1\right\}$	$\left(\mathrm{P}\right)$	has two elements
$\left(\mathrm{II}\right)$	$\left\{x\in \left[-\frac{5\pi }{18},\frac{5\pi }{18}\right]:\sqrt{3}\tan 3x=1\right\}$	$\left(\mathrm{Q}\right)$	has three elements
$\left(\mathrm{III}\right)$	$\left\{x\in \left[-\frac{6\pi }{5},\frac{6\pi }{5}\right]:2\cos \left(2x\right)=\sqrt{3}\right\}$	$\left(\mathrm{R}\right)$	has four elements
$\left(\mathrm{IV}\right)$	$\left\{x\in \left[-\frac{7\pi }{4},\frac{7\pi }{4}\right]:\sin x-\cos x=1\right\}$	$\left(\mathrm{S}\right)$	has five elements
		$\left(\mathrm{T}\right)$	has six elements

The correct option is:

Let P be the point on the parabola $y^2=4x$ which is at the shortest distance from the center S of the circle $x^2+y^2-4x-16y+64=0.$ Let Q be the point on the circle dividing the line segment SP internally. Then -

A cylindrical vessel of height \(500 \mathrm{~mm}\) has an orifice (small hole) at its bottom. The orifice is initially closed and water is filled in it upto height \(H\). Now the top is completely sealed with a cap and the orifice at the bottom is opened. Some water comes out from the orifice and the water level in the vessel becomes steady with height of water column being 200 \(\mathrm{mm}\). Find the fall in height (in \(\mathrm{mm}\) ) of water level due to opening of the orifice.
[Take atmospheric pressure \(=1.0 \times 10^5 \mathrm{Nm}^{-2}\), density of water \(=1000 \mathrm{~kg} \mathrm{~m}^{-3}\) and \(g=10\) \(\mathrm{ms}^{-2}\). Neglect any effect of surface tension.]

A parallel plate capacitor has a dielectric slab of dielectric constant K between its plates that covers 1/3 of the area of its plates, as shown in the figure. The total capacitance of the capacitor is C while that of the portion with dielectric in between is ${\mathrm{C}}_1$ . When the capacitor is charged, the plate area covered by the dielectric gets charge ${\mathrm{Q}}_1$ and the rest of the area gets charge ${\mathrm{Q}}_2$ . The electric field in the dielectric is ${\mathrm{E}}_1$ and that in the other portion is ${\mathrm{E}}_2$ . Choose the correct option/options, ignoring edge effects.

A positive point charge of \(10^{-8} \mathrm{C}\) is kept at a distance of 20 cm from the center of a neutral conducting sphere of radius 10 cm . The sphere is then grounded and the charge on the sphere is measured. The grounding is then removed and subsequently the point charge is moved by a distance of 10 cm further away from the center of the sphere along the radial direction. Taking \(\frac{1}{4 \pi \epsilon_0}=9 \times 10^9 \mathrm{Nm}^2 / \mathrm{C}^2\) (where \(\epsilon_0\) is the permittivity of free space), which of the following
statements is/are correct: :

An \(\alpha\)-particle and a proton are accelerated from rest by a potential difference of \(100 \mathrm{~V}\). After this, their de Broglie wavelengths are \(\lambda_\alpha\) and \(\lambda_p\) respectively. The ratio \(\frac{\lambda_p}{\lambda_\alpha}\), to the nearest integer, is

If $f\left(x\right)=\left|\begin{array}{ccc}\cos 2x& \cos 2x& \sin 2x\\ -\cos x& \cos x& -\sin x\\ \sin x& \sin x& \cos x\end{array}\right|,$ then

The circle passing through the point \((-1,0)\) and touching the \(Y\)-axis at \((0,2)\), also passes through the point