In the chemical reaction between stoichiometric quantities of ${\mathrm{KMnO}}_4$ $\mathrm{KI}$ in weakly basic solution, what is the number of moles of ${\mathrm{I}}_2$ released for $4$ ${\mathrm{KMnO}}_4$ consumed?

The geometries of the ammonia complexes of $\mathrm{N}{\mathrm{i}}^{2+}\mathrm{ },\mathrm{ }\mathrm{P}{\mathrm{t}}^{2+}\mathrm{ }\mathrm{a}\mathrm{n}\mathrm{d}\mathrm{ }\mathrm{Z}{\mathrm{n}}^{2+}$ respectively, are :

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

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Properties such as boiling point, freezing point and vapour pressure of a pure solvent change when solute molecules are added to get homogeneous solution. These are called colligative properties. Applications of colligative properties are very useful in day-to-day life. One of its examples is the use of ethylene glycol and water mixture as anti-freezing liquid in the radiator automobiles.
A solution \(M\) is prepared by mixing ethanol and water. The mole fraction of ethanol in the mixtrue is \(0.9\)
Given Freezing point depression constant of water \(\left(k^{\text {water }}\right)=1.86 \mathrm{~K} \mathrm{~kg} \mathrm{~mol}^{-1}\)
Freezing point depression constant of ethanol \(\left(k_f^{\text {ethanol }}\right)=2.0 \mathrm{Kkg} \mathrm{mol}^{-1}\)
Boiling point elevation constant of water \(\left(k_b^{\text {water }}\right)=0.52 \mathrm{Kkg} \mathrm{mol}^{-1}\)
Boiling point elevation constant of ethanol \(\left(k_b^{\text {ethanol }}\right)=1.2 \mathrm{Kkg} \mathrm{mol}^{-1}\)
Standard freezing point of water \(=273 \mathrm{~K}\)
Standard freezing point of ethanol \(=155.7 \mathrm{~K}\)
Standard boiling point of water \(=373 \mathrm{~K}\)
Standard boiling point of ethanol \(=351.5 \mathrm{~K}\)
Vapour pressure of pure water \(=328 \mathrm{~mm}\) of \(\mathrm{Hg}\)
Vapour pressure of pure ethanol \(=40 \mathrm{~mm}\) of \(\mathrm{Hg}\)
Molecular weight of water \(=18 \mathrm{~g} \mathrm{~mol}^{-1}\)
Molecular weight of ethanol \(=46 \mathrm{~g} \mathrm{~mol}^{-1}\)
In answering the following questions, consider the solutions to be ideal dilute solutions and solutes to be non-volatile and non-dissociative.

Question:
Water is added to the solution \(M\) such that the mole fraction of water in the solution becomes \(0.9\) The boiling point of this solutions is

Isomers of hexane, based on their branching, can be divided into three distinct classes as shown in the figure.

The correct order of their boiling point is

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 plot of ${\mathrm{logk}}_{\mathrm{f}}$ versus $\frac{1}{\mathrm{T}}$ for a reversible reaction $\mathrm{A}\left(\mathrm{g}\right)\rightleftharpoons \mathrm{P}\left(\mathrm{g}\right)$ is shown.

Pre-exponential factors for the forward and backward reactions are ${10}^{15} {\mathrm{s}}^{-1}$ and ${10}^{11} {\mathrm{s}}^{-1}$, respectively. If the value of $\mathrm{logK}$ for the reaction at $500 \mathrm{K}$ is $6$, the value of $\left|{\mathrm{logk}}_{\mathrm{b}}\right|$ at $250 \mathrm{K}$ is
$\left[\mathrm{K}=\right.$equilibrium constant of the reaction, ${\mathrm{k}}_{\mathrm{f}}=$rate constant of forward reaction, ${\mathrm{K}}_{\mathrm{b}}=$rate constant of backward reaction]

Consider a disc rotating in the horizontal plane with a constant angular speed \(\omega\) about its centre \(\mathrm{O}\). The disc has a shaded region on one side of the diameter and an unshaded region on the other side as shown in the figure. When the disc is in the orientation as shown, two pebbles \(P\) and \(Q\) are simultaneously projected at an angle towards \(R\). The velocity of projection is in the \(y\) - \(z\) plane and is same for both pebbles with respect to the disc. Assume that (i) they land back on the disc before the disc has completed \(1 / 8\) rotation, (ii) their range is less than half the disc radius, and (iii) \(\omega\) remains constant throughout. Then

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Let \(A\) be the set of all \(3 \times 3\) symmetric matrices all of whose entries are either 0 or 1 . Five of these entries are 1 and four of them are 0 .Question:
The number of matrices \(A\) in \(A\) for which the system of linear equations \(A\left[\begin{array}{l}x \\ y \\ z\end{array}\right]=\left[\begin{array}{l}1 \\ 0 \\ 0\end{array}\right]\) is is inconsistent, is

Let \(\alpha\) and \(\beta\) be the distinct roots of the equation \(x^2+x-1=0\). Consider the set \(T=\{1, \alpha, \beta\}\). For a \(3 \times 3\) matrix \(M=\left(a_{i j}\right)_{3 \times 3}\), define \(R_i=a_{i 1}+a_{i 2}+a_{i 3}\) and \(C_j=a_{1 j}+a_{2 j}+a_{3 j}\) for \(i=1,2,3\) and \(j=1,2,3\).
Match each entry in List-I to the correct entry in List-II.

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:

Highly excited states for hydrogen like atoms (also called Rydberg states) with nuclear charge Ze are defined by their principal quantum number n, where n >> 1. Which of the following statement(s) is (are) true?

Let RS be the diameter of the circle $x^2+y^2=1$ where, $S$ is the point$(1, 0)$. Let $P$be a variable point (other than $R \& S$) on the circle and tangents to the circle at $S \& P$meet at the point$Q$. The normal to the circle at P intersects a line drawn through$Q$ parallel to $RS$ at point $E$ . Then, the locus of $E$ passes through the point (s):