The equation of motion of a particle executing simple harmonic motion is given by \(x=3 \sin \left(6 t+\frac{\pi}{6}\right)\), where \(x\) is in metres and \(t\) is in seconds. The ratio of the potential and kinetic energies of the particle at time \(t=0\) is

The magnitude of the induced emf in a coil of inductance \(30 \mathrm{mH}\) in which the current changes from \(6 \mathrm{~A}\) to \(2 \mathrm{~A}\) in 2 \(\mathrm{s}\) is

A body of mass $10 \mathrm{kg}$ is attached to a wire of $0.3 \mathrm{m}$ length. The breaking stress is $4.8\times {10}^7 \mathrm{N} {\mathrm{m}}^{-2}$. The area of cross-section from the wire is ${10}^{-6}{ \mathrm{m}}^2$. The maximum angular velocity with which it can be rotated in a horizontal circle is

One end of a uniform metal rod of length \(100 \mathrm{~cm}\) is placed in ice and the other end is placed in boiling water. A point of the rod which is at a distance of \(60 \mathrm{~cm}\) from the ice end is maintained at a constant temperature of \(325^{\circ} \mathrm{C}\). If \(2 \mathrm{~g}\) of water is converted into steam per second, the mass of ice melted per second in steady state is
(Latent heat of steam \(=6.75\) times latent heat of fusion of ice)

Two waves are represented by: $x_1=A\sin \left(\omega t+\frac{\pi }{6}\right)$ and $x_2=A\cos \omega t$. Then the phase difference between them is

A coil of mean area $500 {\mathrm{cm}}^2$ and having $1000$ turns is held with its plane perpendicular to a uniform field of $0.4 \mathrm{G}$. If the coil is turned through $180°$ in $\frac{1}{10}$ second, then the average induced emf is $\left(1\mathrm{G}={10}^{-4} \mathrm{T}\right)$

A projectile is given an initial velocity of \((\hat{\mathbf{i}}+2 \hat{\mathbf{j}}) \mathrm{ms}^{-1}\). The equation of its path is \(\left(g=10 \mathrm{~ms}^{-2}\right)\)

As shown in the figure, two infinitely long straight parallel wires \(P\) and \(Q\) carrying equal currents in opposite directions are arranged parallel to \(\mathrm{Y}\)-axis. If the magnetic field due to wire \(\mathrm{P}\) at the origin ' \(\mathrm{O}\) ' of the co-ordinate system is B, then match the resultant magnetic fields at various points given in column \(A\) with the points given in column \(B\).
\(\begin{array}{ll}
\text{Column A} & \text{Column B} \\
\text{A)} \frac{B}{4} & \text{i)} (0,0) \\
\text{B)} \frac{B}{2} & \text{ii)} (a, 0) \\
\text{C)} \frac{2 \mathrm{~B}}{3} & \text{iii)} (2a, 0) \\
\text{D)} 2 \mathrm{~B} & \text{iv)} (3 \mathrm{a}, 0) \\
\end{array}\)


The correct answer is

If the uncertainty in momentum and uncertainty in the position of a particle are equal, then the uncertainty in its velocity would be given by

If \(3 f(\cos x)+2 f(\sin x)=5 x\), then \(f^{\prime}(\cos x)+f^{\prime}(\sin x)=\)

The following graph is obtained for an ideal solution containing a non-volatile solute, \(x\) - and \(y\)-axis represent, respectively

Two point charges \(+6 \mu \mathrm{C}\) and \(+10 \mu \mathrm{C}\) kept at certain distance repel each other with a force of 30 N . If each charge is given an additional charge of \(-8 \mu \mathrm{C}\), the two charges

The mean deviation from the median for the following data is