A rod of mass $m$ and length $L,$ pivoted at one of its ends, is hanging vertically. A bullet of the same mass moving at speed $v$ strikes the rod horizontally at a distance $x$ from its pivoted end and gets embedded in it. The combined system now rotates with an angular speed $\omega$ about the pivot. The maximum angular speed ${\omega }_{\mathrm{M}}$ is achieved for $x=x_{\mathrm{M}}$. Then

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The capacitor of capacitance \(C\) can be charged (with the help of a resistance \(R\) ) by a voltage source \(V\), by closing switch \(S_1\) while keeping switch \(S_2\) open. The capacitor can be connected in series with an inductor \(L\) by closing switch \(S_2\) and opening \(S_1\).

Question :
Initially, the capacitor was uncharged. Now, switch \(S_1\) is closed and \(S_2\) is kept open. If time constant of this circuit is \(\tau\), then

Two identical uniform discs roll without slipping on two different surfaces AB and CD (see figure) starting at A and C with linear speeds \(v_1\) and \(v_2\), respectively, and always remain in contact with the surfaces. If they reach B and D with the same linear speed and \(v_1=3 m / s\), then \(v_2\) in \(m / s\) is \(\left( g =10 \frac{m}{ s ^2}\right)\)

Answer the following by appropriately matching the lists based on the information given in the paragraph.
A musical instrument is made using four different metal strings, $1,\mathrm{ }2,\mathrm{ }3$ and $4$ with mass per unit length $\mathrm{\mu },2\mathrm{\mu },3\mathrm{\mu }$ and $4\mathrm{\mu }$ respectively. The instrument is played by vibrating the strings by varying the free length in between the range ${\mathrm{L}}_0$ and $2{\mathrm{L}}_0.$ It is found that in string- $1\left(\mathrm{\mu }\right)$ at free length ${\mathrm{L}}_0$ and tension ${\mathrm{T}}_0$ the fundamental mode frequency is ${\mathrm{f}}_0.$
List-I gives the above four strings while list-II lists the magnitude of some quantity.

	List I		List II
$\left(\mathrm{a}\right)$	String $-1\left(\mathrm{\mu }\right)$	$\left(\mathrm{p}\right)$	$1$
$\left(\mathrm{b}\right)$	String $-2\left(2\mathrm{\mu }\right)$	$\left(\mathrm{q}\right)$	$\frac{1}{2}$
$\left(\mathrm{c}\right)$	String $-3\left(3\mathrm{\mu }\right)$	$\left(\mathrm{r}\right)$	$\frac{1}{\sqrt{2}}$
$\left(\mathrm{d}\right)$	String $-4\left(4\mathrm{\mu }\right)$	$\left(\mathrm{s}\right)$	$\frac{1}{\sqrt{3}}$
		$\left(\mathrm{t}\right)$	$\frac{3}{16}$
		$\left(\mathrm{u}\right)$	$\frac{1}{16}$

The length of the strings $1,\mathrm{ }2,\mathrm{ }3$ and $4$ are kept fixed at ${\mathrm{L}}_0,\frac{3{\mathrm{L}}_0}{2},\frac{5{\mathrm{L}}_0}{4}$ and $\frac{7{\mathrm{L}}_0}{4},$ respectively. Strings $1,\mathrm{ }2,\mathrm{ }3$ and $4$ are vibrated at their $1^{\mathrm{s}\mathrm{t}},\mathrm{ }{3}^{\mathrm{r}\mathrm{d}},\mathrm{ }{5}^{\mathrm{t}\mathrm{h}}$ and ${14}^{\mathrm{t}\mathrm{h}}$ harmonics, respectively such that all the strings have same frequency. The correct match for the tension in the four strings in the units of ${\mathrm{T}}_0$ will be.

