The efficiency of a Carnot engine operating with a hot reservoir kept at a temperature of 1000 K is 0.4. It extracts 150 J of heat per cycle from the hot reservoir. The work extracted from this engine is being fully used to run a heat pump which has a coefficient of performance 10 . The hot reservoir of the heat pump is at a temperature of 300 K . Which of the following statements is/are correct:

Light of wavelength ${\lambda }_{ph}$ falls on a cathode plate inside a vacuum tube as shown in the figure. The work function of the cathode surface is $\varphi$ and the anode is a wire mesh of conducting material kept at a distance d from the cathode. A potential difference V is maintained between the electrodes. If the minimum de Broglie wavelength of the electrons passing through the anode is ${\lambda }_e,$ which of the following statement(s) is(are) true ?

Two identical concave mirrors each of focal length \(f\) are facing each other as shown in the schematic diagram. The focal length \(f\) is much larger than the size of the mirrors. A glass slab of thickness \(t\) and refractive index \(n_0\) is kept equidistant from the mirrors and perpendicular to their common principal axis. A monochromatic point light source \(S\) is embedded at the center of the slab on the principal axis, as shown in the schematic diagram. For the image to be formed on \(S\) itself, which of the following distances between the two mirrors is/are correct:

Consider a hydrogen-like ionized atom with atomic number $Z$ with a single electron. In the emission spectrum of this atom, the photon emitted in the $n=2 \mathrm{t}\mathrm{o} n= 1$ transition has energy $74.8 \mathrm{eV}$ higher than the photon emitted in the $n=3 \mathrm{t}\mathrm{o} n=2$ transition. Given that the ionization energy of the hydrogen atom is $13.6 \mathrm{eV}$, what is the value of $Z$?

An incompressible liquid is kept in a container having a weightless piston with a hole. A capillary tube of inner radius $0.1\mathrm{mm}$ is dipped vertically into the liquid through the airtight piston hole, as shown in the figure. The air in the container is isothermally compressed from its original volume $V_0$to$\frac{100}{101}{V}_0$with the movable piston.Considering air as an ideal gas, the height $\left(h\right)$ of the liquid column in the capillary above the liquid level in $\mathrm{cm}$is _____.
[Given: Surface tension of the liquid is $0.075\mathrm{N}{\mathrm{m}}^{-1}$, atmospheric pressure is ${10}^5\mathrm{N}{\mathrm{m}}^{-2}$, acceleration due to gravity $\left(g\right)$ is $10\mathrm{m}{\mathrm{s}}^{-2}$, density of the liquid is ${10}^3\mathrm{kg}{\mathrm{m}}^{-3}$and contact angle of capillary surface with the liquid is zero]

Two equilateral-triangular prisms \(\mathrm{P}_1\) and \(\mathrm{P}_2\) are kept with their sides parallel to each other, in vacuum, as shown in the figure. A light ray enters prism \(P_1\) at an angle of incidence \(\theta\) such that the outgoing ray undergoes minimum deviation in prism \(P_2\). If the respective refractive indices of \(\mathrm{P}_1\) and \(\mathrm{P}_2\) are \(\sqrt{\frac{3}{2}}\) and \(\sqrt{3}\), then \(\theta=\sin ^{-1}\left[\sqrt{\frac{3}{2}} \sin \left(\frac{\pi}{\beta}\right)\right]\), where the value of \(\beta\) is _______ .

A train with cross-sectional area $S_t$ is moving with speed $v_t$ inside a long tunnel of cross-sectional area $S_0\left(S_0=4S_t\right)$. Assume that almost all the air (density $\rho$) in front of the train flows back between its sides and the walls of the tunnel. Also, the airflow with respect to the train is steady and laminar. Take the ambient pressure and that inside the train to be $P_0$. If the pressure in the region between the sides of the train and the tunnel walls is $P$, then $\left(P_0-P\right)=\frac{7}{2N}\rho v_t^2.$ The value of $N$ is _______.

A particle of mass $m$ is initially at rest at the origin. It is subjected to a force and starts moving along the x - axis. Its kinetic energy changes with time as $dK/dt=\gamma t,$ where $\gamma$ is a positive constant of appropriate dimensions. Which of the following statements is (are) true?

The correct order of acidity for the following compounds is :

In an atom, the total number of electrons having quantum numbers $\mathrm{n}=4,\mathrm{ }\left|{\mathrm{m}}_{\mathcal{l}}\right|=1$ and ${\mathrm{m}}_{\mathrm{s}}=–1/2$ is

Match the following.

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?

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In a thin rectangular metallic strip a constant current \(I\) flows along the positive \(x\)-direction, as shown in the figure. The length, width and thickness of the strip are \(l, w\) and \(d\), respectively.

A uniform magnetic field \(\vec{B}\) is applied on the strip along the positive \(y\)-direction. Due to this, the charge carriers experience a net deflection along the \(z\)-direction. This results in accumulation of charge carriers on the surface \(P Q R S\) and appearance of equal and opposite charges on the face opposite to \(P Q R S\). A potential difference along the \(z\)-direction is thus developed. Charge accumulation continues until the magnetic force is balanced by the electric force. The current is assumed to be uniformly distributed on the cross section of the strip and carried by electrons.






Question:

Consider two different metallic strips (\(1\) and \(2\)) of the same material. Their lengths are the same, widths are \(w_{1}\) and \(w_{2}\) and thicknesses are \(d_{1}\) and \(d_{2}\), respectively. Two points \(K\) and \(M\) are symmetrically located on the opposite faces parallel to the \(x-y\) plane (see figure). \(V_{1}\) and \(V_{2}\) are the potential differences between \(K\) and \(M\) in strips \(1\) and \(2\), respectively. Then, for a given current \(I\) flowing through them in a given magnetic field strength \(B\), the correct statement(s) is(are)