Liquids $\mathrm{A}$ and $\mathrm{B}$ form ideal solution for all compositions of $\mathrm{A}$ and $\mathrm{B}$ at ${25}^{\circ }C$. Two such solutions with $0.25$ and $0.50$ mole fractions of $\mathrm{A}$ have the total vapor pressures of $0.3$ and $0.4$ bar, respectively. What is the vapor pressure of pure liquid $\mathrm{B}$ in bar?

All the compounds listed in Column I react with water. Match the result of the respective reactions with the appropriate options listed in Column II.

Among \(H_2, H e_2^{+}, L i_2, B e_2, B_2, C_2, N_2, O_2^{-}\), and \(F_2\), the number of diamagnetic species is -
(Atomic number : \(H=1, H e=2, L i=3, B e=4, B=5,\) \(C=6, N =7, O=8, F=9\) )

Aluminium reacts with sulfuric acid to form aluminium sulfate and hydrogen. What is the volume of hydrogen gas in liters $\left(\mathrm{L}\right)$ produced at $300 \mathrm{K}$ and $1.0$ atm pressure, when $5.4 \mathrm{g}$ of aluminium and $50.0 \mathrm{mL}$ of $5.0 \mathrm{M}$ sulfuric acid are combined for the reaction? (Use molar mass of aluminium as $27.0 \mathrm{g} {\mathrm{mol}}^{-1}$, $\mathrm{R}=0.082 \mathrm{atm} \mathrm{L} {\mathrm{mol}}^{-1}{\mathrm{K}}^{-1}$)

LIST-I contains reactions and LIST-II contains major products

List - I	List - II
(A)	(P)
(B)	(Q)
(C)	(R)
(D)	(S)
	(T)

Match each reaction in LIST-I with one or more products in LIST-II and choose the correct option

A positron is emitted from \({ }_{11}^{23} \mathrm{Na}\). The ratio of the atomic mass and atomic number of the resulting nuclide is

Among the following complexes, the total number of diamagnetic species is_____
\(\left[\mathrm{Mn}\left(\mathrm{NH}_3\right)_6\right]^{3+},\left[\mathrm{MnCl}_6\right]^{3-},\left[\mathrm{FeF}_6\right]^{3-},\left[\mathrm{CoF}_6\right]^{3-},\)\(\left[\mathrm{Fe}\left(\mathrm{NH}_3\right)_6\right]^{3+}\), and \(\left[\mathrm{Co}(\mathrm{en})_3\right]^{3+}\)
[Given, atomic number: \(\mathrm{Mn}=25, \mathrm{Fe}=26, \mathrm{Co}=27\);
\(\text { en } \left.=\mathrm{H}_2 \mathrm{NCH}_2 \mathrm{CH}_2 \mathrm{NH}_2\right]\)

The pair(s) of diamagnetic ions is(are)

The value of \(\lim _{x \rightarrow 0} \frac{1}{x^3} \int_0^x \frac{t \log (1+t)}{t^4+4} d t\) is

The number of optically active products obtained from the complete ozonolysis of the given compound is :

If $g\left(x\right)={\int }_{\sin x}^{\sin 2x}{\sin }^{-1}tdt,$  then

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Electrical resistance of certain materials, known as superconductors, changes abruptly from a non-zero value to zero as their temperature is lowered below a critical temperature \(T_C(0)\). An interesting property of superconductors is that their critical temperature becomes smaller than \(T_C(0)\) if they are placed in a magnetic field i.e., the critical temperature \(T_C(B)\) is a function of the magnetic field strength \(B\). The dependence of \(T_C(B)\) on \(B\) is shown in the figure.

Question :
A superconductor has \(T_C(0)=100 \mathrm{~K}\). When a magnetic field of \(7.5\) Tesla is applied, its \(T_C\) decreases to \(75 \mathrm{~K}\). For this material one can definitely say that when (Note : \(\mathrm{T}\) = Tesla)

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A dense collection of equal number of electrons and positive ions is called neutral plasma. Certain solids containing fixed positive ions surrounded by free electrons can be treated as neutral plasma. Let \(N\) be the number density of free electrons, each of mass \(m\). When the electrons are subjected to an electric field, they are displaced relatively away from the heavy positive ions.If the electric field becomes zero, the electrons being to oscillate about the positive ions with a natural angular frequency \(\omega_p\), which is called the plasma frequency. To sustain the oscillations, a time varying electric field needs to be applied that has an angular frequency \(\omega\), where a part of the energy is absorbed and a part of it is reflected. As \(\omega\) approaches \(\omega_p\), all the free electrons are set to resonance together and all the energy is reflected. This is the explanation of high reflectivity of metals.
Question :
Taking the electronic charge as \(e\) and the permittivity as \(\varepsilon_0\), use dimensional analysis to determine the correct expression for \(\omega_p\).