The correct statement(s) regarding,
(i) $\mathrm{H}\mathrm{C}\mathrm{l}\mathrm{O}$
(ii) $\mathrm{H}\mathrm{C}\mathrm{l}{\mathrm{O}}_2$
(iii) $\mathrm{H}\mathrm{C}\mathrm{l}{\mathrm{O}}_3$ and
(iv) $\mathrm{H}\mathrm{C}\mathrm{l}{\mathrm{O}}_4$ is /are

The oxidation number of \(\mathrm{Mn}\) in the product of alkaline oxidative fusion of \(\mathrm{MnO}_2\) is

In the following monobromination reaction, the number of possible chiral products is

For one mole of a van der Waal's gas when \(b=0\) and \(\mathrm{T}=300 \mathrm{~K}\), the PV vs, \(1 / \mathrm{V}\) plot is shown below. The value of the van der Waal's constant \(a\) (atm. liter \({ }^{2} \mathrm{~mol}^{-2}\) ) is :

Match the rate expressions in LIST-I for the decomposition of \(\mathrm{X}\) with the corresponding profiles provided in LIST-II. \(X_s\) and \(\mathrm{k}\) are constants having appropriate units.

Consider the following reaction,
\(2 \mathrm{H}_2(\mathrm{~g})+2 \mathrm{NO}(\mathrm{g}) \rightarrow \mathrm{N}_2(\mathrm{~g})+2 \mathrm{H}_2 \mathrm{O}(\mathrm{g})\)
which follows the mechanism given below:
\(2 \mathrm{NO}(\mathrm{g}) \stackrel{k_1}{\underset{k_{-1}}{\rightleftharpoons}} \mathrm{N}_2 \mathrm{O}_2(\mathrm{~g})\) \(\quad\) (fast equlibrium)
\(\mathrm{N}_2 \mathrm{O}_2(\mathrm{~g})+\mathrm{H}_2(\mathrm{~g}) \xrightarrow{k_2} \mathrm{~N}_2 \mathrm{O}(\mathrm{g})+\mathrm{H}_2 \mathrm{O}(\mathrm{g})\) \(\quad\)(slow reaction)
\(\mathrm{N}_2 \mathrm{O}(\mathrm{g})+\mathrm{H}_2(\mathrm{~g}) \xrightarrow{k_3} \mathrm{~N}_2(\mathrm{~g})+\mathrm{H}_2 \mathrm{O}(\mathrm{g})\) \(\quad\) (fast reaction)
The order of the reaction 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:
Tangents are drawn from the point \(P(3,4)\) to the ellipse \(\frac{x^2}{9}+\frac{y^2}{4}=1\) touching the ellipse at points \(A\) and \(B\).Question:
The coordinates of \(A\) and \(B\) are

Let \(f(x)\) be differentiable on the interval \((0, \infty)\) such that \(f(1)=1\), and \(\lim _{t \rightarrow x} \frac{t^2 f(x)-x^2 f(t)}{t-x}=1\) for each \(x>0\). Then, \(f(x)\) is

A free hydrogen atom after absorbing a photon of wavelength ${\mathrm{\lambda }}_{\mathrm{a}}$ gets excited from the state $\mathrm{n}=1$ to the state $\mathrm{n}=4.$ Immediately after that the electron jumps to $\mathrm{n}=\mathrm{m}$ state by emitting a photon of wavelength ${\mathrm{\lambda }}_{\mathrm{e}}.$ Let the change in momentum of atom due to the absorption and the emission are $\mathrm{\Delta }{\mathrm{p}}_{\mathrm{a}}$ and $\mathrm{\Delta }{\mathrm{p}}_{\mathrm{e}},$ respectively. If $\frac{{\mathrm{\lambda }}_{\mathrm{a}}}{{\lambda }_e}=\frac{1}{5}.$ Which of the option(s) is/are correct?
[Use $\mathrm{h}\mathrm{c}=1242\mathrm{e}\mathrm{V}\mathrm{n}\mathrm{m},1\mathrm{n}\mathrm{m}={10}^{-9}\mathrm{m},$ $\mathrm{h}$ and $\mathrm{c}$ are Planck's constant and speed of light, respectively]

Let $H:\frac{x^2}{a^2 }-\frac{y^2}{b^2}=1$ , where $a>b>0$, be a hyperbola in the xy-plane whose conjugate axis $LM$ subtends an angle of ${60}^o$ at one of its vertices $N$. Let the area of the triangle $LMN$ be $4\sqrt{3}$.

LIST-I	LIST-II
A. The length of the conjugate axis of $H$ is	P. $8$
B. The eccentricity of $H$ is	Q. $\frac{4}{\sqrt{3}}$
C. The distance between the foci of $H$ is	R. $\frac{2}{\sqrt{3}}$
D. The length of the latus rectum of $H$ is	S. $4$

The correct option is:

A rectangular sheet of fixed perimeter with sides having their lengths in the ratio $8:15$ is converted into an open rectangular box by folding after removing squares of equal area from all four corners. If the total area of removed squares is $100$, the resulting box has maximum volume. Then the lengths of the sides of the rectangular sheet are:

Inradius of a circle which is inscribed in an isosceles triangle one of whose angle is \(2 \pi / 3\), is \(\sqrt{3}\), then area of triangle is