The acidic hydrolysis of ether (X) shown below is fastest when

The reaction of $\mathrm{Pb}{\left({\mathrm{NO}}_3\right)}_2$ and $\mathrm{NaCl}$ in water produces a precipitate that dissolves upon the addition of $\mathrm{HCl}$ of appropriate concentration. The dissolution of the precipitate is due to the formation of

The compound (s) that exhibit (s) geometrical isomerism is (are)

In the scheme given below, the total number of intramolecular aldol condensation products formed from ' \(Y\) ' is

Among \(\mathrm{V}(\mathrm{CO})_6, \mathrm{Cr}(\mathrm{CO})_5, \mathrm{Cu}(\mathrm{CO})_3,\)\(\operatorname{Mn}(\mathrm{CO})_5, \mathrm{Fe}(\mathrm{CO})_5,\left[\mathrm{Co}(\mathrm{CO})_3\right]^{3-},\left[\mathrm{Cr}(\mathrm{CO})_4\right]^{4-}\), and \(\operatorname{Ir}(\mathrm{CO})_3\), the total number of species isoelectronic with \(\mathrm{Ni}(\mathrm{CO})_4\) is ________.
[Given, atomic number:\(\mathrm{V}=23, \mathrm{Cr}=24, \mathrm{Mn}=25,\) \(\mathrm{Fe}=26, \mathrm{Co}=27, \mathrm{Ni}=28, \mathrm{Cu}=29,\) \(\mathrm{Ir}=77\)]

For an ideal gas, consider only P-V work in going from an initial state \(\mathrm{X}\) to the final state \(\mathrm{Z}\). The final state \(\mathrm{Z}\) can be reached by either of the two paths shown in the figure. Which of the following choice(s) is (are) correct?
[Take \(\Delta \mathrm{S}\) as change in entropy and w as work done].

An organic compound $\left({\mathrm{C}}_8{\mathrm{H}}_{10}{\mathrm{O}}_2\right)$ rotates plane-polarized light. It produces pink color with neutral ${\mathrm{FeCl}}_3$ solution. What is the total number of all the possible isomers for this compound?

In the following reactions, $\mathbf{P},\mathbf{Q},\mathbf{R}$ and $\mathbf{S}$ are the major products.




The correct statement about $\mathbf{P},\mathbf{Q},\mathbf{R}$ and $\mathbf{S}$ is

The total number of contributing structures showing hyperconjugation (involving \(\mathrm{C}-\mathrm{H}\) bonds) for the following carbocation is

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The key feature of Bohr's theory of spectrum of hydrogen atom is the quantization of angular momentum when an electron is revolving around a proton. We will extend this to a general rotational motion to find quantized rotational energy of a diatomic molecule assuming it to be rigid. The rule to be applied is Bohr's quantization condition.
Question:
It is found that the excitation frequency from ground to the first excited state of rotation for the \(\mathrm{CO}\) molecule is close to \(\frac{4}{\pi} \times 10^{11} \mathrm{~Hz}\). Then the moment of inertia of CO molecule about its centre of mass is close to (Take \(h=2 \pi \times 10^{-34} J-s\) )

Three numbers are chosen at random, one after another with replacement, from the set \(S=\{1,2,3, \ldots, 100\}\). Let \(p_{1}\) be the probability that the maximum of chosen numbers is at least 81 and \(p_{2}\) be the probability that the minimum of chosen numbers is at most 40 .
The value of $\frac{625}{4}{p}_1$ is

For non-negative integers $n,$ let
$f\left(n\right)=\frac{\sum _{k=0}^n\sin \left(\frac{k+1}{n+2}\pi \right)\sin \left(\frac{k+2}{n+2}\pi \right)}{\sum _{k=0}^n{\sin }^2\left(\frac{k+1}{n+2}\pi \right)}$
Assuming ${\cos }^{-1}x$ takes values in $\left[0, \pi \right],$ which of the following options is/are correct?

Consider the equation ${\int }_1^e\frac{{\left({\log }_{e}x\right)}^{{\displaystyle \frac{1}{2}}}}{x{\left(a-{\left({\log }_{e}x\right)}^{{\displaystyle \frac{3}{2}}}\right)}^2}dx=1,a\in \left(-\infty ,0\right)\cup \left(1,\infty \right)$. Which of the following statements is/are TRUE?