Calculate the time (in \(minutes\)) interval between \(33 \,\%\) decay and \(67\, \%\) decay if half-life of a substance is \(20\, minutes.\)

Consider a rectangular sheet of solid material of length \(\ell=9 \mathrm{~cm}\) and width \(\mathrm{d}=4 \mathrm{~cm}\). The coefficient of linear expansion is \(\alpha=3.1 \times 10^{-5} \mathrm{~K}^{-1}\) at room temperature and one atmospheric pressure. The mass of sheet \(\mathrm{m}=0.1 \mathrm{~kg}\) and the specific heat capacity \(\mathrm{C}_{\mathrm{v}}=900 \mathrm{~J} \mathrm{~kg}^{-1} \mathrm{~K}^{-1}\). If the amount of heat supplied to the material is \(8.1 \times 10^2 \mathrm{~J}\) then change in area of the rectangular sheet is :-

The position of a projectile launched from the origin at \(t = 0\) is given by \(\vec r = \left( {40\hat i + 50\hat j} \right)\,m\) at \(t = 2\,s\). If the projectile was launched at an angle \(\theta\) from the horizontal, then \(\theta\) is (take \(g = 10\, ms^{-2}\))

A moving coil galvanometer has a coil with \(175\, turns\) and area \(1\, cm^2\) It uses a torsion band of torsion constant \(10^{-6}\, N\, -m/rad\). The coil is placed in a magnetic field \(B\) parallel to its plane. The coil deflects by \(10\) for a current of \(1\, mA\). The value of \(B\) (in Tesla) is approximately

For a thin symmetric prism made of glass (refractive index \(1.5\)), the ratio of incident angle and minimum deviation will be _______.

Two bodies of masses \(m_{1}=5\,kg\) and \(m _{2}=3\,kg\) are connected by a light string going over a smooth light pulley on a smooth inclined plane as shown in the figure. The system is at rest. The force exerted by the inclined plane on the body of mass \(m _{1}\) will be\(....N\) [Take \(g=10\,ms ^{-2}\) ]

Refer the figure given below. \(\mu_1\) and \(\mu_2\) are refractive indices of air and lens material. The height of image will be _______ cm.

A cell of emf \(90\,V\) is connected across series combination of two resistors each of \(100\,\Omega\) resistance. A voltmeter of resistance \(400\,\Omega\) is used to measure the potential difference across each resistor. The reading of the voltmeter will be \(.........\,V\)

The ratio of speed of sound in hydrogen gas to the speed of sound in oxygen gas at the same temperature is

A force is applied to a steel wire ' \(A\) ', rigidly clamped at one end. As a result elongation in the wire is \(0.2\,mm\). If same force is applied to another steel wire ' \(B\) ' of double the length and a diameter \(2.4\) times that of the wire ' \(A\) ', the elongation in the wire ' \(B\) ' will be \(............\times 10^{-2}\,mm\) (wires having uniform circular cross sections)

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:

Four particles each of mass \(1 \mathrm{~kg}\) are placed at four corners of a square of side \(2 \mathrm{~m}\). Moment of inertia of system about an axis perpendicular to its plane and passing through one of its vertex is_________ \(\mathrm{kgm}^2\).

Some energy levels of a molecule are shown in the figure. The ratio of the wavelengths \(r\) =\(\frac{{{\lambda _1}}}{{{\lambda _2}}}\) is given by