A mixture of 2 moles of helium gas (atomic mass \(=4 \mathrm{amu}\) ) and 1 mole of argon gas (atomic mass \(=40 \mathrm{amu}\) ) is kept at \(300 \mathrm{~K}\) in a container. The ratio of the rms speeds \(\left(\frac{v_{\mathrm{rms}}(\text { helium })}{v_{\mathrm{rms}}(\operatorname{argon})}\right)\) is

Consider a star of mass \(m_2 \mathrm{~kg}\) revolving in a circular orbit around another star of mass \(m_1 \mathrm{~kg}\) with \(m_1 \gg m_2\). The heavier star slowly acquires mass from the lighter star at a constant rate of \(\gamma \mathrm{kg} / \mathrm{s}\). In this transfer process, there is no other loss of mass. If the separation between the centers of the stars is \(r\), then its relative rate of change \(\frac{1}{r} \frac{\mathrm{~d} r}{\mathrm{~d} t}\left(\right.\) in s \(\left.^{-1}\right)\) is given by:
[Note: The original question was awarded bonus marks in the JEE Advanced 2025 exam. We have changed the options to make the question valid and solvable.]

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The key feature of Bohr's theory of spectrum of hydrogen atom is the quantization of angular momentum when an electron is revolving around a proton. We will extend this to a general rotational motion to find quantized rotational energy of a diatomic molecule assuming it to be rigid. The rule to be applied is Bohr's quantization condition.
Question:
It is found that the excitation frequency from ground to the first excited state of rotation for the \(\mathrm{CO}\) molecule is close to \(\frac{4}{\pi} \times 10^{11} \mathrm{~Hz}\). Then the moment of inertia of CO molecule about its centre of mass is close to (Take \(h=2 \pi \times 10^{-34} J-s\) )

A horizontal force$F$is applied at the centre of mass of a cylindrical object of mass$m$and radius$R$, perpendicular to its axis as shown in the figure. The coefficient of friction between the object and the ground is$\mu$. The centre of mass of the object has an acceleration$a$. The acceleration due to gravity is$g$. Given that the object rolls without slipping, which of the following statement(s) is (are) correct?

One mole of a monatomic ideal gas undergoes four thermodynamic processes as shown schematically in the PV - diagram below. Among these four processes, one is isobaric, one is isochoric, one is isothermal and one is adiabatic. Match the processes mentioned in List-1 with the corresponding statements in List-II.

	LIST -I		LIST -II
A)	In process $I$	P)	Work done by the gas is zero
B)	In process $II$	Q)	Temperature of the gas remains unchanged
C)	In process $III$	R)	No heat is exchanged between the gas and its surroundings
D)	In process $IV$	S)	Work done by the gas is $6P_0{V}_0$

In a particular system of units, a physical quantity can be expressed in terms of the electric charge $e$, electron mass $m_e$. Planck's constant $h$, and Coulomb's constant $k=\frac{1}{4\pi {\epsilon }_0}$, where ${\epsilon }_0$ is the permittivity of vacuum. In terms of these physical constants, the dimension of the magnetic field is $\left[B\right]={\left[e\right]}^{\alpha }{\left[m_e\right]}^{\beta }{\left[h\right]}^{\gamma }{\left[k\right]}^{\delta }$. The value of $\alpha +\beta +\gamma +\delta$ is _______.

A \(0.1 \mathrm{~kg}\) mass is suspended from a wire of negligible mass. The length of the wire is \(1 \mathrm{~m}\) and its cross-sectional area is \(4.9 \times 10^{-7} \mathrm{~m}^2\). If the mass is pulled a little in the vertically downward direction and released, it performs simple harmonic motion of angular frequency \(140 \mathrm{rad} \mathrm{s}^{-1}\). If the Young's modulus of the material of the wire is \(n \times 10^9 \mathrm{Nm}^{-2}\), the value of \(n\) is

A field line is shown in the figure. This field cannot represent

The reduction potential $\left({\mathrm{E}}^0\right.$, in $\left.\mathrm{V}\right)$ of ${\mathrm{MnO}}_4^{-}\left(\mathrm{aq}\right)/\mathrm{Mn}\left(\mathrm{s}\right)$ is
\(\left[\right.\)Given: \(E _{\left( MnO _4^{-}( aq ) / MnO _2(s)\right)}^{ o }=1.68 V ;\) \( E _{\left( MnO _2(s) / Mn ^{2+}( aq )\right)}^{ o }=\) \(1.21 V ; E _{\left( Mn ^{2+}( aq ) / Mn ( s )\right)}^{ o }\) \(=-1.03 V]\)
Truncate/round-off the value to TWO decimal places.

For the circuit shown in the figure

The correct match of the group reagents in List-I for precipitating the metal ion given in List-II from solutions, is

List - I	List - II
(P) Passing \(H _2 S\) in the presence of \(NH _4 OH\)	(1) \(Cu ^{2+}\)
(Q) \(\left( NH _4\right)_2 CO _3\) in the presence of \(NH _4 OH\)	(2) \(Al ^{3+}\)
(R) \(NH _4 OH\) in the presence of \(NH _4 Cl\)	(3) \(Mn ^{2+}\)
(S) Passing \(H _2 S\) in the presence of dilute HCl	(4) \(Ba ^{2+}\)
	(5) \(Mg ^{2+}\)

The number of real solutions of the equation ${\sin }^{-1}\left(\sum _{i=1}^{\infty }{x}^{i+1}-x\sum _{i=1}^{\infty }{\left(\frac{x}{2}\right)}^i\right)=\frac{\pi }{2}-{\cos }^{-1}\left(\sum _{i=1}^{\infty }{\left(-\frac{x}{2}\right)}^i-\sum _{i=1}^{\infty }{\left(-x\right)}^i\right)$ lying in the interval $\left(-\frac{1}{2},\frac{1}{2}\right)$ is____.
(Here, the inverse trigonometric functions ${\sin }^{-1}x and {\cos }^{-1}x$ assume values in $\left[-\frac{\pi }{2},\frac{\pi }{2}\right] \mathrm{a}\mathrm{n}\mathrm{d} \left[0,\pi \right]$
respectively.)

For any $3\times 3$ matrix $M$, let $\left|M\right|$ denote the determinant of $M$. Let $I$ be the $3\times 3$ identity matrix. Let $E$ and $F$ be two $3\times 3$ matrices such that $(I-EF)$ is invertible. If $G=(I-EF{)}^{-1}$, then which of the following statements is (are) TRUE ?