The radius of the orbit of an electron in a Hydrogen-like atom is \(4.5 a _0\), where \(a _0\) is the Bohr radius. Its orbital angular momentum is \(\frac{3 h}{2 \pi}\). It is given that \(h\) is Planck's constant and R is Rydberg constant. the possible wavelength \((s)\), when the atom de-excites, is (are)

A length - scale $\left(\mathcal{l}\right)$ depends on the permittivity $\left(\mathrm{\epsilon }\right)$ of a dielectric material, Boltzmann constant $k_B$ , the absolute temperature T, the number per unit volume (n) of certain charged particles, and the charge (q) carried by each of the particles, Which of the following expressions (s) for $\mathcal{l}$ is (are) dimensionally correct?

An electric dipole with dipole moment $\frac{{\mathrm{p}}_0}{\sqrt{2}}\left(\widehat{\mathrm{i}}+\widehat{\mathrm{j}}\right)$ is held fixed at the origin $\mathrm{O}$ in the presence of an uniform electric field of magnitude ${\mathrm{E}}_0.$ If the potential is constant on a circle of radius $\mathrm{R}$ centered at the origin as shown in figure, then the correct statement(s) is/are:
( ${\mathrm{\epsilon }}_0$ is permittivity of free space, $\mathrm{R}>>$ dipole size)

A disk of radius \(\frac{a}{4}\) having a uniformly distributed charge \(6 \mathrm{C}\) is placed in the \(x-y\) plane with its centre at \(\left(\frac{-a}{2}, 0,0\right)\). A rod of length a carrying a uniformly distributed charge \(8 \mathrm{C}\) is placed on the \(x\)-axis from \(x=\frac{a}{4}\) to \(x=\frac{5 a}{4}\). Two point charges \(-7 \mathrm{C}\) and \(3 \mathrm{C}\) are placed at \(\left(\frac{a}{4}, \frac{-a}{4}, 0\right)\) and \(\left(\frac{-3 a}{4}, \frac{3 a}{4}, 0\right)\), respectively. Consider a cubical surface formed by six surfaces \(x=\pm \frac{a}{2}, y=\pm \frac{a}{2}, z=\pm \frac{a}{2}\).
The electric flux through this cubical surface is

A rocket is launched normal to the surface of the Earth, away from the Sun, along the line joining the Sun and the Earth. The Sun is $3\times {10}^5$ times heavier than the Earth and is at a distance $2.5\times {10}^4$ times larger than the radius of the Earth. The escape velocity from Earth's gravitational field is $v_e=11.2 km s^{-1}$ . The minimum initial velocity $\left(v_s\right)$ required for the rocket to be able to leave the Sun-Earth system is closest to
(Ignore the rotation and revolution of the Earth and the presence of any other planet)

A steady current \(I\) goes through a wire loop \(P Q R\) having shape of a right angle triangle with \(P Q=3 x, P R=4 x\) and \(Q R=5 x\). If the magnitude of the magnetic field at \(P\) due to this loop is \(k\left(\frac{\mu_0 I}{48 \pi x}\right)\), find the value of \(k\).

A charge $q$ is surrounded by a closed surface consisting of an inverted cone of height $h$ and base radius $R$, and a hemisphere of radius $R$ as shown in the figure. The electric flux through the conical surface is$\frac{nq}{6{\epsilon }_0}$  (in $\mathrm{SI}$ units). The value of $n$ is _________.

Let $x, y$ and $z$ be positive real numbers. Suppose $x, y$ and $z$ are the lengths of the sides of a triangle opposite to its angles $X, Y$ and $Z$ respectively. If $\tan \frac{X}{2}+\tan \frac{Z}{2}=\frac{2y}{x+y+z}$, then which of the following statements is/are TRUE?

A colorless aqueous solution contains nitrates of two metals, $\mathrm{X}$ and $\mathrm{Y}$. When it was added to an aqueous solution of $\mathrm{NaCl}$, a white precipitate was formed. This precipitate was found to be partly soluble in hot water to give a residue $\mathrm{P}$ and a solution $\mathrm{Q}$. The residue $\mathrm{P}$ was soluble in aq. ${\mathrm{NH}}_3$ and also in excess sodium thiosulfate. The hot solution $\mathrm{Q}$ gave a yellow precipitate with $\mathrm{KI}$. The metals $\mathrm{X}$ and $\mathrm{Y}$, respectively, are

Among the following, the correct statement(s) is(are)

Let S be the set of all non - zero real numbers $\alpha$ such that the quadratic equation $\alpha x^2-x+\alpha =0$ has two distinct real roots $x_1 and x_2$ satisfying the inequality $\left|x_1-x_2\right|<1.$ Which of the following intervals is(are) a subset(s) of S?

A flat surface of a thin uniform disk $A$of radius$R$ is glued to a horizontal table. Another thin uniform disk$B$ of mass $M$and with the same radius $R$rolls without slipping on the circumference of $A$, as shown in the figure. A flat surface of $B$ also lies on the plane of the table. The center of mass of$B$ has fixed angular speed $\omega$about the vertical axis passing through the center of $A$. The angular momentum of $B$ is $nM\omega R^2$with respect to the center of $A$. Which of the following is the value of $n$?

An ideal gas is in thermodynamic equilibrium. The number of degrees of freedom of a molecule of the gas is $n$. The internal energy of one mole of the gas is $U_n$ and the speed of sound in the gas is $v_n$. At a fixed temperature and pressure, which of the following is the correct option?