Consider a spherical gaseous cloud of mass density $\mathrm{\rho }\left(\mathrm{r}\right)$ in free space where $\mathrm{r}$ is the radial distance from its center. The gaseous cloud is made of particles of equal mass m moving in circular orbits about the common center with the same kinetic energy $\mathrm{K}$ . The force acting on the particles is their mutual gravitational force. If $\mathrm{\rho }\left(\mathrm{r}\right)$ is constant in time, the particle number density $\mathrm{n}\left(\mathrm{r}\right)=\mathrm{\rho }\left(\mathrm{r}\right)/\mathrm{m}$ is: [ $\mathrm{G}$ is universal gravitational constant]

A circuit with an electrical load having impedance \(Z\) is connected with an AC source as shown in the diagram. The source voltage varies in time as \(V(t)=300 \sin (400 t) \mathrm{V}\), where \(t\) is time in s. List-I shows various options for the load. The possible currents \(i(t)\) in the circuit as a function of time are given in List-II.

Choose the option that describes the correct match between the entries in List-I to those in List-II.

	List-I		List-II
\((P)\)		\((1)\)
\((Q)\)		\((2)\)
\((R)\)		\((3)\)
\((S)\)		\((4)\)
		\((5)\)

Consider a system of three connected strings, \(S_1, S_2\) and \(S_3\) with uniform linear mass densities \(\mu \mathrm{kg} / \mathrm{m}\), \(4 \mu \mathrm{~kg} / \mathrm{m}\) and \(16 \mu \mathrm{~kg} / \mathrm{m}\), respectively, as shown in the figure. \(S_1\) and \(S_2\) are connected at the point \(P\), whereas \(S_2\) and \(S_3\) are connected at the point \(Q\), and the other end of \(S_3\) is connected to a wall. A wave generator O is connected to the free end of \(S_1\). The wave from the generator is represented by \(y=y_0\) \(\cos (\omega t-k x) \mathrm{cm}\), where \(y_0, \omega\) and \(k\) are constants of appropriate dimensions. Which of the following statements is/are correct:

Consider an electric field \(\mathbf{E}=E_0 \hat{\mathbf{x}}\) where \(\bar{E}_0\) is a constant. The flux through the shaded area ( as shown in the figure) due to this field is

A double slit setup is shown in the figure. One of the slits is in medium $2$ of refractive index $n_2$. The other slit is at the interface of this medium with another medium $1$ of refractive index $n_1\left(\ne n_2\right)$. The line joining the slits is perpendicular to the interface and the distance between the slits is $d$. The slit widths are much smaller than $d$. A monochromatic parallel beam of light is incident on the slits from medium $1$. A detector is placed in medium $2$ at a large distance from the slits, and at an angle $\theta$ from the line joining them, so that $\theta$ equals the angle of refraction of the beam. Consider two approximately parallel rays from the slits received by the detector.
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Which of the following statement(s) is(are) correct?

A particle, of mass ${10}^{-3}\mathrm{ kg}$ and charge $1.0 \mathrm{C}$, is initially at rest. At time $t=0$, the particle comes under the influence of an electric field $\overrightarrow{E}\left(t\right)=E_0\sin \left(\omega t\right) \hat{\mathrm{i}}$ where $E_0=1.0\mathrm{ N} {\mathrm{C}}^{-1}$ and $\omega ={10}^3\mathrm{ rad} {\mathrm{s}}^{-1}$. Consider the effect of only the electrical force on the particle. Then the maximum speed, in $\mathrm{m} {\mathrm{s}}^{-1}$, attained by the particle at subsequent times is ____________.

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Phase space diagrams are useful tools in analyzing all kinds of dynamical problems. They are especially useful in studying the changes in motion as initial position and momentum are changed. Here we consider some simple dynamical systems in one-dimension. For such systems, phase space is a plane in which position is plotted along horizontal axis and momentum is plotted along vertical axis. The phase space diagram is \(x(t) v s p(t)\) curve in this plane. The arrow on the
curve indicates the time flow. For example, the phase space diagram for a particle moving with constant velocity is a straight line as shown in the figure. We use the sign convention in which position or momentum upwards (or to right) is positive and downwards (or to left) is negative.

Question :
The phase space diagram for simple harmonic motion is a circle centered at the origin. In the figure, the two circles represent the same oscillator but for different initial conditions, and \(E_1\) and \(E_2\) are the total mechanical energies respectively. Then,

In the figure, a ladder of mass m is shown leaning against a wall. It is in static equilibrium making an angle $\mathrm{\theta }$ with the horizontal floor. The coefficient of friction between the wall and the ladder is ${\mathrm{\mu }}_1$ and that between the floor and the ladder is ${\mathrm{\mu }}_2$ . The normal reaction of the wall on the ladder is ${\mathrm{N}}_1$ and that of the floor is ${\mathrm{N}}_2$ If the ladder is about to slip, then

Let \(A B C\) and \(A B C^{\prime}\) be two non-congruent triangles with sides \(A B=4\), \(A C=A C^{\prime}=2 \sqrt{2}\) and angle \(B=30^{\circ}\). The absolute value of the difference between the areas of these triangles is

For a > b > c > 0, the distance between (1, 1) and the point of intersection of the lines ax + by + c = 0 and bx + ay + c = 0 is less than $2\sqrt{2}$.Then

A spring - block system is resting on a frictionless floor as shown in the figure. The spring constant is $2.0N{m}^{-1}$ and the mass of the block is $2.0kg$ . Ignore the mass of the spring. Initially the spring is in an unstretched condition. Another block of mass 1.0kg moving with a speed of $2.0m{s}^{-1}$ collides elastically with the first block. The collision is such that the 2.0kg block does not hit the wall. The distance, in metres, between the two blocks when the spring returns to its unstretched position for the first time after the collision is _________.

A bi-convex lens is formed with two thin plano-convex lenses as shown in the figure. Refractive index \(n\) of the first lens is \(1.5\) and that of the second lens is \(1.2\). Both the curved surface are of the same radius of curvature \(R=14\) \(\mathrm{cm}\). For this bi-convex lens, for an object distance of 40 \(\mathrm{cm}\), the image distance will be

A Conductor (shown in the figure) carrying constant current $\mathrm{I}$ is kept in the $\mathrm{x}$ - $\mathrm{y}$ plane in a uniform magnetic field $\overrightarrow{\mathrm{B}}$ . If $\mathrm{F}$ is the magnitude of the total magnetic force acting on the conductor, then the correct statement(s) is (are)