A cube of unit volume contains \(35 \times 10^7\) photons of frequency \(10^{15} \mathrm{~Hz}\). If the energy of all the photons is viewed as the average energy being contained in the electromagnetic waves within the same volume, then the amplitude of the magnetic field is \(\alpha \times 10^{-9} \mathrm{~T}\). Taking permeability of free space \(\mu_0=4 \pi \times 10^{-7} \mathrm{Tm} / \mathrm{A}\), Planck's constant \(h=6 \times 10^{-34} \mathrm{Js}\) and \(\pi=\frac{22}{7}\), the value of \(\alpha\) is_______

A uniformly charged thin spherical shell of radius \(R\) carries uniform surface charge density of \(\sigma\) per unit area. It is made of two hemispherical shells, held together by pressing them with force \(F\) (see figure). \(F\) is proportional to

A spherical portion has been removed from a solid sphere having a charge distributed uniformly in its volume as shown in the figure. The electric field inside the emptied space is

Two ideal batteries of emf ${\mathrm{V}}_1$ and ${\mathrm{V}}_2$ and three resistances ${\mathrm{R}}_1,\mathrm{}{\mathrm{R}}_2$ and ${\mathrm{R}}_3$ are connected as shown in the figure. The current in resistance ${\mathrm{R}}_2$ would be zero if

Paragraph :
When liquid medicine of density \(\rho\) is to be put in the eye, it is done with the help of a dropper. As the bulb on the top of the dropper is pressed, a drop forms at the opening of the dropper. We wish to estimate the size of the drop.
We first assume that the drop formed at the opening is spherical because that requires a minimum increase in its surface energy. To determine the size, we calculate the net vertical force due to the surface tension \(T\) when the radius of the drop is \(R\). When this force becomes smaller than the weight of the drop, the drop gets detached from the dropper.
Question :
If \(r=5 \times 10^{-4} \mathrm{~m}, \rho=10^3 \mathrm{~kg} \mathrm{~m}^{-3}\), \(g=10 \mathrm{~ms}^{-2}, T=0.11 \mathrm{Nm}^{-1}\), the radius of the drop when it detaches from the dropper is approximately

The dimensions of a cone are measured using a scale with a least count of \(2 \mathrm{~mm}\). The diameter of the base and the height are both measured to be \(20.0 \mathrm{~cm}\). The maximum percentage error in the determination of the volume is _______ .

Column I gives a list of possible set of parameters measured in some experiments. The variations of the parameters in the form of graphs are shown in Column II. Match the set of parameters given in Column I with the graphs given in Column II. Indicate your answer by darkening the appropriate bubble of the \(4 \times 4\) matrix given in the ORS.

For any $3\times 3$ matrix $M$, let $\left|M\right|$ denote the determinant of $M$. Let $E=\left[\begin{array}{ccc}1& 2& 3\\ 2& 3& 4\\ 8& 13& 18\end{array}\right], P=\left[\begin{array}{lll}1& 0& 0\\ 0& 0& 1\\ 0& 1& 0\end{array}\right] \& F=\left[\begin{array}{ccc}1& 3& 2\\ 8& 18& 13\\ 2& 4& 3\end{array}\right]$ If $Q$ is a nonsingular matrix of order $3\times 3$, then which of the following statements is (are) TRUE ?

Let \(\mathbf{a}=-\hat{\mathbf{i}}-\hat{\mathbf{k}}, \mathbf{b}=-\hat{\mathbf{i}}+\hat{\mathbf{j}}\) and \(\mathbf{c}=\hat{\mathbf{i}}+2 \hat{\mathbf{j}}+3 \hat{\mathbf{k}}\) be three given vectors. If \(\mathbf{r}\) is a vector such that \(\mathbf{r} \times \mathbf{b}=\mathbf{c} \times \mathbf{b}\) and \(\mathbf{r} \cdot \mathbf{a}=0\), then the value of \(\mathbf{r} \cdot \mathbf{b}\) is

Answer the following by appropriately matching the lists based on the information given in the paragraph.
In a thermodynamics process on an ideal monatomic gas, the infinitesimal heat absorbed by the gas is given by $\mathrm{T}\mathrm{\Delta }\mathrm{X},$ where $\mathrm{T}$ is temperature of the system and $\mathrm{\Delta }\mathrm{X}$ is the infinitesimal change in a thermodynamic quantity $\mathrm{X}$ of the system. For a mole of monatomic ideal gas $\mathrm{X}=\frac{3}{2}\mathrm{R}\mathrm{l}\mathrm{n}\left(\frac{\mathrm{T}}{{\mathrm{T}}_{\mathrm{A}}}\right)+\mathrm{R}\mathrm{l}\mathrm{n}\left(\frac{\mathrm{V}}{{\mathrm{V}}_{\mathrm{A}}}\right).$ Here, $\mathrm{R}$ is gas constant, $\mathrm{V}$ is volume of gas, ${\mathrm{T}}_{\mathrm{A}}$ and ${\mathrm{V}}_{\mathrm{A}}$ are constants.
The List-I below gives some quantities involved in a process and List-II gives some possible values of these quantities.

	List I		List II
$\left(\mathrm{a}\right)$	Work done by the system in process $1\to 2\to 3$	$\left(\mathrm{p}\right)$	$\frac{1}{3}\mathrm{R}{\mathrm{T}}_0\mathrm{l}\mathrm{n}2$
$\left(\mathrm{b}\right)$	Change in internal energy in process $1\to 2\to 3$	$\left(\mathrm{q}\right)$	$\frac{1}{3}\mathrm{R}{\mathrm{T}}_0$
$\left(\mathrm{c}\right)$	Heat absorbed by the system in process $1\to 2\to 3$	$\left(\mathrm{r}\right)$	$\mathrm{R}{\mathrm{T}}_0$
$\left(\mathrm{d}\right)$	Heat absorbed by the system in process $1\to 2$	$\left(\mathrm{s}\right)$	$\frac{4}{3}\mathrm{R}{\mathrm{T}}_0$
		$\left(\mathrm{t}\right)$	$\frac{1}{3}\mathrm{R}{\mathrm{T}}_0\left(3+\mathrm{l}\mathrm{n}2\right)$
		$\left(\mathrm{u}\right)$	$\frac{5}{6}\mathrm{R}{\mathrm{T}}_0$

If the process carried out on one mole of monatomic ideal gas is as shown in figure in the $\mathrm{T}$ - $\mathrm{V}$ -diagram with ${\mathrm{P}}_0{\mathrm{V}}_0=\frac{1}{3}\mathrm{R}{\mathrm{T}}_0,$ the correct match is,

A person blows into open-end of a long pipe. As a result, a high pressure pulse of air travels down the pipe.

When this pulse reaches the other end of the pipe,

\(C_V\) and \(C_p\) denote the molar specific heat capacities of a gas at constant volume and constant pressure, respectively. Then

Paragraph:
Box \(1\) contains three cards bearing numbers \(1,2,3\); box \(2\) contains five cards bearing numbers \(1,2,3,4,5\); and box \(3\) contains seven cards bearing numbers \(1,2,3,4,5,6,7\). A card is drawn from each of the boxes. Let \(x_{i}\) be the number on the card drawn from the \(i^{t h}\) box, \(i=1,2,3\).

Question:
The probability that \(x_{1}, x_{2}, x_{3}\) are in an arithmetic progression, is