The correct order of acidity for the following compounds is :

The decomposition reaction $2{\mathrm{N}}_2{\mathrm{O}}_5\left(\mathrm{g}\right)\stackrel{ ∆ }{\to }2{\mathrm{N}}_2{\mathrm{O}}_4\left(\mathrm{g}\right)+{\mathrm{O}}_2\left(\mathrm{g}\right)$ is started in a closed cylinder under the isothermal isochoric condition at an initial pressure of $1 \mathrm{atm}$. After $\mathrm{Y}\times {10}^3\mathrm{ }\mathrm{s}$ , the pressure inside the cylinder is found to be $1.45\mathrm{ }\mathrm{a}\mathrm{t}\mathrm{m}$ . If the rate constant of the reaction is $5\times {10}^{-4}{\mathrm{s}}^{-1}$, assuming ideal gas behaviour, the value of $\mathrm{Y}$ is

Consider the strong electrolytes ${\mathrm{Z}}_{\mathrm{m}}{\mathrm{X}}_{\mathrm{n}}, {\mathrm{U}}_{\mathrm{m}}{\mathrm{Y}}_{\mathrm{p}}$ and ${\mathrm{V}}_{\mathrm{m}}{\mathrm{X}}_{\mathrm{n}}$. Limiting molar conductivity of ${\mathrm{U}}_{\mathrm{m}}{\mathrm{Y}}_{\mathrm{p}}$ and ${\mathrm{V}}_{\mathrm{m}}{\mathrm{X}}_{\mathrm{n}}$ are $250$ and $440 \mathrm{S} {\mathrm{cm}}^2 {\mathrm{mol}}^{-1}$, respectively. The value of $\left(\mathrm{m}+\mathrm{n}+\mathrm{p}\right)$ is Given:

Ion	\(Z ^{ n +}\)	\(U ^{ p +}\)	\(V ^{ n +}\)	\(X ^{ m }\)	\(Y ^{ m -}\)
\(\lambda^0(\text{S cm} ^2\) \(\text{mol}^{-1})\)	50.0	25.0	100.0	80.0	100.0

${\lambda }^0$ is the limiting molar conductivity of ions The plot of molar conductivity $\left(\Lambda \right)$ of ${\mathrm{Z}}_{\mathrm{m}}{\mathrm{X}}_{\mathrm{n}}$ vs ${\mathrm{c}}^{\frac{1}{2}}$ is given below. If the numerical value has more than two decimal places, truncate/round-off the value to TWO decimal places.

Among the following, the state function(s) is (are)

Reagent(s) which can be used to bring about the following transformation is(are)

In the given reaction scheme, $\mathbf{P}$ is a phenyl alkyl ether, $\mathbf{Q}$  is an aromatic compound; $\mathbf{R}$ and $\mathbf{S}$ are the major products.

The correct statement about $\mathbf{S}$ is

The increasing order of atomic radii of the following group $13$ elements is

Let $X=\left\{\left(x, y\right)\in \mathrm{\mathbb{Z}}\times \mathrm{\mathbb{Z}} : \frac{x^2}{8}+\frac{y^2}{20}<1 \mathrm{and} y^2<5\mathrm{x}\right\}$. Three distinct point $P, Q$ and $R$ are randomly chosen from $X$. Then the probability that $P, Q$ and $R$ form a triangle whose area is a positive integer, is

Among the following metal carbonyls, the \(\mathrm{C}-\mathrm{O}\) bond order is lowest in

The spin only magnetic moment value (in Bohr magneton units) of \(\mathrm{Cr}(\mathrm{CO})_6\) is

An ideal gas of density $\rho =0.2 \mathrm{kg} {\mathrm{m}}^{-3}$ enters a chimney of height $h$ at the rate of $\alpha =0.8 \mathrm{kg} {\mathrm{s}}^{-1}$ from its lower end, and escapes through the upper end as shown in the figure. The cross-sectional area of the lower end is $A_1=0.1 {\mathrm{m}}^2$ and the upper end is $A_2=0.4 {\mathrm{m}}^2$. The pressure and the temperature of the gas at the lower end are $600 \mathrm{Pa}$ and $300 \mathrm{K}$, respectively, while its temperature at the upper end is $150 \mathrm{K}$. The chimney is heat insulated so that the gas undergoes adiabatic expansion. Take $g=10 \mathrm{m} {\mathrm{s}}^{-2}$ and the ratio of specific heats of the gas $\gamma =2$. Ignore atmospheric pressure.

Which of the following statement(s) is(are) correct?

The left and right compartments of a thermally isolated container of length \(L\) are separated by a thermally conducting, movable piston of area \(A\). The left and right compartments are filled with \(\frac{3}{2}\) and 1 moles of an ideal gas, respectively. In the left compartment the piston is attached by a spring with spring constant k and natural length \(\frac{2 L}{5}\). In thermodynamic equilibrium, the piston is at a distance \(\frac{L}{2}\) from the left and right edges of the container as shown in the figure. Under the above conditions, if the pressure in the right compartment is \(P=\frac{\mathrm{k} L}{\mathrm{~A}} \alpha\), then the value of \(\alpha\) is \(\qquad\)

Columns 1, 2 and 3 contain conics, equations of tangents to the conics and points of contact, respectively.
 

Column 1	Column 2	Column 3
(I) $x^2+y^2=a^2$	(i) $my=m^{2}x+a$	(P) $\left(\frac{a}{m^2},\frac{2a}{m}\right)$
(II) $x^2+a^2{y}^2=a^2$	(ii) $y=mx+a \sqrt{m^2+1}$	(Q) $\left(\frac{-ma}{\sqrt{m^2+1}},\frac{a}{\sqrt{m^2+1}}\right)$
(III) $y^2=4ax$	(iii) $y=mx+ \sqrt{a^2{m}^2-1}$	(R) $\left(\frac{-a^{2}m}{\sqrt{a^2{m}^2+1}},\frac{1}{\sqrt{a^2{m}^2+1}}\right)$
(IV) $x^2-a^2{y}^2=a^2$	(iv) $y=mx+\sqrt{a^2{m}^2+1}$	(S) $\left(\frac{-a^{2}m}{\sqrt{a^2{m}^2-1}},\frac{-1}{\sqrt{a^2{m}^2-1}}\right)$

For $a=\sqrt{2}$ , if a tangent is drawn to a suitable conic (Column 1) at the point of contact $\left(-1, 1\right)$ , then which of the following options is the only Correct combination for obtaining its equation?