The time period of a particle in simple harmonic motion is \(8 \mathrm{~s}\). At \(t=0\), it is at the mean position. The ratio of the distances travelled by it in the first and second seconds is

A hydrogen atom at the grotund level absorbs a photon and is excited to $n=4$ level. The potential energy of the electron in the excited state is

\(\Delta \lambda\) is the difference between the wavelength of \(k_\alpha\) line and the minimum wavelength of the continuous X-ray spectrum when the X-ray tube is operated at a voltage \(V\). If the operating voltage is changed to \(V / 3\), then the above difference is \(\Delta \lambda^{\prime}\). Then

The component of a vector \(\mathbf{P}=3 \hat{i}+4 \hat{j}\) alog the direction \((\hat{i}+2 \hat{j})\) is

A bullet of mass \(m\) is fired from a rifle of mass \(M\). If \(\mathbf{v}\) be the velocity of the bullet, velocity acquired by the rifle is

The de Broglie wavelength of the most energetic photoelectrons emitted from a photosensitive metal of work function \(\phi\), when light frequency \(v\) incidents on it is \(\lambda\). Then \(\mathrm{v}=\) ( \(\mathrm{h}\) - Planck's constant, \(\mathrm{m}\) - mass of electron)

A circular coil of radius \(10 \mathrm{~cm}\) and resistance of \(2 \Omega\) is placed with its plane perpendicular to the horizontal component of the earth's magnetic field. It is rotated about its vertical diameter through \(180^{\circ}\) in \(0.25 \mathrm{~s}\). If the magnitude of the induced emf is \(3.8 \times 10^{-3} \mathrm{~V}\), then the number of turns of the coil is
(Horizontal component of earth's magnetic field at the place is \(3 \times 10^{-5} \mathrm{~T}\) )

\(\int(\sqrt{1+\sin (2 x)}) d x=\)

An ideal monoatomic gas expands at constant pressure. The work done by the gas on its environment is \(200 \mathrm{~J}\), then the heat absorbed by the gas during the process is

The transformed equation of \(3 x^2+3 y^2+2 x y=2\), when the coordinate axes are rotated through an angle of \(45^{\circ}\), is

The equation of a chord \(A B\) of an ellipse \(2 x^2+y^2=1\) is \(x-y+1=0\). If 0 is the origin, then \(\angle \mathrm{AOB}=\)

Match the item given in Column-I (method used for determining colligative property) and Column-II (for the corresponding colligative property) and find the correct order.
\(\begin{array}{|c|c|c|}
\hline & \text { Column I } & \text { Column II } \\
\hline \text { A. } & \text { Beckmann method } & \text { 1. Osmotic pressure } \\
\hline \text { B. } & \begin{array}{l}
\text { Ostwald-Walker method }
\end{array} & \text { 2. Elevation in B.P } \\
\hline \text { C. } & \begin{array}{l}
\text { Berkeley-Hartley method }
\end{array} & \text { 3. Depression in F.P. } \\
\hline \text { D. } & \text { Landsberger method } & \begin{array}{l}
\text { 4. Relative lowering of vapour pressure }
\end{array} \\
\hline
\end{array}\)

A monoatomic gas \(\left(\gamma=\frac{5}{3}\right)\) at a pressure of \(4 \mathrm{~atm}\) is compressed adiabatically so that its temperature rises from \(27^{\circ} \mathrm{C}\) to \(327^{\circ} \mathrm{C}\). The pressure of the gas in its final state is