The half-life of a radioactive nucleus is \(5\) years, The fraction of the original sample that would decay in \(15\) years is

Six equal resistances are connected between points \(P, Q\) and \(R\) as shown in figure. Then net resistance will be maximum between

Imagine that a reactor converts all given mass into energy and that it operates at a power level of \(10^9\, watt\). The mass of the fuel consumed per hour in the reactor will be : (velocity of light, \(c\) is \(3\times10^8\, m/s\))

An electron with mass ' \(m\) ' with an initial velocity \((\mathrm{t}=0) \overrightarrow{\mathrm{v}}=\mathrm{v}_0 \hat{\mathrm{i}} \quad\left(\mathrm{v}_0 \gt 0\right)\) enters a magnetic field \(\vec{B}=B_0 \hat{j}\). If the initial de-Broglie wavelength at \(\mathrm{t}=0\) is \(\lambda_0\) then its value after time ' t ' would be :

If \(B\) is magnetic field and \(\mu_0\) is permeability of free space, then the dimensions of \(\left(B / \mu_0\right)\) is

A reversible engine has an efficiency of \(\frac{1}{4}\). If the temperature of the sink is reduced by \(58^{\circ} {C}\), its efficiency becomes double. Calculate the temperature of the sink. (In \(^{\circ} {C}\))

A particle is executing simple harmonic motion \((SHM)\) of amplitude \(A,\) along the \(x-\) axis, about \(x = 0.\) When its potential energy \((PE)\) equals kinetic energy \((KE),\) the position of the particle will be

The displacement time graph of a particle executing \(S.H.M.\) is given in figure: (sketch is schematic and not to scale) Which of the following statements is are true for this motion? \((A)\) The force is zero \(t=\frac{3 T}{4}\) \((B)\) The acceleration is maximum at \(t=T\) \((C)\) The speed is maximum at \(t =\frac{ T }{4}\) \((D)\) The \(P.E.\) is equal to \(K.E.\) of the oscillation at \(t=\frac{T}{2}\)

A pole is vertically submerged in swimming pool, such that it gives a length of shadow \(2.15 \,m\) within water when sunlight is incident at an angle of \(30^{\circ}\) with the surface of water. If swimming pool is filled to a height of \(1.5\,m\), then the height of the pole above the water surface in centimetres is \(\left(\eta_W=4 / 3\right)........\)

The r.m.s speed of oxygen molecules at \(47^{\circ} C\) is equal to that of the hydrogen molecules kept at _________ \({ }^{\circ} C\). (Mass of oxygen molecule/mass of hydrogen molecule \(=32 / 2\) )

A parallel plate air capacitor has a capacitance \(C\). When it is half filled as show in figure with a dielectric constant \(K = 5\), the percentage increase in the capacitance is _______.

Two cars A and B each of mass \(10^3 kg\) are moving on parallel tracks separated by a distance of 10 m , in same direction with speeds \(72 km / h\) and \(36 km / h\). The magnitude of angular momentum of car \(A\) with respect to car \(B\) is _________ J.s.