Chemical Kinetics · JEE Advanced previous year questions

Consider a reaction $A+R⟶Product$. The rate of this reaction is measured to be $k[A][R]$. At the start of the reaction, the concentration of $R$, $[R{]}_0$, is $10$-times the concentration of $A$, $[A{]}_0$. The reaction can be considered to be a pseudo first order reaction with assumption that $k[R]=k^{\prime }$ is constant. Due to this assumption, the relative error (in $\%$) in the rate when this reaction is $40\%$ complete, is _________. [$k$ and $k^{\prime }$ represent corresponding rate constants]

Consider the following reaction, $$2{\mathrm{H}}_2(\mathrm{g})+2NO(\mathrm{g})\to {\mathrm{N}}_2(\mathrm{g})+2{\mathrm{H}}_2\mathrm{O}(\mathrm{g})$$ which follows the mechanism given below: $$2NO(\mathrm{g})\underset{k_{-1}}{\stackrel{k_1}{\rightleftharpoons }}{\mathrm{N}}_2{\mathrm{O}}_2(\mathrm{g})\text{(fast equlibrium)}$$ $${\mathrm{N}}_2{\mathrm{O}}_2(\mathrm{g})+{\mathrm{H}}_2(\mathrm{g})\stackrel{k_2}{\to }{\mathrm{N}}_2\mathrm{O}(\mathrm{g})+{\mathrm{H}}_2\mathrm{O}(\mathrm{g})\text{(slow reaction)}$$ $${\mathrm{N}}_2\mathrm{O}(\mathrm{g})+{\mathrm{H}}_2(\mathrm{g})\stackrel{k_3}{\to }{\mathrm{N}}_2(\mathrm{g})+{\mathrm{H}}_2\mathrm{O}(\mathrm{g})\text{(fast reaction)}$$ The order of the reaction is ____.

A sample initially contains only U-238 isotope of uranium. With time, some of the U-238 radioactively decays into Pb-206 while the rest of it remains undisintegrated. When the age of the sample is $\mathbf{P}\times 10^8$ years, the ratio of mass of Pb-206 to that of U-238 in the sample is found to be 7. The value of $\mathbf{P}$ is ____. [Given: Half-life of U-238 is $4.5\times 10^9$ years; ${\log }_{e}2=0.693$]