The radius of \(R\) of a nucleus of mass number \(A\) can be estimated by the formula \(R =\left(1.3 \times 10^{-15}\right) A ^{1 / 3}\, m .\) It follows that the mass density of a nucleus is of the order of \(\left( M _{\text {prot. }} \cong M _{\text {neut. }}=1.67 \times 10^{-27} kg \right)\)

A certain elastic conducting material is stretched into a circular loop. It is placed with its plane perpendicular to a uniform magnetic field \(B =0.8\,T\). When released the radius of the loop starts shrinking at a constant rate of \(2\,cm ^{-1}\). The induced emf in the loop at an instant when the radius of the loop is \(10\,cm\) will be \(........mV\).

A butterfly is flying with a velocity \(4 \sqrt{2} \,{m} / {s}\) in North-East direction. Wind is slowly blowing at \(1\) \({m} / {s}\) from North to South. The resultant displacement of the butterfly in \(3\, seconds\) is \(....\,{m}.\)

A charge particle is moving in a uniform magnetic field \((2 \hat{i}+3 \hat{j}) T\). If it has an acceleration of \((\alpha \hat{i}-4 \hat{j}) m / s ^{2}\), then the value of \(\alpha\) will be.

A small cube of side \(1\) mm is placed at the centre of a circular loop of radius \(10\) cm carrying a current of \(2\) A. The magnetic energy stored inside the cube is \(\alpha \times 10^{-14}\) J. The value of \(\alpha\) is _______. (\(\mu_o = 4\pi \times 10^{-7}\) Tm/A, \(\pi = 3.14\))

If the radius of earth is reduced to three-fourth of its present value without change in its mass then value of duration of the day of earth will be _______ hours \(30\) minutes.

Two point charges \(8 \, \mu C\) and \(-2 \, \mu C\) are located at \(x = 2\) cm and \(x = 4\) cm, respectively on the \(x\)-axis. The ratio of electric flux due to these charges through two spheres of radii \(3\) cm and \(5\) cm with their centers at the origin is _______.

For particle \(P\) revolving round the centre \(O\) with radius of circular path \(r\) and angular velocity \(\omega\), as shown in below figure, the projection of \(OP\) on the \(x\)-axis at time \(t\) is \(.................\).

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 ________.

Given below are two statements: \(Statement\) \(I\) : If a capillary tube is immersed first in cold water and then in hot water, the height of capillary rise will be smaller in hot water. \(Statement\) \(II\) : If a capillary tube is immersed first in cold water and then in hot water, the height of capillary rise will be smaller in cold water. In the light of the above statements, choose the most appropriate from the options given below

Given below are two statements : one is labelled as Assertion A and the other is labelled as Reason \(R\). Assertion A : Efficiency of a reversible heat engine will be highest at \(-273^{\circ} C\) temperature of cold reservoir. Reason \(R\) : The efficiency of Carnot's engine depends not only on temperature of cold reservoir but it depends on the temperature of hot reservoir too and is given as \(\eta=\left(1-\frac{ T _2}{ T _1}\right)\). In the light of the above statements, choose the correct answer from the options given below