Scheme $1$ and $2$ describe the conversion of $\mathrm{P}$ to $\mathrm{Q}$ and $\mathrm{R}$ to $\mathrm{S},$ respectively. Scheme $3$ describes the synthesis of $\mathrm{T}$ from $\mathrm{Q}$ and $\mathrm{S}.$ The total number of $\mathrm{B}\mathrm{r}$ atoms in a molecule of $\mathrm{T}$ is
Scheme $1$:

Scheme $2$:

Scheme$3$:

The solubility of barium iodate in an aqueous solution prepared by mixing 200 mL of 0.010 M barium nitrate with 100 mL of 0.10 M sodium iodate is \(\boldsymbol{X} \times 10^{-6} \mathrm{~mol} \mathrm{dm}^{-3}\). The value of \(\boldsymbol{X}\) is \(\qquad\) .
Use: Solubility product constant \(\left(K_{\mathrm{sp}}\right)\) of barium iodate \(=1.58 \times 10^{-9}\)

The mole fraction of a solute in a solution is $0.1$. At $298 \mathrm{K}$, the molarity of this solution is the same as its molality. The density of this solution at $298 \mathrm{K}$ is $2.0\mathrm{ }\mathrm{g}\mathrm{ }\mathrm{c}{\mathrm{m}}^{–3}$. The ratio of the molecular weights of the solute and solvent, $\left(\frac{\mathrm{M}{\mathrm{W}}_{\mathrm{s}\mathrm{o}\mathrm{l}\mathrm{u}\mathrm{t}\mathrm{e}}}{\mathrm{M}{\mathrm{W}}_{\mathrm{s}\mathrm{o}\mathrm{l}\mathrm{v}\mathrm{e}\mathrm{n}\mathrm{t}}}\right)$, is:

The electrochemical cell shown below is a concentration cell. \(\mathrm{M} \mid \mathrm{M}^{2+}\) (saturated solution of a sparingly soluble salt, \(\left.\mathrm{MX}_{2}\right) \| \mathrm{M}^{2 \perp}\left(0.001 \mathrm{~mol} \mathrm{dm}^{-3}\right) \mid \mathrm{M}\).
The emf of the cell depends on the difference in concentrations of \(\mathrm{M}^{2+}\) ions at the two electrodes. The emf of the cell at \(298 \mathrm{~K}\) is \(0.059 \mathrm{~V}\).
Question:
The solubility product \(\left(\mathrm{K}_{s p} ; \mathrm{mol}^{3} \mathrm{dm}^{-9}\right)\) of \(\mathrm{MX}_{2}\) at \(298 \mathrm{~K}\) based on the information available for the given concentration cell is (take \(2.303 \times \mathrm{R} \times 298 / \mathrm{F}=0.059 \mathrm{~V}\) )

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]

The freezing point (in \({ }^{\circ} \mathrm{C}\) ) of solution containing \(0.1 \mathrm{~g}\) of \(\mathrm{K}_3\left[\mathrm{Fe}(\mathrm{CN})_6\right]\) (molecular weight 329\()\) in \(100 \mathrm{~g}\) of water \(\left(K_f=1.86 \mathrm{~K} \mathrm{~kg} \mathrm{~mol}^{-1}\right)\) is

A solution is prepared by mixing $0.01\mathrm{mol}$ each of ${\mathrm{H}}_2{\mathrm{CO}}_3,{\mathrm{NaHCO}}_3,{\mathrm{Na}}_2{\mathrm{CO}}_3$, and $\mathrm{NaOH}$ in \(100 \mathrm{~mL}\) of water. $\mathrm{pH}$ of the resulting solution is
[Given: ${\mathrm{pK}}_{{\mathrm{a}}_1}$ and ${\mathrm{pK}}_{{\mathrm{a}}_2}$ of ${\mathrm{H}}_2{\mathrm{CO}}_3$ are $6.37$ and $10.32$, respectively. $\log 2=0.30$]

The IUPAC name of the following compound, is

A source (S) of sound has frequency \(240 \mathrm{~Hz}\). When the observer \((\mathrm{O})\) and the source move towards each other at a speed \(v\) with respect to the ground (as shown in Case 1 in the figure), the observer measures the frequency of the sound to be \(288 \mathrm{~Hz}\). However, when the observer and the source move away from each other at the same speed \(v\) with respect to the ground (as shown in Case 2 in the figure), the observer measures the frequency of sound to be \(n \mathrm{~Hz}\). The value of \(n\) is ______

Let \(f_{1}:(0, \infty) \rightarrow \mathbb{R}\) and \(f_{2}:(0, \infty) \rightarrow \mathbb{R}\) be defined by

\[

f_{1}(x)=\int_{0}^{x} \prod_{j=1}^{21}(t-j)^{j} d t, \quad x>0

\]

and

\(

f_{2}(x)=98(x-1)^{50}-600(x-1)^{49}+2450, \quad x>0

\)

where, for any positive integer \(n\) and real numbers \(a_{1}, a_{2}, \ldots, a_{n}, \prod_{i=1}^{n} a_{i}\) denotes the product of \(a_{1}, a_{2}, \ldots, a_{n} .\) Let \(m_{i}\) and \(n_{i}\), respectively, denote the number of points of local minima and the number of points of local maxima of function \(f_{i}, i=1,2\), in the interval \((0, \infty)\).

The value of $2m_1+3n_1+m_1{n}_1$ is _____.

Let \(S_k, k=1,2, \ldots, 100\), denote the sum of the infinite geometric series whose first term is \(\frac{k-1}{k !}\) and the common ratio is \(\frac{1}{k}\). Then the value of \(\frac{100^2}{100 !}+\sum_{k=1}^{100}\left|\left(k^2-3 k+1\right) S_k\right|\) is

The normal at a point \(P\) on the ellipse \(x^2+4 y^2=16\) meets the \(x\)-axis at \(Q\). If \(M\) is the mid-point of the line segment \(P Q\), then the locus of \(M\) intersects the latusrectum of the given ellipse at the points

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