A thin uniform annular disc (see figure) of mass \(M\) has outer radius \(4 R\) and inner radius \(3 R\). The work required to take a unit mass from point \(P\) on its axis to infinity is

Consider a system of three charges \(\frac{q}{3}, \frac{q}{3}\) and \(-\frac{2 q}{3}\) placed at points \(A, B\) and \(C\), respectively, as shown in the figure. Take \(O\) to be the centre of the circle of radius \(R\) and angle \(C A B=60^{\circ}\).

Consider regular polygons with number of sides $n=\mathrm{3,4},5\dots$ as shown in the figure. The center of mass of all the polygons is at height $h$ from the ground. They roll on a horizontal surface about the leading vertex without slipping and sliding as depicted. The maximum increase in height of the locus of the center of mass for each polygon is $\mathrm{\Delta }$ . Then, $\mathrm{\Delta }$ depends on $n$ and $h$ as

A fission reaction is given by ${}_{92}{}^{236}U\to {}_{54}{}^{140}Xe+{}_{38}{}^{94}Sr+x+y$ , where $x$ and $y$ are two particles. Considering ${}_{92}{}^{236}U$ to be at rest, the kinetic energies of the products are denoted by $K_{Xe},\mathrm{ }{K}_{Sr},\mathrm{ }{K}_x\left(2\mathrm{ }MeV\right)$ and $K_y\left(2\mathrm{ }MeV\right)$ , respectively. Let the binding energies per nucleon of ${}_{92}{}^{236}U,\mathrm{ }{}_{54}{}^{140}Xe\mathrm{ }$ and ${}_{38}{}^{94}Sr$ be 7.5 MeV, 8.5 MeV, and 8.5 MeV respectively. Considering different conservation laws, the correct option(s) is (are)

The \(\beta\)-decay process, discovered around 1900 , is basically the decay of a neutron \((n)\). In the laboratory, a proton \((p)\) and an electron \(\left(e^{-}\right)\)are observed as the decay products of the neutron. Therefore, considering the decay of a neutron as a two-body decay process, it was predicted theoretically that the kinetic energy of the electron should be a constant. But experimentally, it was observed that the electron kinetic energy has continuous spectrum. Considering a three-body decay process, i.e.
\(n \rightarrow p+e^{-}+\bar{v}_{e}\), around 1930, Pauli explained the observed electron energy spectrum. Assuming the anti-neutrino \(\left(\bar{v}_{e}\right)\) to be massless and possessing negligible energy, and the neutron to be at rest, momentum and energy conservation principles are applied. From this calculation, the maximum kinetic energy of the electron is \(0.8 \times 10^{6} \mathrm{eV}\). The kinetic energy carried by the proton is only the recoil energy.

Question:
If the anti-neutrino had a mass of \(3 \mathrm{eV} / \mathrm{c}^{2}\) (where \(\mathrm{c}\) is the speed of light) instead of zero mass, what should be the range of the kinetic energy, \(K\), of the electron?

Paragraph :
A thermally insulating cylinder has a thermally insulating and frictionless movable partition in the middle, as shown in the figure below. On each side of the partition, there is one mole of an ideal gas, with specific heat at constant volume, \(C_{V}=2 R\). Here, \(R\) is the gas constant. Initially, each side has a volume \(V_{0}\) and temperature \(T_{0}\). The left side has an electric heater, which is turned on at very low power to transfer heat \(Q\) to the gas on the left side. As a result the partition moves slowly towards the right reducing the right side volume to \(V_{0} / 2\). Consequently, the gas temperatures on the left and the right sides become \(T_{L}\) and \(T_{R}\), respectively. Ignore the changes in the temperatures of the cylinder, heater and the partition.

