In the diagram shown below, \(m_1\) and \(m_2\) are the masses of two particles and \(x_1\) and \(x_2\) are their respective distances from the origin \(O\).

The centre of mass of the system is

If \(A\) is the angle of prism, \(r\) angle of refraction, then the condition for minimum deviation is

Forces A and B act at a point. The sum of their magnitudes is 50 N and the magnitude of their resultant is 20 N. If the resultant is at \(90^\circ\) with the smaller force, the magnitudes of A and B, in N are

What is the ratio of the mean free paths of the molecules of two gases \(A\) and \(B\) having the molecular diameters \(2 \mathrm{~A}^0\) and \(3 \mathrm{~A}^0\) respectively under the identical conditions of pressure, temperature and volume?

Suppose refractive index \(\mu\) is given as \(\mu=A+\frac{B}{\lambda^{2}}\), where \(A\) and \(B\) are constants and \(\lambda\) is wavelength, then dimensions of \(B\) are same as that of

A rectangular coil of 300 turns has an average area of \(25 \mathrm{~cm} \times 10 \mathrm{~cm}\). The coil rotates with a speed of 50 cps in a uniform magnetic field of strength \(4 \times 10^{-2} \mathrm{~T}\) about an axis perpendicular to the fiel(d)The peak value of the induced emf (in volt) is

In intrinsic semiconductors at room temperature, number of electrons and holes are

Choose the correct metal/ ion from the brackets which ..........
A. has chemical reactivity similar to that of the first few members of the Lanthanoids \((\mathrm{Zn}, \mathrm{Ca}, \mathrm{Fe}, \mathrm{Cu})\).
B. has stable \(4 \mathrm{f}^7\) electronic configuration, but acts as a strong reducing agent and converts to \(\mathrm{M}^{3+}\) state. \(\left(\mathrm{Eu}^{2+}, \mathrm{Ce}^{2+}, \mathrm{Pr}^{2+}, \mathrm{Dy}^{2+}\right)\)
C. is a colorless ion \(\left(\mathrm{Tm}^{3+}, \mathrm{Lu}^{3+}, \mathrm{Gd}^{3+}, \mathrm{Sm}^{3+}\right)\).
D. shows stable +2 oxidation state and is diamagnetic \((\mathrm{Ce}, \mathrm{Sm}, \mathrm{Ho}, \mathrm{Yb})\)

For an adiabatic change in a system, the condition which is applicable is

An air filled parallel plate condenser has a capacity of \(2 \mathrm{pF}\). The separation of the plates is doubled and the interspace between the plates is filled with wax. If the capacity is increased to \(6 \mathrm{pF}\), then the dielectric constant of wax is

Carnot heat engine takes \(300 \mathrm{~J}\) of heat from a source at \(627^{\circ} \mathrm{C}\) and gives some part of it to \(\operatorname{sink}\) at \(27^{\circ} \mathrm{C}\). Work done by engine in one cycle is

\(\int \dfrac{x d x}{2(1+x)^{3 / 2}}\) is equal to

A series LCR circuit having \(\mathrm{R}=44 \Omega, \mathrm{~L}=2 \mathrm{H}\) and \(\mathrm{C}=25 \mu \mathrm{~F}\) is connected to a variable frequency of 220 V . What is the average power transferred to the circuit in one complete cycle if the frequency of supply equals the natural frequency of the circuit?