A steady current $I$ flows along an infinitely long hollow cylindrical conductor of radius $R$. This cylinder is placed co-axially inside an infinite solenoid of radius $2R$. The solenoid has $n$ turns per unit length and carries a steady current $I$. Consider a point $P$ at a distance $r$ from the common axis. The correct statement(s) is(are):

$S_1$ and $S_2$ are two identical sound sources of frequency $656 \mathrm{Hz}$. The source $S_1$ is located at $O$ and $S_2$ moves anti-clockwise with a uniform speed $4\sqrt{2} \mathrm{m} {\mathrm{s}}^{-1}$ on a circular path around $O$, as shown in the figure. There are three points $P, Q$ and $R$ on this path such that $P$ and $R$ are diametrically opposite while $Q$ is equidistant from them. A sound detector is placed at point $P$. The source $S_1$ can move along direction $OP$.
[Given: The speed of sound in air is $324 \mathrm{m} {\mathrm{s}}^{-1}$]

When only $S_2$ is emitting sound and it is at $Q$, the frequency of sound measured by the detector in $\mathrm{Hz}$ is _____.

A projectile of mass 200 g is launched in a viscous medium at an angle \(60^{\circ}\) with the horizontal, with an initial velocity of \(270 \mathrm{~m} / \mathrm{s}\). It experiences a viscous drag force \(\vec{F}=-c \vec{v}\) where the drag coefficient \(c=0.1 \mathrm{~kg} / \mathrm{s}\) and \(\vec{v}\) is the instantaneous velocity of the projectile. The projectile hits a vertical wall after 2 s . Taking \(e=2.7\), the horizontal distance of the wall from the point of projection (in m ) is

A pulse of light of duration 100 ns is absorbed completely by a small object initially at rest. Power of the pulse is 30 mW and the speed of light is 3 ×10⁸ ms^-1. The final momentum of the object is

A field line is shown in the figure. This field cannot represent

A thin uniform cylindrical shell, closed at both ends, is partially filled with water. It is floating vertically in water in half-submerged state. If \(\rho_{c}\) is the relative density of the material of the shell with respect to water, then the correct statement is that the shell is

Two satellites $P$ and $Q$ are moving in different circular orbits around the Earth(radius $R$). The heights of $P$ and $Q$ from the Earth surface are $h_P$ and $h_Q$, respectively, where $h_P=\frac{R}{3}$. The accelerations of $P$ and $Q$ due to Earth’s gravity are $g_P$ and $g_Q$, respectively. If $\frac{g_P}{g_Q}=\frac{36}{25}$, what is the value of $h_Q$?

Match the statements/expressions given in Column I with the values given in Column II.

$\mathrm{M}{\mathrm{X}}_2$ dissociates in ${\mathrm{M}}^{2+}$ and ${\mathrm{X}}^{-}$ ions in an aqueous solution, with a degree of dissociation $\left(\mathrm{\alpha }\right)$ of 0.5. The ratio of the observed depression of freezing point of the aqueous solution to the value of the depression of freezing point in the absence of ionic dissociation is

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Consider the lines: \(L_1: \frac{x+1}{3}=\frac{y+2}{1}=\frac{z+1}{2}, L_2: \frac{x-2}{1}=\frac{y+2}{2}=\frac{z-3}{3}\)Question:
The shortest distance between \(L_1\) and \(L_2\) is

In a high school, a committee has to be formed from a group of $6$ boys
$M_1, M_2, M_3, M_4, M_5, M_6$ and $5$ girls $G_1, G_2, G_3, G_4, G_5$.
(i) Let ${\alpha }_1$ be the total number of ways in which the committee can be formed such that the committee has $5$ members, having exactly $3$ boys and $2$ girls.
(ii) Let ${\alpha }_2$ be the total number of ways in which the committee can be formed such that the committee has at least $2$ members, and having an equal number of boys and girls.
(iii) Let ${\alpha }_3$ be the total number of ways in which the committee can be formed such that the committee has $5$ members, at least $2$ of them being girls.
(iv) Let ${\alpha }_4$ be the total number of ways in which the committee can be formed such that the committee has $4$ members, having at least $2$ girls and such that both $M_1$ and $G_1$ are NOT in the committee together.

LIST-I	LIST-II
A. The value of ${\alpha }_1$ is	P. $136$
B. The value of ${\alpha }_2$ is	Q. $189$
C. The value of ${\alpha }_3$ is	R. $192$
D. The value of ${\alpha }_4$ is	S. $200$
	T. $381$
	U. $461$

The correct option is:

Let \(z_k=\cos \left(\frac{2 k \pi}{10}\right)+i \sin \left(\frac{2 k \pi}{10}\right) ; k=1,2, \ldots, 9\)

List - I	List - II
(A) For each \(z_k\) there exists a \(z_j\) such \(z_k \cdot z_j=1\)	(P) True
(B) There exists a \(k \in\{1,2, \ldots, 9\}\) such that \(z_1 \cdot z=z_k\) has no solution z in the set of complex numbers	(Q) False
(C) \(\frac{\left|1-z_1\right|\left|1-z_2\right|\ldots\left|1-z_9\right|}{10}\) equals	(R) 1
(D) \(1-\sum_{k=1}^9 \cos \left(\frac{2 k \pi}{10}\right)\) equals	(S) 2

For the reaction sequence given below, the correct statement(s) is(are)

(In the options, \(X\) is any atom other than carbon and hydrogen, and it is different in \(\mathbf{P}, \mathbf{Q}\) and \(\mathbf{R}\) )