 The diameter of a cylinder is measured using a vernier calipers with no zero error. It is found that the zero of the vernier scale lies between $5.10 \mathrm{cm}$ and $5.15 \mathrm{cm}$ of the main scale. The vernier scale has $50$  divisions equivalent to $2.45 \mathrm{cm}$. The ${24}^{\mathrm{th}}$ division of the vernier scale exactly coincides with one of the main scale divisions. The diameter of the cylinder is, 

Two batteries of different emfs and different internal resistances are connected as shown. The voltage across \(A B\) in volt is

A particle, of mass ${10}^{-3}\mathrm{ kg}$ and charge $1.0 \mathrm{C}$, is initially at rest. At time $t=0$, the particle comes under the influence of an electric field $\overrightarrow{E}\left(t\right)=E_0\sin \left(\omega t\right) \hat{\mathrm{i}}$ where $E_0=1.0\mathrm{ N} {\mathrm{C}}^{-1}$ and $\omega ={10}^3\mathrm{ rad} {\mathrm{s}}^{-1}$. Consider the effect of only the electrical force on the particle. Then the maximum speed, in $\mathrm{m} {\mathrm{s}}^{-1}$, attained by the particle at subsequent times is ____________.

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:

The reagent(s) for the following conversion,

is/are

Let \(\alpha, \beta\) and \(\gamma\) be real numbers such that the system of linear equations
\[
\begin{array}{c}
x+2 y+3 z=\alpha \\
4 x+5 y+6 z=\beta \\
7 x+8 y+9 z=\gamma-1
\end{array}
\]
is consistent. Let \(|M|\) represent the determinant of the matrix
\[
M=\left[\begin{array}{ccc}
\alpha & 2 & \gamma \\
\beta & 1 & 0 \\
-1 & 0 & 1
\end{array}\right]
\]
Let \(P\) be the plane containing all those \((\alpha, \beta, \gamma)\) for which the above system of linear equations is consistent, and \(D\) be the square of the distance of the point \((0,1,0)\) from the plane \(P\).
The value of $\left|M\right|$ is

Suppose that $\overrightarrow{p}, \overrightarrow{q} and \overrightarrow{r}$ are three non - coplanar vectors in ${\mathbb{R}}^3$ . Let the components of a vector $\overrightarrow{s}$ along \(p, q\) and \(r\) be 4,3 and 5 respectively. If the components of $\left(-\overrightarrow{p}+ \overrightarrow{q}+ \overrightarrow{r}\right), \left(\overrightarrow{p}- \overrightarrow{q}+ \overrightarrow{r}\right) and \left(- \overrightarrow{p}- \overrightarrow{q}+ \overrightarrow{r}\right)$ are x, y and z, respectively, then the value of $2x+y+z$ is

Three objects \(A, B\) and \(C\) are kept in a straight line on a frictionless horizontal surface. These have masses \(m, 2 m\) and \(m\), respectively. The object \(A\) moves towards \(B\) with a speed \(9 \mathrm{~ms}^{-1}\) and makes an elastic collision with it. Thereafter, \(B\) makes completely inelastic collision with \(C\). All motions occur on the same straight line. Find the final speed (in \(\mathrm{ms}^{-1}\) ) of the object \(C\).

Consider an electrochemical cell: $\text{A}\left(\text{s}\right)\left|{\text{ A}}^{\text{n+}}\left(\text{aq, 2 M}\right)\right|\left|{\text{ B}}^{\text{2n+}}\left(\text{aq, 1 M}\right)\right|\text{ B}\left(\text{s}\right)$. The value of ${\text{ΔH}}^{\text{o}}$ for the cell reaction is twice that of ${\text{ΔG}}^{\text{o}}$ at $\text{300 K}\text{.}$ If the emf of the cell is zero, the magnitude of $\mathrm{\Delta }{\text{S}}^o$ $\left(\text{in} {\text{J}}^{-1}{\text{K}}^{-1} {\text{mol}}^{-1}\right)$ of the cell reaction per mole of $\text{B}$ formed at $\text{300}\text{K}$ is ____
(Given: R $\text{=}\text{8}\text{.3}{\text{JK}}^{-\text{1}}{\text{mol}}^{-1}.$ $\text{H,}\text{S}$ and $\text{G}$ are enthalpy, entropy and Gibbs energy, respectively.)