Let $m$ be the minimum possible value of ${\log }_3\left(3^{y_1}+3^{y_2}+3^{y_3}\right)$ , where $y_1, y_2, y_3$ are real numbers for which $y_1+y_2+y_3=9$. Let $M$ be the maximum possible value of $\left({\log }_3{x}_1+{\text{log}}_3 x_2+{\text{log}}_3 x_3\right)$, where $x_1, x_2, x_3$ are positive real numbers for which $x_1+x_2+x_3=9$. Then the value of ${\text{log}}_2 \left(m^3\right)+{\text{log}}_3\left(M^2\right)$ is

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Let \(p\) be an odd prime number and \(T_p\) be the following set of \(2 \times 2\) matrices
\[
T_p=\left\{A=\left[\begin{array}{ll}
a & b \\
c & a
\end{array}\right] ; a, b, c \in\{0,1,2, \ldots, p-1\}\right\}
\]Question:
The number of \(A\) in \(T_p\) such that \(\operatorname{det}(A)\) is not divisible by \(p\), is

Consider the matrix, \(P=\left(\begin{array}{lll}2 & 0 & 0 \\ 0 & 2 & 0 \\ 0 & 0 & 3\end{array}\right)\)
Let the transpose of a matrix \(X\) be denoted by \(\mathrm{X}^T\). Then the number of \(3 \times 3\) invertible matrices Q with integer entries, such that \(Q^{-1}=Q^T\) and \(P Q=Q P\) is

Let \((x, y, z)\) be points with integer coordinates satisfying the system of homogeneous equations \(3 x-y-z=0,-3 x+z=0,-3 x+2 y+z=0\). Then, the number of such points for which \(x^2+y^2+z^2 \leq 100\) is

Let $S$ be the of all column matrices $\left[\begin{array}{c}{b}_1\\ b_2\\ b_3\end{array}\right]$ such that $b_1, b_2, b_3\in \mathrm{ℝ}$ and the system of equations (in real variables)
$\begin{array}{c}-x+2y+5z=b_1\\ 2x-4y+3z=b_2\\ x-2y+2z=b_3\end{array}$
has at least one solution. Then, which of the following system(s) (in real variables) has (have) at least one solution of each $\left[\begin{array}{c}{b}_1\\ b_2\\ b_3\end{array}\right]ϵ S$ ?

Let $f\left(x\right)=7{\tan }^{8}x+7{\tan }^{6}x-3{\tan }^{4}x-3{\tan }^{2}x$ for all $x\in \left(-\frac{\pi }{2},\frac{\pi }{2}\right).$ Then the correct expression(S) is(are)

The following integral $\underset{\frac{\mathrm{\pi }}{4}}{\overset{\frac{\mathrm{\pi }}{2}}{\int }}{\left(2\mathrm{ }\mathrm{c}\mathrm{o}\mathrm{s}\mathrm{e}\mathrm{c}\mathrm{ }\mathrm{x}\right)}^{17}\mathrm{ }\mathrm{d}\mathrm{x}$ is equal to

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\(RCONH _2\) is converted into \(RNH _2\) by means of Hofmann bromamide degradation.

In this reaction, \(R \mathrm{CONHBr}\) is formed from which this reaction has derived its name. Electron donating group at phenyl activates the reaction. Hofmann degradation reaction is an intramolecular reaction.
Question:
How can the conversion of (i) to (ii) be brought about?

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

Consider a thin square sheet of side \(L\) and thickness \(t\), made of a material of resistivity \(\rho\). The resistance between two opposite faces, shown by the shaded areas in the figure is

Extraction of metal from the ore cassiterite involves

The molar conductivity of a solution of a weak acid HX (0.01 M) is 10 times smaller than the molar conductivity of a solution of a weak acid HY (0.10 M). If ${\mathrm{\lambda }}_{{\mathrm{X}}^{-}}^0\approx {\mathrm{\lambda }}_{{\mathrm{Y}}^{-}}^0$ , the difference in their $\mathrm{p}{\mathrm{K}}_{\mathrm{a}}$ values, $\mathrm{p}{\mathrm{K}}_{\mathrm{a}}\left(\mathrm{H}\mathrm{X}\right)-\mathrm{p}{\mathrm{K}}_{\mathrm{a}}\left(\mathrm{H}\mathrm{Y}\right)$ , is (Consider degree of ionization of both acids to << 1)

A thin rod of mass $M$ and length $a$ is free to rotate in horizontal plane about a fixed vertical axis passing through point $O$. A thin circular disc of mass $M$ and of radius $\frac{a}{4}$ is pivoted on this rod with its centre at a distance $\frac{a}{4}$ from the free end so that it can rotate freely about its vertical axis, as shown in the figure. Assume that both the rod and the disc have uniform density and they remain horizontal during the motion. An outside stationary conserver finds the rod rotating with an angular velocity $\Omega$ and the disc rotating about its vertical axis with angular velocity $4\Omega$. The total angular momentum of the system about the point $O$ is $\left[\left(\frac{M{a}^2\Omega }{48}\right)n\right]$. The value of $n$ is _________ .