Question :
The value of $\frac{Q}{R{T}_0}$ is:

A proton is fired from very far away towards a nucleus with charge \(Q=120 e\), where \(e\) is the electronic charge. It makes a closest approach of \(10 \mathrm{fm}\) to the nucleus. The de Broglie wavelength (in units of \(\mathrm{fm}\) ) of the proton at its start is: (take the proton mass, \(m_{p}=(5 / 3) \times 10^{-27} \mathrm{~kg} ; h / e=4.2 \times\) \(10^{-15} \mathrm{~J} . \mathrm{S} / \mathrm{C}\);

\(\frac{1}{4 \pi \varepsilon_{0}}=9 \times 10^{9} \mathrm{~m} / \mathrm{F} ; 1 \mathrm{fm}=10^{-15} \mathrm{~m}\)

Let $\mathrm{f}\mathrm{ }:\mathbb{R}\to \mathbb{R}\mathrm{ }$ be a function defined by $\mathrm{f}\left(\mathrm{x}\right)=\mathrm{ }\left\{\begin{array}{cc}\left[\mathrm{x}\right],& \mathrm{x}\mathrm{ }\le 2\\ 0,& \mathrm{x}>2\end{array}\right.$ , where $\left[\mathrm{x}\right]$ is the greatest integer less than or equal to $\mathrm{x}$ . If $\mathrm{I}=\mathrm{ }\underset{-1}{\overset{2}{\int }}\frac{\mathrm{x}\mathrm{f}\left({\mathrm{x}}^2\right)}{2+\mathrm{f}(\mathrm{x}+1)}\mathrm{ }\mathrm{d}\mathrm{x},$ then the value of $(4\mathrm{I}-1)$ is

A physical quantity $\overrightarrow{S}$ is defined as $\overrightarrow{S}=\frac{\left(\overrightarrow{E}\times \overrightarrow{B}\right)}{{\mu }_0}$. where $\overrightarrow{E}$ is electric field, $\overrightarrow{B}$ is magnetic field and ${\mu }_0$ is the permeability of free space. The dimensions of $\overrightarrow{S}$ are the same as the dimensions of which of the following quantity/quantities?

Let $\overrightarrow{a}=2\widehat{i}+\widehat{j}-\widehat{k}$ and $\overrightarrow{b}=\widehat{i}+2\widehat{j}+\widehat{k}$ be two vectors. Consider a vector $\overrightarrow{c}=\alpha \overrightarrow{a}+\beta \overrightarrow{b}, \alpha , \beta \in R.$ If the projection of $\overrightarrow{c}$ on the vector $\left(\overrightarrow{a}+\overrightarrow{b}\right)$ is $3\sqrt{2},$ then the minimum value of $\left(\overrightarrow{c}-\left(\overrightarrow{a}\times \overrightarrow{b}\right)\right).\overrightarrow{c}$ equals

Consider the parabola \(y^2=8 x\). Let \(\Delta_1\) be the area of the triangle formed by the end points of its latusrectum and the point \(P\left(\frac{1}{2}, 2\right)\) on the parabola and \(\Delta_2\) be the area of the triangle formed by drawing tangents at \(P\) and at the end points of the latusrectum. Then, \(\frac{\Delta_1}{\Delta_2}\) is

The reaction of ${\mathrm{Xe}}_{}$ and ${\mathrm{O}}_2{\mathrm{F}}_2$ gives a $\mathrm{Xe}$ compound $\mathrm{P}$. The number of moles of $\mathrm{HF}$ produced by the complete hydrolysis of $1 \mathrm{mol}$ of $\mathrm{P}$ is _____ .

Paragraph: When potassium iodide is added to an aqueous solution of potassium ferricyanide, a reversible reaction is observed in which a complex \(\mathbf{P}\) is formed. In a strong acidic medium, the equilibrium shifts completely towards \(\mathbf{P}\). Addition of zinc chloride to \(\mathbf{P}\) in a slightly acidic medium results in a sparingly soluble complex Q.
Question: The number of zinc ions present in the molecular formula of \(\mathbf{Q}\) is