A horse rider covers half the distance with \(5\,m / s\) speed. The remaining part of the distance was travelled with speed \(10\,m / s\) for half the time and with speed \(15 m / s\) for other half of the time. The mean speed of the rider averaged over the whole time of motion is \(\frac{x}{7}\,m / s\). The value of \(x\) 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

In the isothermal expansion of \(10\,g\) of gas from volume \(V\) to \(2V\) the work done by the gas is \(575\,J\). What is the root mean square speed of the molecules of the gas at that temperature ..... \(m/s\)?

\(ABC\) is a plane lamina of the shape of an equilateral triagnle. \(D,\) \(E\) are mid points of \(AB\), \(AC\) and \(G\) is the centroid of the lamina. Moment of inertia of the lamina about an axis passing through \(G\) and perpendicular to the plane \(ABC\) is \(I _{0} .\) If part \(ADE\) is removed, the moment of inertia of the remaining part about the same axis is \(\frac{ NI _{0}}{16}\) where \(N\) is an integer. Value of \(N\) is

A microscope was initially placed in air (refractive index \(1)\). It is then immersed in oil (refractive index \(2)\). For a light whose wavelength in air is \(\lambda\), calculate the change of microscope's resolving power due to oil and choose the correct option.

The current flowing through the \(1 \Omega\) resistor is \(\frac{\mathrm{n}}{10} \ A.\) The value of \(n\) is _______.

The current \(I_{1}\) (in \(A\)) flowing through \(1\; \Omega\) resistor in the following circuit is:

A body of mass ' \(m\) ' is projected with a speed ' \(u\) ' making an angle of \(45^{\circ}\) with the ground. The angular momentum of the body about the point of projection, at the highest point is expressed as \(\frac{\sqrt{2} \mathrm{mu}^3}{\mathrm{Xg}}\). The value of ' \(\mathrm{X}\) ' is _______.

Given below are two statements: Statement \(I :\) In a diatomic molecule, the rotational energy at a given temperature obeys Maxwell's distribution. Statement \(II :\) In a diatomic molecule, the rotational energy at a given temperature equals the translational kinetic energy for each molecule. In the light of the above statements, choose the correct answer from the options given below:

A water drop of diameter \(cm\) is broken into \(64\) equal droplets. The surface tension of water is \(0.075\,N / m\). In this process the gain in surface energy will be ...........\(J\)

The ratio of intensities at two points \(P\) and \(Q\) on the screen in a Young's double slit experiment where phase difference between two wave of same amplitude are \(\pi / 3\) and \(\pi / 2\), respectively are

A potential \(V_0\) is applied across a uniform wire of resistance \(R\). The power dissipation is \(P_1\). The wire is then cut into two equal halves and a potential of \(V _0\) is applied across the length of each half. The total power dissipation across two wires is \(P_2\). The ratio \(P_2: P_1\) is \(\sqrt{x}: 1\). The value of \(x\) is \(.............\).

If the \(rms\) speed of oxygen molecules at \(0^{\circ} {C}\) is \(160\; {m} / {s}\), find the rms speed of hydrogen molecules at \(0^{\circ} {C}\). (In \({m}/{s}\))