A force \(\overrightarrow{ F }=4 \hat{ i }+3 \hat{ j }+4 \hat{ k }\) is applied on an intersection point of \(x =2\) plane and \(x\) -axis. The magnitude of torque of this force about a point \((2,3,4)\) is.......... . (Round off to the Nearest Integer)

A body of mass ' \(m\) ' connected to a massless and unstretchable string goes in verticle circle of radius ' \(R\) 'under gravity \(g\). The other end of the string is fixed at the center of circle. If velocity at top of circular path is \(n \sqrt{g R}\), where, \(n \geqslant 1\), then ratio of kinetic energy of the body at bottom to that at top of the circle is

An ideal gas enclosed in a vertical cylindrical container supports a freely moving piston of mass \(M\). The piston and the cylinder have equal cross sectional area \(A\). When the piston is in equilibrium, the volume of the gas is \(V_0\) and its pressure is \(P_ 0\). The piston is slightly displaced from the equilibrium position and released. Assuming that the system is completely isolated from its surrounding, the piston executes a simple harmonic motion with frequency

The half-life of \({ }^{198} {Au}\) is \(3 \,days.\) If atomic weight of \({ }^{198} {Au}\) is \(198\, {g} / {mol}\) then the activity of \(2 \,{mg}\) of \({ }^{198} {Au}\) is ..... \(\times 10^{12}\,disintegration/second\)

The water is filled upto height of \(12\, {m}\) in a tank having vertical sidewalls. A hole is made in one of the walls at a depth \('{h}'\) below the water level. The value of \('h'\) for which the emerging stream of water strikes the ground at the maximum range is \(.....\,{m}\).

A body of mass \(M\) at rest explodes into three pieces, in the ratio of masses \(1: 1: 2\). Two smaller pieces fly off perpendicular to each other with velocities of \(30 \,ms ^{-1}\) and \(40 \,ms ^{-1}\) respectively. The velocity of the third piece will be ............... \(\,ms ^{-1}\)

A conducting bar moves on two conducting rails as shown in the figure. A constant magnetic field B exists into the page. The bar starts to move from the vertex at time \(t=0\) with a constant velocity. If the induced EMF is \(\mathrm{E} \propto \mathrm{t}^{\mathrm{n}}\), then value of n is ________.

A particle executes \(S.H.M.,\) the graph of velocity as a function of displacement is :-

A current carrying rectangular loop PQRS is made of uniform wire. The length \(PR = QS =5\,cm\) and \(PQ = RS =100\,cm\). If ammeter current reading changes from I to \(2 I\), the ratio of magnetic forces per unit length on the wire \(P Q\) due to wire RS in the two cases respectively \(f_{ PQ }^{ I }: f_{ PQ }^{2 I }\) is :

A rigid massless road of length \(3l\) has tow masses attached at each end as shown in the figure. The rod is pivoted at point \(P\) on the horizontal axis (see figure). When released from initial horizontal position, its instantaneous angular acceleration will be

Two identical bodies \(A\) and \(B\) of equal masses have initial velocities \(\vec{v_1} = 4\hat{i}\) m/s and \(\vec{v_2} = 4\hat{j}\) m/s respectively. The body \(A\) has acceleration \(\vec{a_1} = 6\hat{i} + 6\hat{j}\) m/s\(^2\) while the acceleration of the other body \(B\) is zero. The centre of mass of the two bodies moves in __________ path.

A null point is found at \(200\,cm\) in potentiometer when cell in secondary circuit is shunted by \(5\,\Omega\). When a resistance of \(15\,\Omega\) is used for shunting null point moves to \(300\,cm\). The internal resistance of the cell is \(..............\,\Omega\).

The de Broglie wavelength of a molecule in a gas at room temperature \((300\,K )\) is \(\lambda_1\). If the temperature of the gas is increased to \(600\,K\), then the de Broglie wavelength of the same gas molecule becomes \(..............\).