A glass tube of uniform internal radius \((r)\) has a valve separating the two identical ends. Initially, the valve is in a tightly closed position. End 1 has a hemispherical soap bubble of radius \(r\). End 2 has sub-hemispherical soap bubble as shown in figure. Just after opening the valve,

A transparent slab of thickness d has a refractive index n (z) that increases with z. Here z is the vertical distance inside the slab, measured from the top. The slab is placed between two media with uniform refractive indices $n_{1}and{n}_2\left(>n_1\right),$ as shown in the figure. A ray of light is incident with angle ${\theta }_i$ from medium 1 and emerges in medium 2 with refraction angle ${\theta }_f$ with a lateral displacement $\mathcal{l}$ . Which of the following statement(s) is (are) true?

List I contains four combinations of two lenses ( 1 and 2 ) whose focal lengths (in cm ) are indicated in the figures. In all cases, the object is placed 20 cm from the first lens on the left, and the distance between the two lenses is 5 cm . List II contains the positions of the final images.

	List - I		List - II
(I)		(P)	Final image is formed at 7.5 cm on the right side of lens 2.
(II)		(Q)	Final image is formed at 60.0 cm on the right side of lens 2.
(III)		(R)	Final image is formed at 30.0 cm on the left side of lens 2.
(IV)		(S)	Final image is formed at 6.0 cm on the right side of lens 2.
		(T)	Final image is formed at 30.0 cm on the right side of lens 2.

Which one of the following options is correct?

Two non-conducting solid spheres of radii $R$ and $2R$, having uniform volume charge densities ${\rho }_1$ and ${\rho }_2$, respectively, touch each other. The net electric field at a distance $2R$ from the centre of the smaller sphere, along the line joining the centres of the spheres, is zero. The ratio $\frac{{\rho }_1}{{\rho }_2}$ can be

An ideal gas undergoes a four step cycle as shown in the $P-V$ diagram below. During this cycle, heat is absorbed by the gas in:

A monochromatic light wave is incident normally on a glass slab of thickness $d$, as shown in the figure. The refractive index of the slab increases linearly from $n_1$ to $n_2$ over the height $h$. Which of the following statement(s) is(are) true about the light wave emerging out of the slab?

A source, approaching with speed $u$ towards the open end of a stationary pipe of length $L$, is emitting a sound of frequency $f_s$. The farther end of the pipe is closed. The speed of sound in air is $v$ and $f_0$ is the fundamental frequency of the pipe. For which of the following combination(s) of $u$ and $f_s$, will the sound reaching the pipe lead to a resonance?

Let ${\tan }^{-1}\left(x\right)\in \left(-\frac{\pi }{2}, \frac{\pi }{2}\right)$, for $x\in \mathrm{ℝ}$. Then the number of real solutions of the equation $\sqrt{1+\cos \left(2x\right)}=\sqrt{2}{\tan }^{-1}\left(\tan x\right)$ in the set $\left(-\frac{3\pi }{2}, -\frac{\pi }{2}\right)\cup \left(-\frac{\pi }{2}, \frac{\pi }{2}\right)\cup \left(\frac{\pi }{2}, \frac{3\pi }{2}\right)$ is equal to

Paragraph :

Consider an evacuated cylindrical chamber of height \(h\) having rigid conducting plates at the ends and an insulating curved surface as shown in the figure. A number of spherical balls made of a light weight and soft material and coated with a conducting material are placed on the bottom plate. The balls have a radius \(r \ll h\). Now a high voltage source \((HV)\) is connected across the conducting plates such that the bottom plate is at \(+V_{0}\) and the top plate at \(-V_{0} .\) Due to their conducting surface, the balls will get charged, will become equipotential with the plate and are repelled by it. The balls will eventually collide with the top plate, where the coefficient of restitution can be taken to be zero due to the soft nature of the material of the balls. The electric field in the chamber can be considered to be that of a parallel plate capacitor. Assume that there are no collisions between the balls and the interaction between them is negligible. (Ignore gravity)



Question :

Which one of the following statements is correct?

The focal length of a thin biconvex lens is \(20 \mathrm{~cm}\). When an object is moved from a distance of \(25 \mathrm{~cm}\) in front of it to \(50 \mathrm{~cm}\), the magnification of its image changes from \(m_{25}\) to \(m_{50}\). The ratio \(\frac{m_{25}}{m_{50}}\) is

A metal surface is illuminated by light of two different wavelengths 248 nm and 310 nm. The maximum speeds of the photoelectrons corresponding to these wavelengths are u₁ and u₂, respectively. If the ratio $u_1:u_2=2:1$ , the work function of the metal is nearly

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:

A particle is moved along a path $\mathrm{A}\mathrm{B}-\mathrm{B}\mathrm{C}-\mathrm{C}\mathrm{D}-\mathrm{D}\mathrm{E}-\mathrm{E}\mathrm{F}-\mathrm{F}\mathrm{A},$ as shown in figure, in presence of a force $\overrightarrow{\mathrm{F}}=\left(\mathrm{\alpha }\mathrm{y}\widehat{\mathrm{i}}+2\mathrm{\alpha }\mathrm{x}\widehat{\mathrm{j}}\right)\mathrm{N},$ where $x$ and $\mathrm{y}$ are in meter and $\mathrm{\alpha }=-1\mathrm{N}/{\mathrm{m}}^{-1}.$ The work done on the particle by this force $\overrightarrow{\mathrm{F}}$ will be ____ Joule.