Two coils have mutual inductance \(0.002 \ \mathrm{H}\). The current changes in the first coil according to the relation \(\mathrm{i}=\mathrm{i}_0 \sin \omega \mathrm{t}\), where \(\mathrm{i}_0=5 \mathrm{~A}\) and \(\omega=50 \pi\) \(\mathrm{rad} / \mathrm{s}\). The maximum value of \(\mathrm{emf}\) in the second coil is \(\frac{\pi}{\alpha} \mathrm{V}\). The value of \(\alpha\) is_______.

A thermometer graduated according to a linear scale reads a value \(x_0\) when in contact with boiling water, and \(x_0/3\) when in contact with ice. What is the temperature of an object in \({}^oC\), if this thermometer in the contact with the object reads \(x_0/2\) ?

A \(16\  \Omega\) wire is bend to form a square loop. A \(9 \mathrm{~V}\) battery with internal resistance \(1\  \Omega\) is connected across one of its sides. If a \(4\  \mu \mathrm{F}\) capacitor is connected across one of its diagonals, the energy stored by the capacitor will be \(\frac{x}{2} \ \mu \mathrm{J}\). where \(x=\) ________.

In an experiment to determine the period of a simple pendulum of length \(1\, m\), it is attached to different spherical bobs of radii \(r_1\) and \(r_2\) . The two spherical bobs have uniform mass distribution. If the relative difference in the periods, is found to be \(5\times10^{-4}\, s\), the difference in radii, \(\left| {{r_1} - {r_2}} \right|\) is best given by .... \(cm\)

In order to determine the Young's Modulus of a wire of radius \(0.2\, cm\) (measured using a scale of least count \(=0.001\, cm )\) and length \(1 \,m\) (measured using a scale of least count \(=1\, mm\) ), a weight of mass \(1\, kg\) (measured using a scale of least count \(=1 \,g\) ) was hanged to get the elongation of \(0.5\, cm\) (measured using a scale of least count \(0.001\, cm\) ). What will be the fractional error in the value of Young's Modulus determined by this experiment? (in \(\%\))

Given below are two statements : one is labelled as Assertion (A) and the other is labelled as Reason (R).
Assertion-(A) : If Young's double slit experiment is performed in an optically denser medium than air, then the consecutive fringes come closer.
Reason-(R) : The speed of light reduces in an optically denser medium than air while its frequency does not change.
In the light of the above statements, choose the most appropriate answer from the options given below :

Two coaxial solenoids of different radii carry current \(I\) in the same direction. Let \(\;{\overrightarrow {\;F} _1}\) be the magnetic force on the inner solenoid due to the outer one and \(\;{\overrightarrow {\;F} _2}\) be the magnetic force on the outer solenoid due to the inner one. Then

A body of mass \(m\) is moving in a circular orbit of radius \(R\) about a planet of mass \(M\). At some instant, it splits into two equal masses. The first mass moves in a circular orbit of radius \(\frac{R}{2}\) , and the other mass, in a circular orbit of radius \(\frac{3R}{2}\). The difference between the final and initial total energies is

A bottle has an opening of radius \(a\) and length \(b\). A cork of length band radius \(\left( {a + \Delta a} \right)\) where \(\left( {\Delta a <  < a} \right)\) is compressed to fit into the opening completely (see figure). If the bulk modulus of cork is \(B\) and frictional coefficient between the bottle and cork is \(\mu \) then the force needed to push the cork into the bottle is

In the following circuit, the switch \(S\)  is closed at \(t = 0.\)  The charge on the capacitor  \(C_1\)  as a function of time will be given by \(\left( {{C_{eq}}\, = {\kern 1pt} \,\frac{{{C_1}{C_2}}}{{{C_1} + {C_2}}}} \right).\)

A particle of mass \(4\, {M}\) at rest disintegrates into two particles of mass \({M}\) and \(3 \,{M}\) respectively having non zero velocities. The ratio of de-Broglie wavelength of particle of mass \(M\) to that of mass \(3\, {M}\) will be :

An object of mass 8 kg is hanging from one end of a uniform rod \(C D\) of mass 2 kg and length 1 m pivoted at its end \(C\) on a vertical wall as shown in figure. It is supported by a cable \(A B\) such that the system is in equilibrium. The tension in the cable is:
(Take \(g=10 m s ^{-2}\) )

\(A\) and \(B\) are two sources generating sound waves. A listener is situated at \(C\) . The frequency of the source at \(A\) is \(500\,Hz\) . \(A,\) now, moves towards \(C\) with a speed \(4\,m/s.\) The number of beats heard at \(C\) is \(6.\) When \(A\) moves away from \(C\) with speed \(4\,m/s,\) the number of beats heard at \(C\) is \(18.\) The speed of sound is \(340\,m/s.\) The frequency of the source at \(B\) is ..... \(Hz\)