The number of rotational degrees of freedom of a monatomic molecule is

If the average translational kinetic energy of a molecule in a gas is equal to the kinetic energy of an electron accelerating from rest through \(10 \mathrm{~V}\), then the temperature of the gas molecule is
\[
\left(\text { Boltzmann constant }=1.38 \times 10^{-23} \mathrm{JK}^{-1}\right)
\]

Two identical condensers \(M\) and \(N\) are connected in series with a battery. The space between the plates of \(M\) is completely filled with a dielectric medium of dielectric constant 8 and a copper plate of thickness \(\frac{d}{2}\) is introduced between the plates of \(N\). ( \(d\) is the distance between the plates). Then potential differences across \(M\) and \(N\) are, respectively, in the ratio

The number of vibrational degrees of freedom of a diatomic molecule is

Minimum excitation potential of Bohr's first orbit of Hydrogen atom is_____.

In the Uranium radioactive series, the initial nucleus is \({ }_{92}^{238} \mathrm{U}\) and final nucleus is \({ }_{82}^{206} \mathrm{~Pb}\). When the Uranium nucleus decays to lead, the number of \(\alpha\)-particles emitted is .......... and the number of \(\beta\)-particles emitted is ____.

A wire of length 0.5 m and area of cross-section \(4 \times 10^{-6} \mathrm{~m}^2\) at a temperature of \(100^{\circ} \mathrm{C}\) is suspended vertically by fixing its upper end to the ceiling. The wire is then cooled to \(0^{\circ} \mathrm{C}\), but is prevented from contracting, by attaching a mass at the lower end. If the mass of the wire is negligible, then the value of the mass attached to the wire is
[Young's modulus of material of the wire \(=10^{11} \mathrm{~N} \mathrm{~m}^{-2}\); coefficient of linear expansion of the material of the wire \(=10^{-5} \mathrm{~K}^{-1}\) and acceleration due to gravity \(\left.=10 \mathrm{~m} \mathrm{~s}^{-2}\right]\)

Identify the correct statement from the following :

A basket contains 12 apples in which 3 are rotten. If 3 apples are drawn at random simultaneously from it, then the probability of getting atmost one rotten apple is

If \(f(x)=(2 k+1) x-3-k e^{-x}+2 e^x\) is monotonically increasing for all \(x \in R\), then the least value of \(k\) is

An unbiased coin is tossed 3 times. If the third toss gets head, then the probability of getting at least one more head is

If the function $f\left(x\right)$, defined below, is continuous on the interval $[0,8]$, then $f\left(x\right)=\left\{\begin{array}{l}{x}^2+ax+b,0\le x<2\\ 3x+2,2\le x\le 4\\ 2ax+5b,4<x\le 8\end{array}\right.$

If the sum of all the coefficients of \(\left(\alpha x^2-2 x+1\right)^{2019}\) is equal to the sum of all the coefficients of \((x-\alpha y)^{2019}\), then \(\alpha=\)