JEE (Advanced) 2020 Paper-1

Shown in the figure is a semicircular metallic strip that has thickness t and resistivity $$\rho$$. Its inner radius is $$R_{1}$$ and outer radius is $$R_{2}$$. If a voltage $$V_{0}$$ is applied between its two ends, a current I flows in it. In addition, it is observed that a transverse voltage $$\triangle V$$ develops between its inner and outer surfaces due to purely kinetic effects of moving electrons (ignore any role of the magnetic field due to the current). Then (figure is schematic and not drawn to scale)

As shown schematically in the figure, two vessels contain water solutions (at temperature 𝑇) of
potassium permanganate ($$KMnO_{4}$$) of different concentrations $$n_{1}$$ and $$n_{2} (n_{1} > n_{2})$$ molecules per unit volume with $$\triangle n = (n_{1} − n_{2}) << n_{1}$$. When they are connected by a tube of small length l and cross-sectional area S, $$KMnO_{4}$$ starts to diffuse from the left to the right vessel through the tube. Consider the collection of molecules to behave as dilute ideal gases and the difference in their partial pressure in the two vessels causing the diffusion. The speed 𝑣 of the molecules is limited by the viscous force $$−\beta v on each molecule, where $$\beta$$ is a constant. Neglecting all terms of the order $$\left(\tirangle n\right)^{2}$$, which of the following is/are correct? ($$k_{B}$$ is the Boltzmann constant)

Put a uniform meter scale horizontally on your extended index fingers with the left one at 0.00 cm and the right one at 90.00 cm. When you attempt to move both the fingers slowly towards the center, initially only the left finger slips with respect to the scale and the right finger does not. After some distance, the left finger stops and the right one starts slipping. Then the right finger stops at a distance $$x_{R}$$ from the center (50.00 cm) of the scale and the left one starts slipping again. This happens because of the difference in the frictional forces on the two fingers. If the coefficients of static and dynamic friction between the fingers and the scale are 0.40 and 0.32, respectively, the value of $$x_{R}$$ (in
cm) is ______.

When water is filled carefully in a glass, one can fill it to a height h above the rim of the glass due to the surface tension of water. To calculate h just before water starts flowing, model the shape of the water above the rim as a disc of thickness h having semicircular edges, as shown schematically in the figure. When the pressure of water at the bottom of this disc exceeds what can be withstood due to the surface tension, the water surface breaks near the rim and water starts flowing from there. If the density of water, its surface tension and the acceleration due to gravity are $$10^{3}kg m^{−3}, 0.07 Nm^{−1}$$ and $$10 ms^{−2}$$, respectively, the value of h (in mm) is _________.

One end of a spring of negligible unstretched length and spring constant k is fixed at the origin (0,0). A point particle of mass m carrying a positive charge q is attached at its other end. The entire system is kept on a smooth horizontal surface. When a point dipole $$\overrightarrow{p}$$ pointing towards the charge q is fixed at the origin, the spring gets stretched to a length l and attains a new equilibrium position (see figure below). If the point mass is now displaced slightly by $$\triangle l << l$$ from its equilibrium position and released, it is found to oscillate at frequency $$\frac{1}{\delta}\sqrt{\frac{k}{m}}$$. The value of $$\delta$$ is ______.

Consider one mole of helium gas enclosed in a container at initial pressure $$p_{1}$$ and volume $$v_[1]$$. It expands isothermally to volume $$4𝑉_{1}$$. After this, the gas expands adiabatically and its volume becomes $$32𝑉_{1}$$. The work done by the gas during isothermal and adiabatic expansion processes are $$𝑊_{𝑖𝑠𝑜}$$ and $$𝑊_{𝑎𝑑𝑖𝑎}$$, respectively. If the ratio $$\frac{𝑊_{𝑖𝑠𝑜}}{𝑊_{𝑎𝑑𝑖𝑎}}$$ 𝑓 In2, then 𝑓 is ________.

A stationary tuning fork is in resonance with an air column in a pipe. If the tuning fork is moved with a speed of $$2 ms^{−1}$$ in front of the open end of the pipe and parallel to it, the length of the pipe should be changed for the resonance to occur with the moving tuning fork. If the speed of sound in air is $$320 ms^{−1}$$, the smallest value of the percentage change required in the length of the pipe is
____________.

A circular disc of radius 𝑅 carries surface charge density $$\sigma (𝑟) = \sigma 0 (1 − \frac{r}{R}\right)$$, where $$\sigma_{0}$$ is a constant and 𝑟 is the distance from the center of the disc. Electric flux through a large spherical surface that encloses the charged disc completely is $$\phi_{0}$$. Electric flux through another spherical surface of radius $$\frac{R}{4}$$ and concentric with the disc is $$\phi$$. Then the ratio $$\frac{\phi_{0}}{\phi}$$ is_________.

If the distribution of molecular speeds of a gas is as per the figure shown below, then the ratio of the most probable, the average, and the root mean square speeds, respectively, is

Which of the following liberates O2 upon hydrolysis?