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?

A current carrying infinitely long wire is kept along the diameter of a circular wire loop, without touching it, the correct statement(s) is(are)

An \(\alpha\)-particle and a proton are accelerated from rest by a potential difference of \(100 \mathrm{~V}\). After this, their de Broglie wavelengths are \(\lambda_\alpha\) and \(\lambda_p\) respectively. The ratio \(\frac{\lambda_p}{\lambda_\alpha}\), to the nearest integer, is

A thin rod of mass $M$ and length $a$ is free to rotate in horizontal plane about a fixed vertical axis passing through point $O$. A thin circular disc of mass $M$ and of radius $\frac{a}{4}$ is pivoted on this rod with its centre at a distance $\frac{a}{4}$ from the free end so that it can rotate freely about its vertical axis, as shown in the figure. Assume that both the rod and the disc have uniform density and they remain horizontal during the motion. An outside stationary conserver finds the rod rotating with an angular velocity $\Omega$ and the disc rotating about its vertical axis with angular velocity $4\Omega$. The total angular momentum of the system about the point $O$ is $\left[\left(\frac{M{a}^2\Omega }{48}\right)n\right]$. The value of $n$ is _________ .

A block $M$ hangs vertically at the bottom end of a uniform rope of constant mass per unit length. The top end of the rope is attached to a fixed rigid support at O. A transverse wave pulse (Pulse 1) of wavelength ${\lambda }_0$ is produced at point O on the rope. The pulse takes time $T_{OA}$ to reach point A. If the wave pulse of wavelength ${\lambda }_0$ is produced at point A (Pulse 2) without disturbing the position of M it takes time $T_{OA}$ to reach point O. Which of the following options is/are correct?

A container of fixed volume has a mixture of one mole of hydrogen and one mole of helium in equilibrium at temperature $\mathrm{T}$ . Assuming the gases are ideal, the correct statement(s) is (are)

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A small spherical monoatomic ideal gas bubble \(\left(\gamma=\frac{5}{3}\right)\) is trapped inside a liquid of density \(\rho_1\) (see figure). Assume that the bubble does not exchange any heat with the liquid. The bubble contains \(n\) moles of gas. The temperature of the gas when the bubble is at the bottom is \(T_0\), the height of the liquid is \(H\) and the atmospheric pressure is \(p_0\) (Neglect surface tension).

Question:
The buoyancy force acting on the gas bubble is (Assume \(R\) is the universal gas constant)

In the following carbocation, \(\mathrm{H} / \mathrm{CH}_3\) that is most likely to migrate to the positively charged carbon is

A beaker of radius $r$ is filled with water$\left(\text{refractive index}\frac{4}{3}\right)$ up to a height $H$ as shown in the figure on the left. The beaker is kept on a horizontal table rotating with angular speed $\omega$. This makes the water surface curved so that the difference in the height of water level at the centre and at the circumference of the beaker is $h\left(h\ll H\text{,}h\ll r\right)$, as shown in the figure on the right. Take this surface to be approximately spherical with a radius of curvature $R.$ Which of the following is/are correct? ($g$ is the acceleration due to gravity)

Let the function $f:\left(0,\pi \right)\to \mathrm{ℝ}$ be defined by $f\left(\theta \right)={\left(\sin \theta +\cos \theta \right)}^2+{\left(\sin \theta -\cos \theta \right)}^4$. Suppose the function $f$ has a local minimum at $\theta$ precisely when $\theta \in \left\{{\lambda }_1\pi ,\dots ,{\lambda }_r\pi \right\},$ where $0<$${\lambda }_1<\cdots <{\lambda }_r<1.$ Then the value of  ${\lambda }_1+\cdots +{\lambda }_r$ is_______

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Consider the function \(f:(-\infty, \infty) \rightarrow(-\infty, \infty)\) defined by \(f(x)=\frac{x^2-a x+1}{x^2+a x+1} ; 0 < a < 2\)Question:
Which of the following is true ?

The order of the oxidation state of the phosphorus atom in $H_{3}P{O}_2, H_{3}P{O}_4, H_{3}P{O}_3$ and $H_4{P}_2{O}_6$ is

Newman projections $\mathrm{P},\mathrm{Q},\mathrm{R}$ and $\mathrm{S}$ are shown below:

Which one of the following options represent identical molecules?