The following integral $\underset{\frac{\mathrm{\pi }}{4}}{\overset{\frac{\mathrm{\pi }}{2}}{\int }}{\left(2\mathrm{ }\mathrm{c}\mathrm{o}\mathrm{s}\mathrm{e}\mathrm{c}\mathrm{ }\mathrm{x}\right)}^{17}\mathrm{ }\mathrm{d}\mathrm{x}$ is equal to

Let $P$ be the plane $\sqrt{3}x+2y+3z=16$ and let
$S=\left\{\alpha \hat{i}+\beta \hat{j}+\gamma \hat{k}: {\alpha }^2+{\beta }^2+{\gamma }^2=1\right.$ and the distance of $\left(\alpha , \beta , \gamma \right)$ from the plane $P$ is $\frac{7}{2}\}$.
Let $\overrightarrow{u}, \overrightarrow{v}$ and $\overrightarrow{w}$ be three distinct vectors in $S$ such that $\left|\overrightarrow{u}-\overrightarrow{v}\right|=\left|\overrightarrow{v}-\overrightarrow{w}\right|=\left|\overrightarrow{w}-\overrightarrow{u}\right|$. Let $V$ be the volume of the parallelepiped determined by vectors $\overrightarrow{u}, \overrightarrow{v}$ and $\overrightarrow{w}$. Then the value of $\frac{80}{\sqrt{3}}V$ is

An engineer is required to visit a factory for exactly four days during the first $15$ days of every month and it is mandatory that no two visits take place on consecutive days. Then the number of all possible ways in which such visits to the factory can be made by the engineer during $1-15$ June $2021$ is_____

Let $H:\frac{x^2}{a^2 }-\frac{y^2}{b^2}=1$ , where $a>b>0$, be a hyperbola in the xy-plane whose conjugate axis $LM$ subtends an angle of ${60}^o$ at one of its vertices $N$. Let the area of the triangle $LMN$ be $4\sqrt{3}$.

LIST-I	LIST-II
A. The length of the conjugate axis of $H$ is	P. $8$
B. The eccentricity of $H$ is	Q. $\frac{4}{\sqrt{3}}$
C. The distance between the foci of $H$ is	R. $\frac{2}{\sqrt{3}}$
D. The length of the latus rectum of $H$ is	S. $4$

The correct option is:

If the line $x=\alpha$ divides the area of the region $R=\{\left(x,y\right)\in R^2:x^3\le y\le x, 0\le x\le 1\}$ into two equal parts, then

The number of distinct solutions of the equation $\frac{5}{4}{\cos }^2\mathrm{ }2\mathrm{x}+{\cos }^4\mathrm{x}+{\sin }^4\mathrm{x}+{\cos }^6\mathrm{x}+{\sin }^6\mathrm{x}=2$ in the interval $[0,\mathrm{ }2\mathrm{\pi }]$ is

In a hotel, four rooms are available. Six persons are to be accommodated in these four rooms in such a way that each of these rooms contains at least one person and at most two persons. Then the number of all possible ways in which this can be done is_____

Paragraph :
A spray gun is shown in the figure where a piston pushes air out of a nozzle. A thin tube of uniform cross section is connected to the nozzle. The other end of the tube is in a small liquid container. As the piston pushes air through the nozzle, the liquid from the container rises into the nozzle and is sprayed out. For the spray gun shown, the radii of the piston and the nozzle are \(20 \mathrm{~mm}\) and \(1 \mathrm{~mm}\), respectively. The upper end of the container is open to the atmosphere.

Question :
If the density of air is \(\rho_{a}\) and that of the liquid \(\rho_{\ell}\), then for a given piston speed the rate (volume per unit time) at which the liquid is sprayed will be proportional to

Let $f: \left(0, \infty \right)\to R$ be a differentiable function such that $f^{\text{'}}\left(x\right)=2-\frac{f\left(x\right)}{x}\forall x\in (0, \infty )$ and $f\left(1\right)\ne 1$. Then,

Methylene blue, from its aqueous solution, is adsorbed on activated charcoal at 25^o C. For this process, the correct statement is 

Let \(\mathbf{a}=\hat{\mathbf{i}}+\hat{\mathbf{j}}+\hat{\mathbf{k}}, \mathbf{b}=\hat{\mathbf{i}}-\hat{\mathbf{j}}+\hat{\mathbf{k}}\) and \(\mathbf{c}=\hat{\mathbf{i}}-\hat{\mathbf{j}}-\hat{\mathbf{k}}\) be three vectors. A vector \(\mathbf{v}\) in the plane of \(\mathbf{a}\) and \(\mathbf{b}\) whose projection of \(\mathbf{c}\) is \(\frac{1}{\sqrt{3}}\), is given by

In a photoemission experiment, the maximum kinetic energies of photoelectrons from metals $P,Q$ and $R$ are $E_P, E_Q$ and $E_R$, respectively, and they are related by $E_P=2E_Q=2E_R$. In this experiment, the same source of monochromatic light is used for metals $P$ and $Q$ while a different source of monochromatic light is used for the metal $R$. The work functions for metals $P,Q$ and $R$ are $4.0 \mathrm{eV}, 4.5 \mathrm{eV}$ and $5.5 \mathrm{eV}$, respectively. The energy of the incident photon used for metal $R$, in $eV$ is ___.

Based on VSEPR model, match the xenon compounds given in List-I with the corresponding geometries and the number of lone pairs on xenon given in List-II and choose the correct option.

List - I	List - II
(P) \(XeF _2\)	(1) Trigonal bipyramidal and two lone pair of electrons
(Q) \(XeF _4\)	(2) Tetrahedral and one lone pair of electrons
(R) \(XeO _3\)	(3) Octahedral and two lone pair of electrons
(S) \(XeO _3 F_2\)	(4) Trigonal bipyramidal and no lone pair of electrons
	(5) Trigonal bipyramidal and three lone pair of electrons