The nuclear activity of a radioactive element becomes \(\left(\frac{1}{8}\right)^{\text {th }}\) of its initial value in \(30\, years.\) The half-life of radioactive element is \(....\,years.\)

A current of \(2\, mA\) was passed through an unknown resistor which dissipated a power of \(4.4\, W\). Dissipated power when an ideal power supply of \(11\, V\) is connected across it is

A collimated beam of light of diameter 2 mm is propagating along x -axis. The beam is required to be expanded in a collimated beam of diameter 14 mm using a system of two convex lenses. If first lens has focal length 40 mm , then the focal length of second lens is _________ mm.

With the help of potentiometer, we can determine the value of emf of a given cell. The sensitivity of the potentiometer is \((A)\) directly proportional to the length of the potentiometer wire \((B)\) directly proportional to the potential gradient of the wire \((C)\) inversely proportional to the potential gradient of the wire \((D)\) inversely proportional to the length of the potentiometer wire Choose the correct option for the above statements:

A ceiling fan having 3 blades of length \(80 \mathrm{~cm}\) each is rotating with an angular velocity of \(1200\  \mathrm{rpm}\). The magnetic field of earth in that region is \(0.5 \ \mathrm{G}\) and angle of dip is \(30^{\circ}\). The emf induced across the blades is \(N \pi \times 10^{-5} \mathrm{~V}\). The value of \(\mathrm{N}\) is _______.

Due to cold weather a \(1\, {m}\) water pipe of cross-sectional area \(1\, {cm}^{2}\) is filled with ice at \(-10^{\circ} {C}\). Resistive heating is used to melt the ice. Current of \(0.5\, {A}\) is passed through \(4\, {k} \Omega\) resistance. Assuming that all the heat produced is used for melting, what is the minimum time required ? (In \({s}\)) (Given latent heat of fusion for water/ice \(=3.33 \times 10^{5}\, {J} {kg}^{-1}\), specific heat of ice \(=2 \times 10^{3}\, {J}\) \({kg}^{-1}\) and density of ice \(=10^{3}\, {kg} / {m}^{3}\)

A current \(i\) is flowing in a straight conductor of length \(L.\) The magnetic induction at a point on its axis at a distance \(\frac {L}{4}\) from its centre will be

Two light waves having the same wavelength \(\lambda\) in vacuum are in phase initially. Then the first wave travels a path \(L _{1}\) through a medium of refractive index \(n_{1}\) while the second wave travels a path of length \(L_{2}\) through a medium of refractive index \(n _{2}\). After this the phase difference between the two waves is:

In the following nuclear rection, \(D \stackrel{\alpha}{\longrightarrow} D _{1} \stackrel{\beta-}{\longrightarrow} D _{2} \stackrel{\alpha}{\longrightarrow} D _{3} \stackrel{\gamma}{\longrightarrow} D _{4}\) Mass number of \(D\) is \(182\) and atomic number is \(74\) . Mass number and atomic number of \(D _{4}\) respectively will be

Two blocks of masses \(10 \; kg\) and \(30 \; kg\) are placed on the same straight line with coordinates \((0,0) \; cm\) and \(( x , 0) \; cm\) respectively. The block of \(10 \; kg\) is moved on the same line through a distance of \(6 \; cm\) towards the other block. The distance through which the block of \(30 \; kg\) must be moved to keep the position of centre of mass of the system unchanged is

In an ac circuit, the instantaneous current is zero, when the instantaneous voltage is maximum. In this case, the source may be connected to: \(A\). pure inductor. \(B\). pure capacitor. \(C\). pure resistor. \(D\). combination of an inductor and capacitor. Choose the correct answer from the options given below:

A thin lens made of glass (refractive index \(= 1.5\)) of focal length \(f=16\; \mathrm{cm}\) is immersed in a liquid of refractive index \(1.42 .\) If its focal length in liquid is \(f_{l},\) then the ratio \(f_{l} /f\) is closest to the integer

The stopping potential for photoelectrons emitted from a surface illuminated by light of wavelength \(6630 \; \mathring A\) is \(0.42 \; V\). If the threshold frequency is \(x \times\) \(10^{13}  / s\), where \(x\) is\(\dots\) (nearest integer). (Given, speed light \(=3 \times 10^{8} \; m / s\), Planck's constant \(=6.63 \times 10^{-34} \; Js\) )