Four identical particles of mass \(M\) are located at the corners of a square of side \(‘a’\). What should be their speed if each of them revolves under the influence of other’s gravitational field in a circular orbit circumscribing the square?

Proton with kinetic energy of \(1\;MeV\) moves from south to north. It gets an acceleration of \(10^{12}\; \mathrm{m} / \mathrm{s}^{2}\) by an applied magnetic field (west to east). The value of magnetic field :.......\(mT\) (Rest mass of proton is \(1.6 \times 10^{-27} \;\mathrm{kg}\) )

Two long current carrying thin wires, both with current \(I\), are held by insulating threads oflength \(L\) and are in equilibrium as shown in the figure, with threads making an angle '\(\theta\)' with the vertical. If wires have mass \(\lambda\) per unit length then the value of \(l\) is 
(\(g =\) gravitational acceleration)

Two known resistance of \(R \Omega\) and \(2 R \Omega\) and and one unknown resistance \(X \Omega\) are connected in a circuit as shown in the figure. If the equivalent resistance between points A and B in the circuit is \(X \Omega\), then the value of X is _________ \(\Omega\).

The ratio of wavelengths of proton and deuteron accelerated by potential \(V_{p}\) and \(V_{d}\) is 1: \(\sqrt{2}\). Then, the ratio of \(V_{p}\) to \(V_{d}\) will be

A piston of mass \(M\) is hung from a massless spring whose restoring force law goes as \(\mathrm{F}=-\mathrm{kx}^3\), where k is the spring constant of appropriate dimension. The piston separates the vertical chamber into two parts, where the bottom part is filled with ' \(n\) ' moles of an ideal gas. An external work is done on the gas isothermally (at a constant temperature T) with the help of a heating filament (with negligible volume) mounted in lower part of the chamber, so that the piston goes up from a height \(\mathrm{L}_0\) to \(\mathrm{L}_1\), the total energy delivered by the filament is (Assume spring to be in its natural length before heating)

A coil is placed perpendicular to a magnetic field of 5000 T . When the field is changed to 3000 T in 2 s , an induced emf of 22 V is produced in the coil. If the diameter of the coil is 0.02 m , then the number of turns in the coil is _______.

The energy stored in the electric field produced by a metal sphere is \(4.5\, J\). lf the sphere contains \(4\,\mu C\) charge, its radius will be.......\(mm\) : [Take : \(\frac{1}{{4\pi {\varepsilon _0}}} = 9 \times {10^9}\,N - {m^2}\,/{C^2}\, ]\)

Sun light falls normally on a surface of area \(36\,cm ^{2}\) and exerts an average force of \(7.2 \times 10^{-9}\,N\) within a time period of \(20\) minutes. Considering a case of complete absorption, the energy flux of incident light is.

Young's moduli of the material of wires \(A\) and \(B\) are in the ratio of \(1: 4\), while its area of cross sections are in the ratio of \(1: 3\). If the same amount of load is applied to both the wires, the amount of elongation produced in the wires \(A\) and \(B\) will be in the ratio of [Assume length of wires \(A\) and \(B\) are same]

Given below are two statements: one is labelled as Assertion (A) and the other is labelled as Reason (R).
Assertion (A) : The radius vector from the Sun to a planet sweeps out equal areas in equal intervals of time and thus areal velocity of planet is constant.
Reason (R) : For a central force field the angular momentum is a constant.
In the light of the above statements, choose the most appropriate answer from the options given below :

Let \(f(x)=\left[x^2-x\right]+|-x+[x]|\), where \(x \in R\) and \([t]\) denotes the greatest integer less than or equal to \(t\). Then, \(f\) is

If the mean deviation about the median of the numbers \( k, 2k, 3k, \dots, 1000k \) is 500, then \( k^{2} \) is equal to :