A projectile is launched at an angle ' \(\alpha\) ' with the horizontal with a velocity \(20 \; ms ^{-1}\). After \(10 s\), its inclination with horizontal is ' \(\beta\) '. The value of \(\tan \beta\) will be : \(\left( g =10 \; ms ^{-2}\right)\)

A capacitance of \(2\ μF\) is required in an electrical circuit across a potential difference of \(1.0\ kV\). A large number of \(1\ μF\) capacitors are available which can withstand a potential difference of not more than \(300\ V\). The minimum number of capacitors required to achieve this is

Interference pattern is observed at \('P'\) due to superimposition of two rays coming out from a source \('S'\) as shown in the figure. The value of \('I'\) for which maxima is obtained at \('P'\) is: ( \(R\) is perfect reflecting surface)

A uniform heavy rod of mass \(20\,kg\). Cross sectional area \(0.4\,m ^{2}\) and length \(20\,m\) is hanging from a fixed support. Neglecting the lateral contraction, the elongation in the rod due to its own weight is \(x \times 10^{-9} m\). The value of \(x\) is (Given. Young's modulus \(Y =2 \times 10^{11} Nm ^{-2}\) અને \(\left.g=10\, ms ^{-2}\right)\)

A convex lens of refractive index 1.5 and focal length \(f=18 cm\) is immersed in water. The difference in focal lengths of the given lens when it is in water and in air is \(\alpha \times f\). The value of \(\alpha\) is ___________.
(refractive index of water \(=4 / 3\) )

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):

Match List\(-I\) with List\(-II\) :

List\(-I\)	List\(-II\)
\((a)\) \(\omega L\,>\,\frac{1}{\omega C}\)	\((i)\) Current is in phase with \(emf\)
\((b)\) \(\omega {L}=\frac{1}{\omega {C}}\)	\((ii)\) Current lags behind the applied \(emf\)
\((c)\) \(\omega {L}\, < \,\frac{1}{\omega {C}}\)	\((iii)\) Maximum current occurs
\((d)\) resonant frequency	\((iv)\) Current leads the \(emf\)

Choose the correct answer from the options given below :

The electric field of an electromagnetic wave in free space is \(\overrightarrow{\mathrm{E}}=57 \cos \left[7.5 \times 10^6 \mathrm{t}-5 \times 10^{-3}(3 x+4 y)\right](4 \hat{i}-3 \hat{j}) N / C\). The associated magnetic field in Tesla is _______.

A particle is moving \(5\) times as fast as an electron. The ratio of the de-Broglie wavelength of the particle to that of the electron is \(1.878 \times 10^{-4}\). The mass of the particle is close to

A cube has side length \(5\) cm and modulus of rigidity \(10^5\) N/m\(^2\). The displacement produced by a force of \(10\) N in the upper face of cube is _____ mm.

A capillary tube of radius 0.1 mm is partly dipped in water (surface tension \(70 \mathrm{dyn} / \mathrm{cm}\) and glass water contact angle \(\simeq 0^{\circ}\)) with \(30^{\circ}\) inclined with vertical. The length of water risen in the capillary is ________ cm.
\(\left(\right.\) Take \(\left.\mathrm{g}=9.8 \mathrm{~m} / \mathrm{s}^2\right)\)

A roller is mad e by joining  together two cones at their vertices \(O\). It is kept on two rails \(AB\) and \(CD\), which are placed asymmetrically (see figure) , with its axis perpendicular to \(CD  \) and its centre \(O\) at the centre ofline joining \(AB\) and \(CD\) (see figure) . It is given a light push so that it \(A\) starts rolling with its centre \(O\) moving parallel to \(CD\) in the direction shown. As it moves, the roller will tend to

In a series \(L-C-R\) circuit, \(C = 10^{-11}\,Farad.\) \(L = 10^{-5}\,Henry\) and \(R =100\,Ohm.\) when a constant \(D.C.\) voltage \(E\) is applied to the circuit, the capacitor acquires a charge \(10^{-9}\,C\) .The \(D.C.\) source is replaced by a sinusoidal voltage source in which the peak voltage \(E_0\) is equal to the constant \(D.C.\) voltage \(E.\) At resonance the peak value of the charge acquired by the capacitor will be