An electron in an excited state of $\mathrm{L}{\mathrm{i}}^{2+}$ ion has angular momentum $\frac{3\mathrm{h}}{2\mathrm{\pi }}$ . The de Broglie wavelength of the electron in this state is $\mathrm{p}\mathrm{\pi }{\mathrm{\alpha }}_0$ (where ${\mathrm{a}}_0$ is the Bohr radius). The value of $\mathrm{p}$ is

A thin conducting rod $MN$ of mass $20 \mathrm{gm}$, length $25 \mathrm{cm}$ and resistance $10\Omega$ is held on frictionless, long, perfectly conducting vertical rails as shown in the figure. There is a uniform magnetic field $B_0=4 \mathrm{T}$ directed perpendicular to the plane of the rod-rail arrangement. The rod is released from rest at time $t=0$ and it moves down along the rails. Assume air drag is negligible. Match each quantity in List-$\mathrm{I}$ with an appropriate value from List-$\mathrm{II}$, and choose the correct option. [Given: The acceleration due to gravity $g=10 {\mathrm{ms}}^{-2}$ and $e^{-1}=0.4$]  

	List-$\mathrm{I}$		List-$\mathrm{II}$
$\mathrm{P}$	At $t=0.2 \mathrm{s}$, the magnitude of the induced emf in Volt	$1$	$0.07$
$\mathrm{Q}$	At $t=0.2 \mathrm{s}$, the magnitude of the magnetic force in Newton	$2$	$0.14$
$\mathrm{R}$	At $t=0.2 \mathrm{s}$, the power dissipated as heat in Watt	$3$	$1.20$
$\mathrm{S}$	The magnitude of terminal velocity of the rod in $\mathrm{m} {\mathrm{s}}^{-1}$	$4$	$0.12$
		$5$	$2.00$

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

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When a particle is restricted to move along \(x\)-axis between \(x=0\) and \(x=a\), where \(a\) is of nanometer dimension, its energy can take only certain specific values. The allowed energies of the particle moving in such a restricted region, correspond to the formation of standing waves with nodes at its ends \(x=0\) and \(x=a\). The wavelength of this standing wave is related to the liner momentum \(p\) of the particle according to the de Broglie relation. The energy of the particle of mass \(m\) is related to its linear momentum as \(E=\frac{p^2}{2 m}\). Thus, the energy of the particle can be denoted by a quantum number \(n\) taking values \(1,2,3, \ldots(n=1\), called the ground state) corresponding to the number of loops in the standing wave.
Use the model described above to answer the following three questions for a particle moving in the line \(x=0\) to \(x=a\) [Take \(h=6.6 \times 10^{-34} \mathrm{Js}\) and \(e=1.6 \times 10^{-19} \mathrm{C}\) ]
Question:
If the mass of the particle is \(m=1.0 \times 10^{-30} \mathrm{~kg}\) and \(a=6.6 \mathrm{~nm}\), the energy of the particle in its ground state is closest to

In order to measure the internal resistance $r_1$ of a cell of emf $\epsilon ,$ a meter bridge of wire resistance $R_0=50 \mathrm{\Omega }$, a resistance $\frac{R_0}{2}$, another cell of emf $\frac{\epsilon }{2}$ (internal resistance $r$) and a galvanometer $G$ are used in a circuit, as shown in the figure. If the null point is found at $l=72 \mathrm{cm},$ then the value of $r_1=\_\_\_\_\mathrm{\Omega }.$

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A wooden cylinder of diameter \(4 r\), height \(H\) and density \(\rho / 3\) is kept on a hole of diameter \(2 r\) of a tank, filled with liquid of density \(\rho\) as shown in the figure.

Question :
If height \(h_2\) of water level is further decreased, then

Statement I In an elastic collision between two bodies, the relative speed of the bodies after collision is equal to the relative speed before the collision.
Statement II In an elastic collision, the linear momentum of the system is conserved.

\(29.2 \%(\mathrm{w} / \mathrm{w}) \mathrm{HCl}\) stock solution has a density of \(1.25 \mathrm{gmL}^{-1}\). The molecular weight of \(\mathrm{HCl}\) is \(36.5 \mathrm{~g} \mathrm{~mol}^{-1}\). The volume \((\mathrm{mL})\) of stock solution required to prepare a 200 \(\mathrm{mL}\) solution of \(0.4 \mathrm{M} \mathrm{HCl}\) is :

Match List I with List II and select the correct answer using the codes given below the lists:

	\(\text{List I}\)		\(\text{List II}\)
a.	Boltzmann constant	p.	\(\left[M L^2 T^{-1}\right]\)
b.	Coefficient of viscosity	q.	\(\left[M L^{-1} T^{-1}\right]\)
c.	Planck constant	r.	\(\left[M L T^{-3} K^{-1}\right]\)
d.	Thermal conductivity	s.	\(\left[M L^2 T^{-2} K^{-1}\right]\)

Let $S=\left(0,1\right)\cup \left(1, 2\right)\cup \left(3, 4\right)$ and $T=\left\{0, 1, 2, 3\right\}$. Then which of the following statements is(are) true?

Consider three sets $E_1=\{1,2,3\},F_1=\{1,3,4\}$ and $G_1=\{2,3,4,5\}$. Two elements are chosen at random, without replacement, from the set $E_1$, and let $S_1$ denote the set of these chosen elements. Let $E_2=E_1-S_1$ and $F_2=F_1\cup S_1$. Now two elements are chosen at random, without replacement, from the set $F_2$ and let $S_2$ denote the set of these chosen elements. Let $G_2=G_1\cup S_2$. Finally, two elements are chosen at random, without replacement, from the set $G_2$ and let $S_3$ denote the set of these chosen elements.
Let $E_3=E_2\cup S_3$. Given that $E_1=E_3$, let $p$ be the conditional probability of the event $S_1=\{1,2\}$. Then the value of $p$ is

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The circle \(x^2+y^2-8 x=0\) and hyperbola \(\frac{x^2}{9}-\frac{y^2}{4}=1\) intersect at the points \(A\) and \(B\).Question:
Equation of the circle with \(A B\) as its diameter is

In the reaction scheme shown below,$\mathrm{Q},\mathrm{R},$ and$\mathrm{S}$ are the major products.

The correct structure of