The correct statement about the following disaccharide is

In an atom, the total number of electrons having quantum numbers $\mathrm{n}=4,\mathrm{ }\left|{\mathrm{m}}_{\mathcal{l}}\right|=1$ and ${\mathrm{m}}_{\mathrm{s}}=–1/2$ is

For 'invert sugar', the correct statement(s) is (are)
(Given: specific rotations of $\left(+\right)-$sucrose, $\left(+\right)-$maltose, $\mathrm{L}-\left(-\right)-$glucose and $\mathrm{L}-\left(+\right)-$fructose in aqueous solution are $+\mathrm{ }{66}^{°},\mathrm{ }+{140}^{°},\mathrm{ }–{52}^{°}$ and $+\mathrm{ }{92}^{°}$, respectively)

An organic compound undergoes first-order decomposition. The time taken for its decomposition to \(1 /\) 8 and \(1 / 10\) of its initial concentration are \(t_{1 / 8}\) and \(t_{1 / 10}\) respectively. What is the value of \(\left[\frac{t_{1 / 8}}{t_{1 / 10}}\right] \times 10\) ? \(\left(\log _{10} 2=0.3\right)\)

The diffusion coefficient of an ideal gas is proportional to its mean free path and mean speed. The absolute temperature of an ideal gas is increased $4$ times and its pressure $2$ times. As a result, the diffusion coefficient of this gas increases $\mathrm{x}$ times. The value of $\mathrm{x}$ is:

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]

Some standard electrode potentials at $298 \mathrm{K}$ are given below: $\begin{array}{ll}{\mathrm{Pb}}^{2+}/\mathrm{Pb}& -0.13 \mathrm{V}\end{array}$ $\begin{array}{ll}{\mathrm{Ni}}^{2+}/\mathrm{Ni}& -0.24 \mathrm{V}\end{array}$ $\begin{array}{ll}{\mathrm{Cd}}^{2+}/\mathrm{Cd}& -0.40 \mathrm{V}\end{array}$ $\begin{array}{ll}{\mathrm{Fe}}^{2+}/\mathrm{Fe}& -0.44 \mathrm{V}\end{array}$ To a solution containing $0.001 \mathrm{M}$ of ${\mathbf{X}}^{2+}$ and $0.1 \mathrm{M}$ of ${\mathbf{Y}}^{2+}$, the metal rods $\mathbf{X}$ and $\mathbf{Y}$ are inserted $($at $298 \mathrm{K})$ and connected by a conducting wire. This resulted in dissolution of $\mathbf{X}$. The correct combination(s) of $\mathbf{X}$ and $\mathbf{Y}$, respectively, is(are) (Given: Gas constant, $\mathrm{R}=8.314 \mathrm{J}{ \mathrm{K}}^{-1}{ \mathrm{mol}}^{-1}$, Faraday constant, $\mathrm{F}=96500 \mathrm{C} {\mathrm{mol}}^{-1}$)

An open-ended U-tube of uniform cross-sectional area contains water (density ${10}^3\mathrm{kg}{\mathrm{m}}^{-3}$ ). Initially the water level stands at $0.29\mathrm{m}$ from the bottom in each arm. Kerosene oil (a water-immiscible liquid) of density $800\mathrm{kg}{\mathrm{m}}^{-3}$ is added to the left arm until its length is $0.1\mathrm{m}$, as shown in the schematic figure below. The ratio $\left(\frac{h_1}{h_2}\right)$ of the heights of the liquid in the two arms is

Match the nuclear processes given in column I with the appropriate option(s) in column II.

	Column I		Column II
A.	Nuclear fusion	P	Absorption of thermal neutrons by ${}_{92}{}^{235}\mathrm{U}$
B.	Fission in a nuclear reactor	Q	${}_{27}{}^{60}\mathrm{C}\mathrm{o}$ nucleus
C.	$\mathrm{\beta }$ -decay	R	Energy production in stars via hydrogen conversion to helium
D.	$\mathrm{\gamma }$ -ray emission	S	Heavy water
		T	Neutrino emission

Paragraph :
Two plane harmonic sound waves are expressed by the equations.
\(
\begin{aligned}
& y_1(x, t)=A \cos (0.5 \pi x-100 \pi t) \\
& y_2(x, t)=A \cos (0.46 \pi x-92 \pi t)
\end{aligned}
\)
(All parameters are in MKS)
Question :
What is the speed of the sound?

Paragraph :
A special metal \(S\) conducts electricity without any resistance. A closed wire loop, made of \(S\), does not allow any change in flux through itself by inducing a suitable current to generate a compensating flux. The induced current in the loop cannot decay due to its zero resistance. This current gives rise to a magnetic moment which in turn repels the source of magnetic field or flux. Consider such a loop, of radius \(a\), with its center at the origin. A magnetic dipole of moment \(m\) is brought along the axis of this loop from infinity to a point at distance \(r(\gg a)\) from the center of the loop with its north pole always facing the loop, as shown in the figure below.

The magnitude of magnetic field of a dipole \(m\), at a point on its axis at distance \(r\), is \(\frac{\mu_{0}}{2 \pi} \frac{m}{r^{3}}\), where \(\mu_{0}\) is the permeability of free space. The magnitude of the force between two magnetic dipoles with moments, \(m_{1}\) and \(m_{2}\), separated by a distance \(r\) on the common axis, with their north poles facing each other, is \(\frac{k m_{1} m_{2}}{r^{4}}\), where \(k\) is a constant of appropriate dimensions. The direction of this force is along the line joining the two dipoles.

Question :
The work done in bringing the dipole from infinity to a distance$r$from the centre of the loop by the given process is proportional to:

A drop of liquid of radius $R={10}^{-2}m$ having surface tension $S=\frac{0.1}{4\pi }N{m}^{-1}$ divides itself into K identical drops. In the process the total change in the surface energy $\mathrm{\Delta }U={10}^{-3}J$ . If $K={10}^{\alpha }$ then the value of $\alpha$ is

Let \(A B C\) be a triangle such that \(\angle A C B=\frac{\pi}{6}\) and let \(a, b\) and \(c\) denote the lengths of the sides opposite to \(A, B\) and \(C\) respectively. The value(s) of \(x\) for which \(a=x^2+x+1, b=x^2-1\) and \(c=2 x+1\) is (are)