A telescope has an objective lens of focal length \(150\,\,cm\) and an eyepiece of focal length \(5 \,\,cm.\) If a \(50\,\,m\) tall tower at a distance of \(1\,\,km\) is observed through this telescope in normal setting, the angle formed by the image of the tower is \(\theta \), then \(\theta \) is close to.....\(^o\)

Seawater at a frequency \(f =9 \times 10^{2} \,Hz ,\) has permittivity \(\varepsilon=80\, \varepsilon_{0}\) and resistivity \(\rho=0.25\, \Omega m.\) Imagine a parallel plate capacitor is immersed in seawater and is driven by an alternating voltage source \(V ( t )= V _{0} \sin (2 \pi ft ) .\) Then the conduction current density becomes \(10^{x}\) times the displacement current density after time \(t =\frac{1}{800} \,s\). The value of \(x\) is ......... .
\(\left(\frac{1}{4 \pi \varepsilon_{0}}=9 \times 10^{9} Nm ^{2} C ^{-2}\right)\)

If electric field intensity of a uniform plane electro magnetic wave is given as \(E =-301.6 \sin ( kz -\omega t ) \hat{a}_{ x }+452.4 \sin ( kz -\omega t )\) \(\hat{a}_{y} \frac{V}{m}\) Then, magnetic intensity \(H\) of this wave in \(Am ^{-1}\) will be [Given: Speed of light in vacuum \(c =3 \times 10^{8} ms ^{-1}\), permeability of vacuum \(\mu_{0}=4 \pi \times 10^{-7} NA ^{-2}\) ]

Given below are two statements Statement \(I\) : When the frequency of an \(a.c.\) source in a series \(LCR\) circuit increases, the current in the circuit first increases, attains a maximum value and then decreases. Statement \(II:\) In a series \(LCR\) circuit, the value of power factor at resonance is one. In the light of given statements, choose the most appropriate answer from the options given below

Two identical solid spheres each of mass \(2\,kg\) and radii \(10\,cm\) are fixed at the ends of a light rod. The separation between the centres of the spheres is \(40\,cm\). The moment of inertia of the system about an axis perpendicular to the rod passing through its middle point is \(........... \times 10^{-3}\,kg - m ^2\)

From the given data, the amount of energy required to break the nucleus of aluminium \({ }_{13}^{27} {Al}\) is \(....\,x \times 10^{-3} {J}\) Mass of neutron \(=1.00866 \,{u}\) Mass of proton \(=1.00726 \,{u}\) Mass of aluminium nucleus \(=27.18846\, {u}\) (Assume \(1\,u\) corresponds to \(x\,J\) of energy) (Round off to the nearest integer)

An element \(\Delta l=\Delta \mathrm{xi}\) is placed at the origin and carries a large current \(\mathrm{I}=10 \mathrm{~A}\). The magnetic field on the \(y\)-axis at a distance of \(0.5 \mathrm{~m}\) from the elements \(\Delta \mathrm{x}\) of \(1 \mathrm{~cm}\) length is _______.

A simple pendulum is placed at a place where its distance from the earth's surface is equal to the radius of the earth. If the length of the string is \(4 \mathrm{~m}\), then the time period of small oscillations will be _______ \(s\). [take \(\mathrm{g}=\pi^2 \mathrm{~ms}^{-2}\)]

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 solid cylinder is released from rest from the top of an inclined plane of inclination \(30^{\circ}\) and length \(60\,cm\). If the cylinder rolls without slipping, its speed upon reaching the bottom of the inclined plane is \(...........ms ^{-1}\). (Given \(g =10\,ms ^{-2}\))

A body cools from \(80^{\circ}\,C\) to \(60^{\circ}\,C\) in \(5\) minutes. The temperature of the surrounding is \(20^{\circ} C\). The time it takes to cool from \(60^{\circ}\,C\) to \(40^{\circ}\,C\) is........... \(s\)

A wheatstone bridge is initially at room temperature and all arms of the bridge have same value of resistances ( \(R_1=R_2=R_3=R_4\) ). When \(R_3\) resistance is heated to some temperature, its resistance value has gone up by \(10 \%\). The potential difference \(\left( V _{ a }- V _{ b }\right)\) (after \(R _3\) is heated) is ___________ V.

An electric bulb rated as \(100 \mathrm{~W}-220 \mathrm{~V}\) is connected to an ac source of rms voltage 220 V. The peak value of current through the bulb is :