A transparent thin film of uniform thickness and refractive index ${\mathrm{n}}_1=1.4$ is coated on the convex spherical surface of radius R at one end of a long solid glass cylinder of refractive index ${\mathrm{n}}_2=1.5$ , as shown in the figure. Rays of light parallel to the axis of the cylinder traversing through the film from air to glass get focused at distance f₁ from the film, while rays of light traversing from glass to air get focused at distance f₂ from the film. Then

A particle of mass $\text{0.2 kg}$ is moving in one dimension under a force that delivers a constant power $\text{0.5 W}$ to the particle. If the initial speed $\left({in\; ms}^{-1}\right)$ of the particle is zero, the speed $\left({in\; ms}^{-1}\right)$ after $\text{5 s}$ is

Two vehicles, each moving with speed u on  the same horizontal straight road, are approaching each other. Wind blows along the road with velocity w. One of these vehicles blows a whistle of frequency f₁. An observer in the other vehicle hears the frequency of the whistle to be f₂. The speed of sound in still air is V. The correct statement (s) is (are) :

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 conducting square loop of side \(L\), mass \(M\) and resistance \(R\) is moving in the \(X Y\) plane with its edges parallel to the \(X\) and \(Y\) axes. The region \(y \geq 0\) has a uniform magnetic field, \(\vec{B}=B_0 k\). The magnetic field is zero everywhere else. At time \(t=0\), the loop starts to enter the magnetic field with an initial velocity \(v_0 \hat{\jmath} \mathrm{~m} / \mathrm{s}\), as shown in the figure. Considering the quantity \(K=\frac{B_0^2 L^2}{R M}\) in appropriate units, ignoring self-inductance of the loop and gravity, which of the following statements is/are correct:

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The mass of a nucleus \({ }_{Z}^{A} X\) is less than the sum of the masses of \((A-Z)\) number of neutrons and \(Z\) number of protons in the nucleus. The energy equivalent to the corresponding mass difference is known as the binding energy of the nucleus. A heavy nucleus of mass \(M\) can break into two light nuclei of masses \(m_{1}\) and \(m_{2}\) only if \(\left(m_{1}+m_{2}\right) \lt M\). Also two light nuclei of masses \(m_{3}\) and \(m_{4}\) can undergo complete fusion and form a heavy nucleus of mass \(M^{\prime}\) only if \(\left(m_{3}+m_{4}\right) \gt M^{\prime}\). The masses of some neutral atoms are given in the table below:


\(\left(1 u=932 M e V / c^{2}\right)\)

Question:

The correct statement is

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,

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When a particle of mass \(m\) moves on the \(x\)-axis in a potential of the form \(V(x)=k x^2\), it performs simple harmonic motion.

The corresponding time period is proportional to \(\sqrt{\frac{m}{k}}\), as can be seen easily using dimensional analysis. However, the motion of a particle can be periodic even when its potential energy increases on both sides of \(x=0\) in a way different from \(k x^2\) and its total energy is such that the particle does not escape to infinity. Consider a particle of mass \(\mathrm{m}\) moving on the \(x\)-axis. Its potential energy is \(V(x)=\alpha x^4(\alpha>0)\) for \(|x|\) near the origin and becomes a constant equal to \(V_0\) for \(|x| \geq X_0\) (see figure).
Question :
If the total energy of the particle is \(E\), it will perform periodic motion only if

The total number of ways in which 5 balls of different colours can be distributed among 3 persons so that each person gets at least one ball is

Paragraph : If the measurement errors in all the independent quantities are known, then it is possible to determine the error in any dependent quantity. This is done by the use of series expansion and truncating the expansion at the first power of the error. For example, consider the relation $z =\frac{x}{y}$. If the errors in $x, y \mathrm{a}\mathrm{n}\mathrm{d} z \mathrm{a}\mathrm{r}\mathrm{e} ∆x, ∆y \mathrm{a}\mathrm{n}\mathrm{d} ∆z$ , respectively, then

$z\pm ∆z=\frac{x\pm ∆x}{y\pm ∆y}=\frac{x}{y}\left(1\pm \frac{∆x}{x}\right){\left(1\pm \frac{∆y}{y}\right)}^{-1}$

The series expansion for ${\left(1\pm \frac{∆y}{y}\right)}^{-1}$, to first power in $∆y/y, \mathrm{i}\mathrm{s} 1\mp \left(∆y/y\right)$. The relative errors in independent variables are always added. So the error in $z$ will be

$\text{Δ}z=z\left(\frac{\text{Δ}x}{x}+\frac{\text{Δ}y}{y}\right).$
The above derivation makes the assumption that \(\Delta r / x \ll 1, \Delta y / y < 1\). Therefore, the higher powers of these quantities are neglected.
Question : Consider the ratio \(r=\frac{(1-a)}{(1+a)}\) to be determined by measuring a dimensionless quantity a If the error in the measurement of a is \(\Delta a(\frac{\Delta a}{a}<<1)\), then what is the error \(\Delta r\) in determining \(r\) ?

Four solid spheres each of diameter \(\sqrt{5} \mathrm{~cm}\) and mass \(0.5 \mathrm{~kg}\) ar placed with their centres at the corners of a square of side \(4 \mathrm{~cm}\). The moment of inertia of the system about the diagonal of the square is \(N \times 10^{-4} \mathrm{~kg}-\mathrm{m}^2\), then \(N\) is

A satellite is moving with a constant speed \(v\) in a circular orbit about the earth. An object of mass \(m\) is ejected from the satellite such that it just escapes from the gravitational pull of the earth. At the time of its ejection, the kinetic energy of the object is

The radius of the orbit of an electron in a Hydrogen-like atom is \(4.5 a _0\), where \(a _0\) is the Bohr radius. Its orbital angular momentum is \(\frac{3 h}{2 \pi}\). It is given that \(h\) is Planck's constant and R is Rydberg constant. the possible wavelength \((s)\), when the atom de-excites, is (are)