A certain pressure \(P\) is applied to \(1\) litre of water and \(2\) litre of a liquid separately. Water gets compressed to \(0.01 \%\) whereas the liquid gets compressed to \(0.03 \%\). The ratio of Bulk modulus of water to that of the liquid is \(\frac{3}{x}\). The value of \(x\) is \(...........\)

A circular disc of mass \(M\) and radius \(R\) is rotating about its axis with angular speed \(\omega_{1}\) If another stationary disc having radius \(\frac{ R }{2}\) and same mass \(M\) is dropped co-axially on to the rotating disc. Gradually both discs attain constant angular speed \(\omega_{2}\). The energy lost in the process is \(p \%\) of the initial energy. Value of \(p\) is

In a coil of resistance \(8 \Omega\), the magnetic flux due to an external magnetic field varies with time as \(\phi=\frac{2}{3}\left(9- t ^{2}\right)\). The value of total heat produced in the coil, till the flux becomes zero, will be \(....J\)

What will be the most suitable combination of three resistors \(A =2\, \Omega, B =4\, \Omega, C =6\, \Omega\) so that \(\left(\frac{22}{3}\right)\Omega\) is equivalent resistance of combination\(?\)

An ideal capacitor of capacitance \(0.2\, μF\) is charged to a potential difference of \(10\,V\). The charging battery is then disconnected. The capacitor is then connected to an ideal inductor of self inductance \(0.5\,mH\). The current at a time when the potential difference across the capacitor is \(5\,V\), is.....\(A\)

An ideal gas occupies a volume of \(2\, m^3\) at a pressure of \(3\times10^6\, Pa\). The energy of  the gas is

A bar magnet is released from rest along the axis of a very long vertical copper tube. After some time the magnet will

If the data \(x_1, x_2, ...., x_{10}\) is such that the mean of first four of these is \(11\), the mean of the remaining six is \(16\) and the sum of squares of all of these is \(2,000\); then the standard deviation of this data is

Let a line \(\mathrm{y}=\mathrm{mx}(\mathrm{m}>0)\) intersect the parabola, \(\mathrm{y}^{2}=\mathrm{x}\) at a point \(\mathrm{P},\) other than the origin. Let the tangent to it at \(P\) meet the \(x\) -axis at the point \(Q\). If area \((\Delta \mathrm{OPQ})=4\) sq. units, then \(\mathrm{m}\) is equal to

A light ray emits from the origin making an angle \(30^{\circ}\) with the positive \(x\)-axis. After getting reflected by the line \(x + y =1\), if this ray intersects \(x\)-axis at \(Q\), then the abscissa of \(Q\) is

Water is flowing at a speed of \(1.5\, ms^{-1}\) through a horizontal tube of cross-sectional area  \(10^{-2}\, m^2\) and you are trying to stop the flow by your palm. Assuming that the water stops immediately after hitting the palm. the minimum force that you must exert should be ......... \(N\) (density of water \(= 10^3\, kgm^{-3}\))

The position vectors of two 1 kg particles, (A) and (B), are given by
\(\overrightarrow{\mathrm{r}}_{\mathrm{A}}=\left(\alpha_1 \mathrm{t}^2 \hat{i}+\alpha_2 \mathrm{t} \hat{j}+\alpha_3 \mathrm{t} \hat{k}\right) \mathrm{m}\) and \(\overrightarrow{\mathrm{r}}_{\mathrm{B}}=(\beta_1 \mathrm{t} \hat{i}+\beta_2 \mathrm{t}^2 \hat{j}\) \(+\beta_3 \mathrm{t} \hat{k}) \mathrm{m}\), respectively; \((\alpha_1=1 \mathrm{~m} / \mathrm{s}^2, \alpha_2=3 \mathrm{n} \mathrm{m} / \mathrm{s},\) \(\alpha_3=2 \mathrm{~m} / \mathrm{s},\) \(\beta_1=2 \mathrm{~m} / \mathrm{s}, \beta_2=-1 \mathrm{~m} / \mathrm{s}^2, \beta_3=4 \mathrm{pm} / \mathrm{s})\), where t is time, n and p are constants. At \(t=1 \mathrm{~s},\left|\overrightarrow{V_A}\right|=\left|\vec{V}_B\right|\) and velocities \(\vec{V}_A\) and \(\vec{V}_B\) of the particles are orthogonal to each other. At \(t=1 \mathrm{~s}\), the magnitude of angular momentum of particle (A) with respect to the position of particle (B) is \(\sqrt{\mathrm{L}} \mathrm{kgm}^2 \mathrm{~s}^{-1}\). The value of L is _______ .

For \(\alpha, \beta, \gamma \neq 0\). If \(\sin ^{-1} \alpha+\sin ^{-1} \beta+\sin ^{-1} \gamma=\pi\) and \((\alpha+\beta+\gamma)(\alpha-\gamma+\beta)=3 \alpha \beta\), then \(\gamma\) equal to