A mass \(0.9\,kg\), attached to a horizontal spring, executes \(SHM\) with an amplitude \(A _{1}\). When this mass passes through its mean position, then a smaller mass of \(124\,g\) is placed over it and both masses move together with amplitude \(A _{2}\). If the ratio \(\frac{ A _{1}}{ A _{2}}\) is \(\frac{\alpha}{\alpha-1}\), then the value of \(\alpha\) will be\(......\)

A biconvex lens of refractive index \(1.5\) has a focal length of \(20 \mathrm{~cm}\) in air. Its focal length when immersed in a liquid of refractive index \(1.6\) will be:

The de Broglie wavelength of a proton and \(\alpha\) -particle are equal. The ratio of their velocities is ...... .

The variation of applied potential and current flowing through a given wire is shown in figure. The length of wire is \(31.4 \,cm\). The diameter of wire is measured as \(2.4 \,cm\). The resistivity of the given wire is measured as \(x \times 10^{-3} \,\Omega cm\). The value of \(x\) is_______ [Take \(\pi=3.14]\)

A body is projected from the ground at an angle of \(45^{\circ}\) with the horizontal. Its velocity after \(2s\) is \(20 \,ms ^{-1}\). The maximum height reached by the body during its motion is \(m\). (use \(g =10\, ms ^{-2}\) )

An electric appliance supplies \(6000\, {J} / {min}\) heat to the system. If the system delivers a power of \(90\, {W}\). How long (in \(sec\)) it would take to increase the internal energy by \(2.5 \times 10^{3}\, {J}\) ?

In Rutherford's alpha-particle scattering experiment, only a few alpha particles rebound back because
A. The size of gold nucleus is very small as compared to the size of gold atom.
B. Alpha particle and gold nucleus have equal charge.
C. The impact parameter is minimum for a few alpha particles.
D. A few alpha particles have very high kinetic energy.
E. Only a few alpha particles undergo head-on collision with the nuclei.
Choose the correct answer from the options given below:

Charge is distributed within a sphere of radius \(R\) with a volume charge density \(\rho (r) = \frac{A}{{{r^2}}}{e^{ - 2r/a}}\) where \(A\) and \(a\) are constants. If \(Q\) is the total charge of this charge distribution, the radius \(R\) is.

A satellite is moving in a low nearly circular orbit around the earth. Its radius is roughly equal to that of the earth's radius \(R _{ e }\). By firing rockets attached to it, its speed is instantaneously increased in the direction of its motion so that is become \(\sqrt{\frac{3}{2}}\) times larger. Due to this the farthest distance from the centre of the earth that the satellite reaches is \(R\), value of \(R\) is\(....R_e\)

A long solenoid of radius \(\mathrm{R}\) carries a time \((t)-\)dependent current \(\mathrm{I}(\mathrm{t})=\mathrm{I}_{0} \mathrm{t}(1-\mathrm{t})\). A ring of radius \(2 \mathrm{R}\) is placed coaxially near its middle. During the time interval \(0 \leq t \leq 1\), the induced current \(\left(\mathrm{I}_{\mathrm{R}}\right)\) and the induced \(\mathrm{EMF}\left(\mathrm{V}_{\mathrm{R}}\right)\) in the ring change as

Light source having wavelength \(331\) nm is used to generate photo-electrons whose stopping potential is \(0.2\) V. The work function of the used metal in the experiment is \(\alpha \times 10^{-19}\) J. The value of \(\alpha\) is _____.
(\(h = 6.62 \times 10^{-34}\) J s, \(e = 1.6 \times 10^{-19}\) C and \(c = 3 \times 10^8\) m/s)

Rods x and y of equal dimensions but of different materials are joined as shown in figure. Temperatures of end points A and F are maintained at \( 100^{\circ}C \) and \( 40^{\circ}C \) respectively. Given the thermal conductivity of rod x is three times of that of rod y, the temperature at junction points B and E are (close to):

At which temperature the r.m.s. velocity of a hydrogen molecule equal to that of an oxygen molecule at \(47^{\circ} \mathrm{C}\) ?