An \(LCR\) series circuit of capacitance \(62.5\,nF\) and resistance of \(50\,\Omega\). is connected to an \(A.C.\) source of frequency \(2.0\,kHz\). For maximum value of amplitude of current in circuit, the value of inductance is \(..........mH\). (take \(\left.\pi^2=10\right)\)

When an object is kept at a distance of \(30\, cm\) from a concave mirror, the image is formed at a distance of \(10\, cm\) from the mirror. If the object is moved with a speed of \(9\, cms ^{-1}\), the speed (in \(cms ^{-1}\) ) with which image moves at that instant is\(....m/s\)

A plane electromagnetic wave propagating in \(\mathrm{x}\)-direction is described by \(\mathrm{E}_{\mathrm{y}}=\left(200\  \mathrm{Vm}^{-1}\right) \sin \left[1.5 \times 10^7 \mathrm{t}-0.05\  \mathrm{x}\right] \text {; }\) The intensity of the wave is :(Use \(\epsilon_0=8.85 \times 10^{-12} \mathrm{C}^2 \mathrm{~N}^{-1} \mathrm{~m}^{-2}\) )

The area of cross-section of a railway track is \(0.01\, {m}^{2}\). The temperature variation is \(10^{\circ} {C}\). Coefficient of linear expansion of material of track is \(10^{-5} /{ }^{\circ} {C}\). The energy stored per meter in the track is ...... \({J} / {m} .\) (Young's modulus of material of track is \(10^{11} \,{Nm}^{-2}\) ))

A driver in a car, approaching a vertical wall notices that the frequency of his car horn, has changed from \(440\, Hz\) to \(480 \,Hz ,\) when it gets reflected from the wall. If the speed of sound in air is \(345 \,m / s ,\) then the speed of the car is \(.......km/hr\)

From the top of a tower, a ball is thrown vertically upward which reaches the ground in \(6 \,s\). A second ball thrown vertically downward from the same position with the same speed reaches the ground in \(1.5\, s\). A third ball released, from the rest from the same location, will reach the ground in........ \(s\).

\(0.08 \mathrm{~kg}\) air is heated at constant volume through \(5^{\circ} \mathrm{C}\). The specific heat of air at constant volume is \(0.17 \mathrm{kcal} / \mathrm{kg}^{\circ} \mathrm{C}\) and \(\mathrm{J}=4.18\) joule \(/ \mathrm{cal}\). The change in its internal energy is approximately.

In finding out refractive index of glass slab the following observations were made through travelling microscope \(50\) vernier scale division \(=\) \(49 \mathrm{MSD} ; 20\) divisions on main scale in each \(\mathrm{cm}\) For mark on paper \(\mathrm{MSR}=8.45 \mathrm{~cm}, \mathrm{VC}=26\) For mark on paper seen through slab \(\mathrm{MSR}=7.12 \mathrm{~cm}, \mathrm{VC}=41\) For powder particle on the top surface of the glass slab \(\mathrm{MSR}=4.05 \mathrm{~cm}, \mathrm{VC}=1\) \((\mathrm{MSR}=\) Main Scale Reading, \(\mathrm{VC}=\) Vernier Coincidence) Refractive index of the glass slab is _______.

A uniform disc of radius \(R\) and mass \(M\) is free to oscillate about the axis \(A\) as shown in the figure. For small oscillations the time period is ______.
(\(g\) is acceleration due to gravity)

A \(0.5\) kg mass is in contact against the inner wall of a cylindrical drum of radius \(4\) m rotating about its vertical axis. The minimum rotational speed of the drum to enable the mass to remain stuck to the wall (without falling) is \(5\) rad/s. The coefficient of friction between the drum's inner wall surface and mass is _______. (Take \(g = 10\) m/s\(^2\))

Match List \(I\) with List \(II\) of Electromagnetic waves with corresponding wavelength range :

List \(I\)	List \(II\)
\((A)\) Microwave	\((I)\) \(400\,nm\) to \(1\,nm\)
\((B)\) Ultraviolet	\((II)\) \(1\,nm\) to \(10^{-3}\,nm\)
\((C)\) \(X-\)Ray	\((III)\) \(1\,mn\) to \(700\,nm\)
\((D)\) Infra-red	\((IV)\) \(0.1\,m\) to \(1\,mm\)

Choose the correct answer from the options given below:

Which of the following relations is true for two unit vectors \(\hat{ A }\) and \(\hat{ B }\) making an angle \(\theta\) to each other\(?\)