The sum of the spin only magnetic moment values (in B.M.) of \(\left[\mathrm{Mn}(\mathrm{Br})_6\right]^{3-}\) and \(\left[\mathrm{Mn}(\mathrm{CN})_6\right]^{3-}\) is _____.

The initial rate of hydrolysis of methyl acetate$\left(1\mathrm{M}\right)$ by a weak acid $(\mathrm{HA}, 1\mathrm{M})$ is $1/{100}^{\mathrm{th}}$ of that by a strong acid $\left(\mathrm{HX}, 1\mathrm{M}\right)$, at $25°\mathrm{C}$. The ${\mathrm{K}}_{\mathrm{a}}$ of $\mathrm{HA}$ is

Paragraph:
In hexagonal system of crystals, a frequently encountered arrangement of atoms is described as a hexagonal prism. Here, the top and bottom of the cell are regular hexagons and three atoms are sandwiched in between them. A space-filling model of this structure, called hexagonal close-packed (HCP), is constituted of a sphere on a flat surface surrounded in the same plane by six identical spheres as closely as possible. Three spheres are then placed over the first layer so that they touch each other and represent the second layer. Each one of the three spheres touches three spheres of the bottom layer. Finally, the second layer is covered with a third layer that is identical to the bottom layer in relative position. Assume radius of every sphere to be ' \(r\) '.
Question:
The volume of this \(\mathrm{HCP}\) unit cell is

According to Molecular Orbital Theory,

Which of the following statement(s) (are) correct regarding the root mean square speed $\left({\mathrm{U}}_{\mathrm{r}\mathrm{m}\mathrm{s}}\right)$ and average translational kinetic energy $\left({\mathrm{\epsilon }}_{\mathrm{a}\mathrm{v}}\right)$ of a molecule in a gas at equilibrium?

The cyanide process of gold extraction involves leaching out gold from its ore with ${\mathrm{C}\mathrm{N}}^{-}$ in the presence of $\mathrm{Q}$ in water to form $\mathrm{R}$ . Subsequently, $\mathrm{R}$ is treated with $\mathrm{T}$ to obtain $\mathrm{A}\mathrm{u}$ and $\mathrm{Z}$ . Choose the correct option(s).

Among the following complexes, the total number of diamagnetic species is_____
\(\left[\mathrm{Mn}\left(\mathrm{NH}_3\right)_6\right]^{3+},\left[\mathrm{MnCl}_6\right]^{3-},\left[\mathrm{FeF}_6\right]^{3-},\left[\mathrm{CoF}_6\right]^{3-},\)\(\left[\mathrm{Fe}\left(\mathrm{NH}_3\right)_6\right]^{3+}\), and \(\left[\mathrm{Co}(\mathrm{en})_3\right]^{3+}\)
[Given, atomic number: \(\mathrm{Mn}=25, \mathrm{Fe}=26, \mathrm{Co}=27\);
\(\text { en } \left.=\mathrm{H}_2 \mathrm{NCH}_2 \mathrm{CH}_2 \mathrm{NH}_2\right]\)

Paragraph:
If a continuous \(f\) defined on the real line \(R\), assume positive and negative values in \(R\), then the equation \(f(x)=0\) has a root in \(R\). For example, if it is known that a continuous function \(f\) on \(R\) is positive at some point and its minimum values is negative, then the equation \(f(x)=0\) has a root in \(R\).
Consider \(f(x)=k e^x-x\) for all real \(x\), where \(k\) is real constant.Question:
The line \(y=x\) meets \(y=k e^x\) for \(k \leq 0\) at

Paragraph: Let \(f:\left[0, \frac{\pi}{2}\right] \rightarrow[0,1]\) be the function defined by \(f(x)=\sin ^2 x\) and let \(g:\left[0, \frac{\pi}{2}\right] \rightarrow[0, \infty)\) be the function defined by \(g(x)=\sqrt{\frac{\pi x}{2}-x^2}\).
Question: The value of \(2 \int_0^{\frac{\pi}{2}} f(x) g(x) d x-\int_0^{\frac{\pi}{2}} g(x) d x\) is

A piece of ice (heat capacity \(=2100 \mathrm{~J} \mathrm{~kg}^{-1}{ }^{\circ} \mathrm{C}^{-1}\) and latent heat \(=3.36 \times 10^5 \mathrm{~J} \mathrm{~kg}^{-1}\) ) of mass \(\mathrm{m}\) gram is at \(-5^{\circ} \mathrm{C}\) at atmospheric pressure. It is given \(420 \mathrm{~J}\) of heat so that the ice starts melting. Finally when the ice-water mixture is in equilibrium, it is found that \(1 \mathrm{~g}\) of ice has melted. Assuming there is no other heat exchange in the process, the value of \(m\) is

Let $M=\left(a_{ij}\right), i, j\in \{1,2,3\}$, be the $3\times 3$ matrix such that $a_{ij}=1$ if $j+1$ is divisible by $i$, otherwise $a_{ij}=0$. Then which of the following statements is(are) true?

In the figure, a ladder of mass m is shown leaning against a wall. It is in static equilibrium making an angle $\mathrm{\theta }$ with the horizontal floor. The coefficient of friction between the wall and the ladder is ${\mathrm{\mu }}_1$ and that between the floor and the ladder is ${\mathrm{\mu }}_2$ . The normal reaction of the wall on the ladder is ${\mathrm{N}}_1$ and that of the floor is ${\mathrm{N}}_2$ If the ladder is about to slip, then

A trinitro compound, $1,3,5$-tris-($4$-nitrophenyl) benzene, on complete reaction with an excess of $\mathrm{Sn}/\mathrm{HCl}$ gives a major product, which on treatment with an excess of ${\mathrm{NaNO}}_2/\mathrm{HCl}$ at $0°\mathrm{C}$ provides $\mathbf{P}$ as the product. $\mathbf{P}$, upon treatment with excess of ${\mathrm{H}}_2\mathrm{O}$ at room temperature, gives the product $\mathbf{Q}$. Bromination of $\mathbf{Q}$ in aqueous medium furnishes the product $\mathbf{R}$. The compound $\mathbf{P}$ upon treatment with an excess of phenol under basic conditions gives the product $\mathbf{S}$. The molar mass difference between compounds $\mathbf{Q}$ and $\mathbf{R}$ is $474 \mathrm{g} {\mathrm{mol}}^{-1}$ and between compounds $\mathbf{P}$ and $\mathbf{S}$ is $172.5 \mathrm{g} {\mathrm{mol}}^{-1}$.
The number of heteroatoms present in one molecule of \(R\) is [Use: Molar mass (in \(\text{g mol} -1\) ): \(H =1,\) \(C =12, N=14, O =16,\) \(Br =80, Cl =35.5\) Atoms other than C and H are considered as heteroatoms]