Which of the following combinations will produce ${\text{H}}_2$ gas?

The option(s) with only amphoteric oxides is(are)

Consider the Bohr's model of a one-electron atom where the electron moves around the nucleus. In the following List-I contains some quantities for the $n^{\mathrm{t}\mathrm{h}}$ orbit of the atom and List-II contains options showing how they depend on $n$ .
 

List - I	List - II
(I) Radius of the $n^{\mathrm{t}\mathrm{h}}$ orbit	$\left(\mathrm{P}\right)$ $\propto n^{-2}$
(II) Angular momentum of the electron in the $n^{th}$ orbit	$\left(\mathrm{Q}\right)$ $\propto n^{-1}$
(III) Kinetic energy of the electron in the $n^{\mathrm{t}\mathrm{h}}$ orbit	$\left(\mathrm{R}\right)$ $\propto n^0$
(IV) Potential energy of the electron in the $n^{\mathrm{t}\mathrm{h}}$ orbit	$\left(\mathrm{S}\right)$ $\propto n^1$
	$\left(\mathrm{T}\right)$ $\propto n^2$
	$\left(\mathrm{U}\right)$ $\propto n^{1/2}$

Which of the following options has the correct combination considering List-I and List-II?

With respect to hypochlorite, chlorate and perchlorate ions, choose the correct statement(s).

If the bond length of \(\mathrm{CO}\) bond in carbon monoxide is \(1.128 \mathrm{~A}^{\circ}\), then what is the value of \(\mathrm{CO}\) bond length in \(\mathrm{Fe}(\mathrm{CO})_5\) ?

When \(20 \mathrm{~g}\) of naphthoic acid \(\left(\mathrm{C}_{11} \mathrm{H}_8 \mathrm{O}_2\right)\) is dissolved in \(50 \mathrm{~g}\) of benzene \(\left(K_f=1.72 \mathrm{~K} \mathrm{~kg} \mathrm{~mol}^{-1}\right)\), a freezing point depression of \(2 \mathrm{~K}\) is observed. The van't Hoff factor \((i)\) is

Match each of the compounds is Column I with its characteristic reaction(s) is Column II.

Two players, $P_1$ and $P_2$, play a game against each other. In every round of the game, each player rolls a fair die once, where the six faces of the die have six distinct numbers. Let $x$ and $y$ denote the readings on the die rolled by $P_1$ and $P_2$, respectively. If $x>y$, then $P_1$ scores $5$ points and $P_2$ scores $0$ point. If $x=y$, then each player scores $2$ points. If $x<y$, then $P_1$ scores $0$ point and $P_2$ scores $5$ points. Let $X_i$ and $Y_i$ be the total scores of $P_1$ and $P_2$, respectively, after playing the $i^{\mathrm{th}}$ round.

	List-$\mathrm{I}$		List-$\mathrm{II}$
$\left(\mathrm{I}\right)$	Probability of $\left(X_2\ge Y_2\right)$ is	$\left(\mathrm{P}\right)$	$\frac{3}{8}$
$\left(\mathrm{II}\right)$	Probability of $\left(X_2>Y_2\right)$ is	$\left(\mathrm{Q}\right)$	$\frac{11}{16}$
$\left(\mathrm{III}\right)$	Probability of $\left(X_3=Y_3\right)$ is	$\left(\mathrm{R}\right)$	$\frac{5}{16}$
$\left(\mathrm{IV}\right)$	Probability of $\left(X_3>Y_3\right)$ is	$\left(\mathrm{S}\right)$	$\frac{355}{864}$
		$\left(\mathrm{T}\right)$	$\frac{77}{432}$

The correct option is:

In a non-right-angled triangle $\mathrm{\Delta }PQR,$ let $p\mathit{,}q\mathit{,}r$ denote the lengths of the sides opposite to the angles at $P,Q,R$ respectively. The median from $R$ meets the side $PQ$ at $S,$ the perpendicular from $P$ meets the side $QR$ at $E$, and $RS$ and $PE$ intersect at $0.$ If $p=\sqrt{3},q=1,$ and the radius of the circumcircle of the $\mathrm{\Delta }PQR$ equals $1,$ then which of the following options is/are correct?

\(\lim _{x \rightarrow \frac{\pi}{4}} \frac{\int_2^{\sec ^2 x} f(t) d t}{x^2-\frac{\pi^2}{16}}\) equals

A current carrying infinitely long wire is kept along the diameter of a circular wire loop, without touching it, the correct statement(s) is(are)

A cylindrical furnace has height $\left(H\right)$ and diameter $\left(D\right)$ both $1 \mathrm{m}$. It is maintained at temperature $360 \mathrm{K}$. The air gets heated inside the furnace at constant pressure $P_a$ and its temperature becomes $T=360 \mathrm{K}$. The hot air with density $\rho$ rises up a vertical chimney of diameter $d=0.1 \mathrm{m}$ and height $h=9 \mathrm{m}$ above the furnace and exits the chimney (see the figure). As a result, atmospheric air of density ${\rho }_a=1.2 \mathrm{kg} {\mathrm{m}}^{-3}$, pressure $P_a$ and temperature $T_a=300 \mathrm{K}$ enters the furnace. Assume air as an ideal gas, neglect the variations in $\rho$ and $T$ inside the chimney and the furnace. Also ignore the viscous effects.
[Given: The acceleration due to gravity $g=10 \mathrm{m} {\mathrm{s}}^{-2}$ and $\pi =3.14$]

Considering the air flow to be streamline, the steady mass flow rate of air exiting the chimney is _____ $\mathrm{gm} {\mathrm{s}}^{-1}$.

Let $\stackrel{r}{7\overbrace{5\dots 5}7}$ denote the $\left(r+2\right)$ digit number where the first and the last digits are $7$ and the remaining $r$ digits are $5$. Consider the sum $S=77+757+7557+\dots +\stackrel{98}{7\overbrace{5\dots 5}7}$. If $S= \frac{7{\displaystyle \stackrel{99}{\overbrace{5\dots 5 }}}7+m}{n}$, where $m$ and $n$ are natural numbers less than $3000$, then the value of $m+n$ is