A plane electromagnetic wave is moving in free space with velocity \(c =3 \times 10^8 m / s\) and its electric field is given as \(\overrightarrow{ E }=54 \sin ( kz -\omega t ) \hat{ j } V / m\), where \(\hat{j}\) is the unit vector along \(y\)-axis. The magnetic field vector \(\overrightarrow{ B }\) of the wave is :

A plane electromagnetic wave travels in free space along the \(x -\) direction. The electric field component of the wave at a particular point of space and time is \(E =6\; Vm^{-1}\) along \(y -\) direction. Its corresponding magnetic filed component, \(B\) would be

List-\(I\)	List-\(II\)
\((a)\) \(MI\) of the rod (length \({L}\), Mass \({M}\), about an axis \(\perp\) to the rod passing through the midpoint)	\((i)\;\frac {8 {ML}^{2}}{3}\)
\((b)\) MI of the rod (length \(L\), Mass \(2 M\), about an axis \(\perp\) to the rod Passing through one its end)	\((ii)\;\frac {{ML}^{2}}{3}\)
\((c)\) MI of the rod (length \(2 {L}\), Mass \({M}\), about an axis \(\perp\) to the rod  Passing through its midpoint)	\((iii)\;\frac {{ML}^{2}}{12}\)
\((d)\) MI of the rod (length \(2 {L}\), Mass \(2 {M}\), about an axis \(\perp\) to the rod  passing through one of its end)	\((iv)\;\frac {2 {ML}^{2}}{3}\)

 Choose the correct answer from the options given below

Read the following statements: \((A)\) Volume of the nucleus is directly proportional to the mass number. \((B)\) Volume of the nucleus is independent of mass number. \((C)\) Density of the nucleus is directly proportional to the mass number. \((D)\) Density of the nucleus is directly proportional to the cube root of the mass number. \((E)\) Density of the nucleus is independent of the mass number. Choose the correct option from the following options.

Two simple pendulums of length \(1\, m\) and \(4\, m\) respectively are both given small displacement in the same direction at the same instant. They will be again in phase after the shorter pendulum has completed number of oscillations equal to

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

An upright object is placed at a distance of \(40\, cm\) in front of a convergent lens of focal length \(20\, cm\). A convergent mirror of focal length \(10\, cm\) is placed at a distance of \(60\, cm\) on the other side of the lens. The position and size of the final image will be

A car is standing \(200\, m\) behind a bus, which is also at rest. The two start moving at the same instant but with different forward accelerations. The bus has acceleration \(2\,m/s^2\) and the car has acceleration \(4\, m/s^2\) . The car will catch up with the bus after a time of

A thin disc of mass \(M\) and radius \(R\) has mass per unit area \(\sigma (r) = kr^2\) where \(r\) is the distance from its centre. Its moment of inertia about an axis going through its centre of mass and perpendicular to its plane is

Statement \(-1\) : The system of linear equations \(x + \left( {\sin \,\alpha } \right)y + \left( {\cos \,\alpha } \right)z = 0\) \(x + \left( {\cos \,\alpha } \right)y + \left( {\sin \alpha } \right)z = 0\) \(x - \left( {\sin \,\alpha } \right)y - \left( {\cos \alpha } \right)z = 0\) has a non-trivial solution for only one value of \(\alpha \) lying in the interval \(\left( {0\,,\,\frac{\pi }{2}} \right)\)  Statement \(-2\) : The equation in \(\alpha \) \(\left| {\begin{array}{*{20}{c}}
  {\cos {\mkern 1mu} \alpha }&{\sin {\mkern 1mu} \alpha }&{\cos {\mkern 1mu} \alpha } \\ 
  {\sin {\mkern 1mu} \alpha }&{\cos {\mkern 1mu} \alpha }&{\sin {\mkern 1mu} \alpha } \\ 
  {\cos {\mkern 1mu} \alpha }&{ - \sin {\mkern 1mu} \alpha }&{ - \cos {\mkern 1mu} \alpha } 
\end{array}} \right| = 0\) has only one solution lying in the interval \(\left( {0\,,\,\frac{\pi }{2}} \right)\)

Which one is the correct option for the two different thermodynamic processes ?

If \({E}\) and \({H}\) represents the intensity of electric field and magnetising field respectively, then the unit of \(E/H\) will be :

If \(\left(\sin ^{-1} x\right)^{2}-\left(\cos ^{-1} x\right)^{2}=a ; 0\,<\,x\,<\,1, a \neq 0\), then the value of \(2 \mathrm{x}^{2}-1\) is :