A large spherical mass $\mathrm{M}$ is fixed at one position and two identical point masses $\mathrm{m}$ are kept on a line passing through the centre of $\mathrm{M}$ (see figure). The point masses are connected by a rigid massless rod of length $\mathrm{l}$ and this assembly is free to move along the line connecting them. All three masses interact only through their mutual gravitational interaction. When the point mass nearer to $\mathrm{M}$ is at a distance $\mathrm{r}=3\mathrm{l}$ from $\mathrm{M}$ , the tension in the rod is zero for $\mathrm{m}=\mathrm{k}\left(\frac{\mathrm{M}}{288}\right)$ . The value of $\mathrm{k}$ is

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Modern trains are based on Maglev technology in which trains are magnetically leviated, which runs its EDS Maglev system.
There are coils on both sides of wheels. Due to motion of train, current induces in the coil of track which levitate it. This is in accordance with Lenz's law. If trains lower down then due to Lenz's law a repulsive force increases due to which train gets uplifted and if it goes much high then there is a net downward force disc to gravity. The advantage of Maglev train is that there is no friction between the train and the track, thereby reducing power consumption and enabling the train to attain very high speeds.
Disadvantage of Maglev train is that as it slows down the electromagnetic forces decreases and it becomes difficult to keep it leviated and as it moves forward according to Lenz law, there is an electromagnetic drag force.
Question:
Which force causes the train to elevate up?

Two identical non-conducting solid spheres of same mass and charge are suspended in air from a common point by two non-conducting, massless strings of same length. At equilibrium, the angle between the strings is $\alpha$. The spheres are now immersed in a dielectric liquid of density $800 \mathrm{kg} {\mathrm{m}}^{-3}$ and dielectric constant $21$. If the angle between the strings remains the same after the immersion, then

Statement 1 Two cylinders, one hollow (metal) and the other solid (wood) with the same mass and identical dimensions are simultaneously allowed to roll without slipping down an inclined plane from the same height. The hollow cylinder will reach the bottom of the inclined plane first.
and Statement 2 By the principle of conservation of energy, the total kinetic energies of both the cylinders are identical when they reach the bottom of the incline.

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

Two non-reactive monoatomic ideal gases have their atomic masses in the ratio 2 : 3. The ratio of their partial pressures, when enclosed in a vessel kept at a constant temperature, is 4 : 3. The ratio of their densities is 

A rocket is launched normal to the surface of the Earth, away from the Sun, along the line joining the Sun and the Earth. The Sun is $3\times {10}^5$ times heavier than the Earth and is at a distance $2.5\times {10}^4$ times larger than the radius of the Earth. The escape velocity from Earth's gravitational field is $v_e=11.2 km s^{-1}$ . The minimum initial velocity $\left(v_s\right)$ required for the rocket to be able to leave the Sun-Earth system is closest to
(Ignore the rotation and revolution of the Earth and the presence of any other planet)

Let $Q$ be the cube with the set of vertices $\left\{\left(x_1, x_2, x_3\right)\in {\mathrm{ℝ}}^3 : x_1, x_2, x_3\in \left\{0, 1\right\}\right\}$. Let $F$ be the set of all twelve lines containing the diagonals of the six faces of the cube $Q$. Let $S$ be the set of all four lines containing the main diagonals of the cube $Q$; for instance, the line passing through the vertices $\left(0, 0, 0\right)$ and $\left(1, 1, 1\right)$ is in $S$. For lines ${\ell }_1$ and ${\ell }_2$, let $d\left({\ell }_1, {\ell }_2\right)$ denote the shortest distance between them. Then the maximum value of $d\left({\ell }_1, {\ell }_2\right)$, as ${\ell }_1$ varies over $F$ and ${\ell }_2$ varies over $S$, is

Let $\mathrm{\omega }\ne 1$ be a cube root of unity. Then the minimum of the set $\{{\left|\mathrm{a}+\mathrm{b}\mathrm{\omega }+\mathrm{c}{\mathrm{\omega }}^2\right|}^2:\mathrm{a},\mathrm{b},\mathrm{c}$ distinct non$–$zero integers $\}$ equals ________

A particle moves in the \(X-Y\) plane under the influence of a force such that its linear momentum is \(\mathbf{p}(t)=A[\hat{\mathbf{i}} \cos k t-\hat{\mathbf{j}} \sin k t]\), where \(A\) and \(k\) are constants. The angle between the force and the momentum is

Let $z$ be a complex number satisfying ${\left|z\right|}^3+2z^2+4\overline{z}-8=0$, where $\overline{z}$ denotes the complex conjugate of $z$. Let the imaginary part of $z$ be nonzero.
Match each entry in List-I to the correct entries in List-II.

List - I	List - II
(P) \(|z|^2\) is equal to	(1) 12
(Q) \(|z-\bar{z}|^2\) is equal to	(2) 4
(R) \(|z|^2+|z+\bar{z}|^2\) is equal to	(3) 8
(S) \(|z+1|^2\) is equal to	(4) 10
	(5) 7

The correct option is

A soft plastic bottle, filled with water of density \(1 \mathrm{gm} / \mathrm{cc}\), carries an inverted glass test-tube with some air (ideal gas) trapped as shown in the figure. The test-tube has a mass of \(5 \mathrm{gm}\), and it is made of a thick glass of density \(2.5 \mathrm{gm} / \mathrm{cc}\). Initially the bottle is sealed at atmospheric pressure \(p_{0}=10^{5} \mathrm{~Pa}\) so that the volume of the trapped air is \(v_{0}=3.3 \mathrm{cc}\). When the bottle is squeezed from outside at constant temperature, the pressure inside rises and the volume of the trapped air reduces. It is found that the test tube begins to sink at pressure \(p_{0}+\Delta p\) without changing its orientation. At this pressure, the volume of the trapped air is \(v_{0}-\Delta v\).
Let \(\Delta v=X\) cc and \(\Delta p=Y \times 10^{3} \mathrm{~Pa}\).

The value of $X$is ____.

Let \(\mathbb{R}^3\) denote the three-dimensional space. Take two points \(P=(1,2,3)\) and \(Q=(4,2,7)\). Let \(\operatorname{dist}(X, Y)\) denote the distance between two points \(X\) and \(Y\) in \(\mathbb{R}^3\). Let
\(\begin{gathered}S=\left\{X \in \mathbb{R}^3:(\operatorname{dist}(X, P))^2-(\operatorname{dist}(X, Q))^2=50\right\} \text { and } \\T=\left\{Y \in \mathbb{R}^3:(\operatorname{dist}(Y, Q))^2-(\operatorname{dist}(Y, P))^2=50\right\} .\end{gathered}\)
Then which of the following statements is (are) TRUE?