A charged particle is introduced at the origin $x=0, y=0, z=0$ with a given initial velocity $\overrightarrow{v}$ . A uniform electric field $\overrightarrow{E}$ and a uniform magnetic field $\overrightarrow{B}$ exist everywhere. The velocity $\overrightarrow{v}$ , electric field $\overrightarrow{E}$ and a uniform magnetic field $\overrightarrow{B}$ are given in the columns below

Column 1	Column 2	Column 3
1. Electron with $\overrightarrow{v}=2\frac{E_0}{B_0}\widehat{x}$	(i) $\overrightarrow{E}=E_0\widehat{z}$	(P) $\overrightarrow{B}=-B_0\widehat{x}$
2. Electron with $\overrightarrow{v}=2\frac{E_0}{B_0}\widehat{y}$	(ii) $\overrightarrow{E}=-E_0\widehat{y}$	(Q) $\overrightarrow{B}=B_0\widehat{x}$
3.Proton with $\overrightarrow{v}$ =0	(iii) $\overrightarrow{E}={-E}_0\widehat{x}$	(R) $\overrightarrow{B}=-B_0\widehat{y}$
4. Proton with $\overrightarrow{v}=2\frac{E_0}{B_0}\widehat{x}$	(iv) $\overrightarrow{E}=E_0\widehat{x}$	(S) $\overrightarrow{B}=-B_0\widehat{z}$

In which case would the particle move in a straight line along the negative direction of y axis

Three plane mirrors form an equilateral triangle with each side of length $L$. There is a small hole at a distance $l>0$ from one of the corners as shown in the figure. A ray of light is passed through the hole at an angle $\theta$ and can only come out through the same hole. The cross section of the mirror configuration and the ray of light lie on the same plane.

Which of the following statement(s) is(are) correct?

A resistance of \(2 \Omega\) is connected across one gap of a meter-bridge (the length of the wire is \(100 \mathrm{~cm}\) ) and an unknown resistance, greater than \(2 \Omega\), is connected across the other gap. When these resistances are interchanged, the balance point shifts by \(20 \mathrm{~cm}\). Neglecting any corrections, the unknown resistance is

Highly excited states for hydrogen like atoms (also called Rydberg states) with nuclear charge Ze are defined by their principal quantum number n, where n >> 1. Which of the following statement(s) is (are) true?

A cylindrical furnace has height $\left(H\right)$ and diameter $\left(D\right)$ both $1\mathrm{m}$. It is maintained at temperature $360\mathrm{K}$. The air getsheated inside the furnace at constant pressure$P_a$and its temperature becomes $T=360\mathrm{K}$. The hot air with density $\rho$rises up a vertical chimney of diameter $d=0.1\mathrm{m}$ and height $h=9\mathrm{m}$ above the furnace and exits the chimney (seethe figure). As a result, atmospheric air of density${\rho }_a=1.2\mathrm{kg}{\mathrm{m}}^{-3}$,pressure $P_a$and temperature $T_a=300\mathrm{K}$ entersthe furnace. Assume air as an ideal gas, neglect the variations in$\rho$and $T$ inside the chimney and the furnace. Also ignore the viscous effects.
[Given: The acceleration due to gravity$g=10\mathrm{m}{\mathrm{s}}^{-2}$and$\pi =3.14$]


When the chimney is closed using a cap at the top, a pressure difference $∆P$develops between the top and the bottom surfaces of the cap. If the changes in the temperature and density of the hot air, due to the stoppage of air flow, are negligible then the value of $∆P$is _____ $\mathrm{N}{\mathrm{m}}^{-2}$.

A steady current \(I\) goes through a wire loop \(P Q R\) having shape of a right angle triangle with \(P Q=3 x, P R=4 x\) and \(Q R=5 x\). If the magnitude of the magnetic field at \(P\) due to this loop is \(k\left(\frac{\mu_0 I}{48 \pi x}\right)\), find the value of \(k\).

A mixture of 2 moles of helium gas (atomic mass \(=4 \mathrm{amu}\) ) and 1 mole of argon gas (atomic mass \(=40 \mathrm{amu}\) ) is kept at \(300 \mathrm{~K}\) in a container. The ratio of the rms speeds \(\left(\frac{v_{\mathrm{rms}}(\text { helium })}{v_{\mathrm{rms}}(\operatorname{argon})}\right)\) is

The correct acidity order of the following is

${\mathit{\text{EDTA}}}^{4-}$ is ethylenediaminetetraacetate ion. The total number of N - Co - O bond angles in ${\left[\mathit{\text{Co}}\left(\mathit{\text{EDTA}}\right)\right]}^{1-}$ complex ion is

Let $\alpha$ be a positive real number. Let $f:\mathrm{ℝ}\to \mathrm{ℝ}$ and $g:\left(\alpha ,\infty \right)\to \mathrm{ℝ}$ be the functions defined by $f\left(x\right)=\sin \left(\frac{\pi x}{12}\right)$ and $g\left(x\right)=\frac{2{\log }_e\left(\sqrt{x}-\sqrt{\alpha }\right)}{{\log }_e\left(e^{\sqrt{x}}-e^{\sqrt{\alpha }}\right)}$.
Then the value of $\underset{x\to {\alpha }^{+}}{\lim }f\left(g\left(x\right)\right)$ is ______.

The total number of chiral molecules formed from one molecule of $P$ on complete ozonolysis $\left({\mathrm{O}}_3,\mathrm{Zn}/H_2\mathrm{O}\right)$ is _____

If $y\left(x\right)$ is the solution of the differential equation $xdy-\left(y^2-4y\right)dx=0$ for $x>0, y\left(1\right)=2$, and the slope of the curve $y=y\left(x\right)$ is never zero, then the value of $10y\left(\sqrt{2}\right)$ is _____ .

A particle of mass $1 \mathrm{kg}$ is subjected to a force which depends on the position as $\overrightarrow{F}=-k\left(x\hat{i}+y\hat{j}\right) \mathrm{kg} \mathrm{m} {\mathrm{s}}^{-2}$ with $k=1 \mathrm{kg} {\mathrm{s}}^{-2}$. At time $t=0$, the particle's position $\overrightarrow{r}=\left(\frac{1}{\sqrt{2}}\hat{i}+\sqrt{2}\hat{j}\right) \mathrm{m}$ and its velocity $\overrightarrow{v}=\left(-\sqrt{2}\hat{i}+\sqrt{2}\hat{j}+\frac{2}{\pi }\hat{k}\right) \mathrm{m} {\mathrm{s}}^{-1}$. Let $v_x$ and $v_y$ denote the $x$ and the $y$ components of the particle's velocity, respectively. Ignore gravity. When $z=0.5 \mathrm{m}$, the value of $\left(x{v}_y-y{v}_x\right)$ is ______ ${\mathrm{m}}^2 {\mathrm{s}}^{-1}$.