In an \(LCR\) series circuit, an inductor \(30 \,{mH}\) and a resistor \(1 \, \Omega\) are connected to an \(AC\) source of angular frequency \(300 \, {rad} / {s}\). The value of capacitance for which, the current leads the voltage by \(45^{\circ}\) is \(\frac{1}{x} \times 10^{-3} \, {F}\). Then the value of \(x\) is ..... .

in circular plate of mass \(M\) and radius \(R\) has its density varying as \(p\left( r \right) = {p_0}\,r\) with \(P_0\) as constant and \(r\) is the distance from its center. The moment of Inertia of the circular plate about an axis perpendicular to the plate and passing through its edge is \(I = aMR^2\) . The value of the coefficient \(a\) is

A bar magnet has total length \(2 l=20\) units and the field point \(P\) is at a distance \(\mathrm{d}=10\) units from the centre of the magnet. If the relative uncertainty of length measurement is \(1 \%\), then uncertainty of the magnetic field at point P is :

The magnitude of torque on a particle of mass \(1\,kg\) is \(2.5\, Nm\) about the origin. If the force acting on it is \(1\,N\), and the distance of the particle from the origin is \(5\,m\), the angle between the force and the position vector is (in radians)

The magnetic flux through a coil perpendicular to its plane is varying according to the relation \(\phi=\) \(\left(5 t^{3}+4 t^2+2 t-5\right) \; Weber\). If the resistant of the coil is \(5 \; ohm\), then the induced current through the coil at \(t =2 \; sec\) will be \(....\,A\)

An object is kept at rest at a distance of 3 R above the earth's surface where R is earth's radius. The minimum speed with which it must be projected so that it does not return to earth is :
(Assume \(\mathrm{M}=\) mass of earth, \(\mathrm{G}=\) Universal gravitational constant)

'\(n\)' moles of an ideal gas undergoes a process \(A \rightarrow B\) as shown in the figure. The maximum temperature of the gas during the  process will be

A body of mass \(1 kg\) begins to move under the action of a time dependent force \(\vec{F}=\left(t \hat{i}+3 t^2 \hat{ j }\right) N\). where \(\hat{i}\) and \(\hat{j}\) are the unit vectors along \(x\) and \(y\) axis. The 10.Power developed by above force, at the time \(t=2 s\). will be \(..............W\).

Consider a water tank as shown in the figure. It's cross-sectional area is \(0.4\, m ^{2}\). The tank has an opening \(B\) near the bottom whose cross-section area is \(1\, cm ^{2}\). A load of \(24\, kg\) is applied on the water at the top when the height of the water level is \(40\, cm\) above the bottom, the velocity of water coming out the opening \(B\) is \(v\, ms ^{-1}\). The value of \(v\), to the nearest integer, is ......\(m/s\). [Take value of \(g\) to be \(10 \,ms ^{-2}\) ]

The electric field a plane electromagnetic wave is given by :
\(E _y=69 \sin \left[0.6 \times 10^3 x -1.8 \times 10^{11} t \right] V / m\).
The expression for magnetic field associated with this electromagnetic wave is _________ T.