An electric dipole is formed by two equal and opposite charges \(q\) with separation \(d\). The charges the same mass \(m\). It is kept in a uniform electric field \(E\). If it slightly rotated from its equilibrium orientation, then its angular frequency \(\omega \) is

Given below are two statements; one is labelled as Assertion \(A\) and the other is labelled as Reason \(R\). Assertion \(A:\) Electromagnets are made of soft iron. Reason \(R:\) Soft iron has high permeability and low retentivity. In the light of above, statements, choose the most appropriate answer from the options given below.

A cylindrical vessel of \(40\) cm radius is completely filled with water and its capacity is \(528\) dm\(^3\) (dm : decimeter). The vessel is placed on a solid block of exactly same height as vessel. If a small hole is made at \(70\) cm below the top of water level, then horizontal range of water falling on the ground in the beginning is __________ cm.

Two blocks of masses \(m\) and \(M,(M \gt m)\), are placed on a frictionless table as shown in figure. A massless spring with spring constant k is attached with the lower block. If the system is slightly displaced and released then
(\(\mu=\) coefficient of friction between the two blocks)

(A) The time period of small oscillation of the two blocks is \(\mathrm{T}=2 \pi \sqrt{\frac{(\mathrm{~m}+\mathrm{M})}{\mathrm{k}}}\)
(B) The acceleration of the blocks is \(\mathrm{a}=\frac{\mathrm{kx}}{\mathrm{M}+\mathrm{m}}\) (\(\mathrm{x}=\) displacement of the blocks from the mean position)
(C) The magnitude of the frictional force on the upper block is \(\frac{m \mu|x|}{M+m}\)
(D) The maximum amplitude of the upper block, if it does not slip, is \(\frac{\mu(M+m) g}{k}\)
(E) Maximum frictional force can be \(\mu(\mathrm{M}+\mathrm{m}) \mathrm{g}\).
Choose the correct answer from the options given below:

In the potentiometer, when the cell in the secondary circuit is shunted with 4 \(\Omega\) resistance, the balance is obtained at the length 120 cm of wire. Now when the same cell is shunted with 12 \(\Omega\) resistance, the balance is shifted to a length of 180 cm. The internal resistance of cell is ___________ \(\Omega\).

The half life period of a radioactive substance is 60 days. The time taken for \(\frac{7}{8}\) th of its original mass to disintegrate will be \(......days\) 

For an electromagnetic wave travelling in free space, the relation between average energy densities due to electric \(\left( U _{ e }\right)\) and magnetic \(\left( U _{ m }\right)\) fields is

Consider a completely full cylindrical water tank of height 1.6 m and cross-sectional area \(0.5 \mathrm{~m}^2\). It has a small hole in its side at a height 90 cm from the bottom. Assume, the cross-sectional area of the hole to be negligibly small as compared to that of the water tank. If a load 50 kg is applied at the top surface of the water in the tank then the velocity of the water coming out at the instant when the hole is opened is : \(\left(\mathrm{g}=10 \mathrm{~m} / \mathrm{s}^2\right)\)

If \(y + 3x = 0\) is the equation of a chord of the circle, \(x^2 + y^2 - 30x = 0,\) then the equation of the circle with this chord as diameter is

If \(A\) and \(B\) are two events such that \(P ( A )=\frac{1}{3}, P ( B )=\frac{1}{5} \) and \(P ( A \cup B )=\frac{1}{2}\), then \(P \left( A \mid B ^{\prime}\right)+ P \left( B \mid A ^{\prime}\right)\) is equal to

For an adiabatic expansion of an ideal gas, the fractional change in its pressure is equal to (where \(\gamma\) is the ratio of specific heats):

The sum of the first \(20\) terms common between the series \(3 +7 + 1 1 + 15+ ... ......\) and \(1 +6+ 11 + 16+ ......\), is

Let a circle \(C\) of radius \(5\) lie below the \(x\)-axis. The line \(L_{1}=4 x+3 y-2\) passes through the centre \(P\) of the circle \(C\) and intersects the line \(L _{2}: 3 x -4 y -11=0\) at \(Q\). The line \(L _{2}\) touches \(C\) at the point \(Q\). Then the distance of \(P\) from the line \(5 x-12 y+51=0\) is