Let $\overrightarrow{\text{PR}}=3\widehat{i}+\widehat{j}-2\widehat{k}\text{and}\overrightarrow{\text{SQ}}=\widehat{i}-3\widehat{j}-4\widehat{k}$ determine diagonals of a parallelogram PQRS and $\overrightarrow{\text{PT}}=\widehat{i}+2\widehat{j}+3\widehat{k}$ be another vector. Then the volume of the parallelepiped determined by the vectors $\overrightarrow{\mathit{\text{PT}}}\text{,}\overrightarrow{\mathit{\text{PQ}}}\text{and}\overrightarrow{\mathit{\text{PS}}}$  is

Let $f:\mathbb{R}\to \mathbb{R}\mathbb{ }\mathrm{a}\mathrm{n}\mathrm{d} g:\mathbb{R}\to \mathbb{R}$ be two non-constant differentiable functions. If $f^{\prime }\left(x\right)=\left(e^{\left(f\left(x\right)\right)-g\left(x\right)}\right)g^{\prime }\left(x\right) \mathrm{f}\mathrm{o}\mathrm{r} \mathrm{a}\mathrm{l}\mathrm{l} x\in \mathbb{R}$, and $f\left(1\right)=g\left(2\right)=1$ then which of the following statement(s) is (are) TRUE?

If \(a, b\) and \(c\) are the sides of a \(\triangle A B C\) such that \(x^2-2(a+b+c) x+3 \lambda(a b+b c+c a)=0\) has real roots, then

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:

Let \(f\) be a real-valued function defined on the interval \((-1,1)\) such that \(e^{-x} f(x)=2+\int_0^x \sqrt{t^4+1} d t, \quad\) for all \(x \in(-1,1)\) and let \(f^{-1}\) be the inverse function of \(f\). Then \(\left(f^{-1}\right)^{\prime}(2)\) is equal to

A number is chosen at random from the set $\left\{1,2,3,\dots ,2000\right\}$. Let $p$ be the probability that the chosen number is a multiple of 3 or $7$. Then the value of $500p$ is ____.

Let $\overline{z}$ denote the complex conjugate of a complex number $z$ and let $i=\sqrt{-1}$. In the set of complex numbers, the number of distinct roots of the equation $\overline{z}-z^2=i\left(\overline{z}+z^2\right)$ is _______.

A student skates up a ramp that makes an angle $30°$ with the horizontal. He/she starts (as shown in the figure) at the bottom of the ramp with speed $v_0$ and wants to turn around over a semicircular path $xyz$ of radius $R$ during which he/she reaches a maximum height $h$ (at point $y$) from the ground as shown in the figure. Assume that the energy loss is negligible and the force required for this turn at the highest point is provided by his/her weight only. Then ($g$ is the acceleration due to gravity)

A small mass \(m\) is attached to a massless string whose other end is fixed at \(P\) as shown in the figure. The mass is undergoing circular motion in the \(x-y\) plane with centre at \(O\) and constant angular speed \(\omega\). If the angular momentum of the system, calculated about \(O\) and \(P\) are denoted by \(\vec{L}_{O}\) and \(\vec{L}_{P}\) respectively, then

Among \(\mathrm{V}(\mathrm{CO})_6, \mathrm{Cr}(\mathrm{CO})_5, \mathrm{Cu}(\mathrm{CO})_3,\)\(\operatorname{Mn}(\mathrm{CO})_5, \mathrm{Fe}(\mathrm{CO})_5,\left[\mathrm{Co}(\mathrm{CO})_3\right]^{3-},\left[\mathrm{Cr}(\mathrm{CO})_4\right]^{4-}\), and \(\operatorname{Ir}(\mathrm{CO})_3\), the total number of species isoelectronic with \(\mathrm{Ni}(\mathrm{CO})_4\) is ________.
[Given, atomic number:\(\mathrm{V}=23, \mathrm{Cr}=24, \mathrm{Mn}=25,\) \(\mathrm{Fe}=26, \mathrm{Co}=27, \mathrm{Ni}=28, \mathrm{Cu}=29,\) \(\mathrm{Ir}=77\)]

Let ${\theta }_1,{\theta }_2,\dots ,{\theta }_{10}$ be positive valued angles (in radian) such that ${\theta }_1+{\theta }_2+\cdots +{\theta }_{10}=2\pi$. Define the complex numbers $z_1=e^{i{\theta }_1},z_k=z_{k-1}{e}^{i{\theta }_k}$ for $k=2,3,\dots ,10$, where $i=\sqrt{-1}$. Consider the statements $P$ and $Q$ given below:
\(P:\left|z_2-z_1\right|+\left|z_3-z_2\right|+\cdots+\left|z_{10}-z_9\right|\) \(+\left|z_1-z_{10}\right| \leq 2 \pi \)
\( Q:\left|z_2^2-z_1^2\right|+\left|z_3^2-z_2^2\right|+\cdots+\left|z_{10}^2-z_9^2\right|\) \(+\left|z_1^2-z_{10}^2\right| \leq 4 \pi\)

A plane passes through \((1,-2,1)\) and is perpendicular to two planes \(2 x-2 y+z=0\) and \(x-y+2 z=4\), then the distance of the plane from the point \((1,2,2)\) is

Paragraph:
In hexagonal system of crystals, a frequently encountered arrangement of atoms is described as a hexagonal prism. Here, the top and bottom of the cell are regular hexagons and three atoms are sandwiched in between them. A space-filling model of this structure, called hexagonal close-packed (HCP), is constituted of a sphere on a flat surface surrounded in the same plane by six identical spheres as closely as possible. Three spheres are then placed over the first layer so that they touch each other and represent the second layer. Each one of the three spheres touches three spheres of the bottom layer. Finally, the second layer is covered with a third layer that is identical to the bottom layer in relative position. Assume radius of every sphere to be ' \(r\) '.
Question:
The volume of this \(\mathrm{HCP}\) unit cell is