The Wheatstone bridge is balanced when $R_3 = 144 \, \Omega$. If $R_2$ and $R_1$ are interchanged, the bridge balances for $R_3 = 169 \, \Omega$. The value of $R_4$ is:

If $m_1$ and $m_2$ ($m_1 \geq m_2$) are the magnification for two positions of the lens between the object and the screen, and $d$ is the distance between the two positions of the lens, the focal length of the lens is:

Two vibrating strings of same length, same cross-sectional area and stretched to same tension are made of material with densities $\rho$ and $2\rho$. Each string is fixed at both ends. If $V_1$ and $V_2$ are speeds of transverse waves in the strings with densities $\rho$ and $2\rho$ respectively, then $\frac{V_1}{V_2}$ is:

A gas is initially at a state of $ (P_0, V_0, T_0) $. If $ W_1 $ is work done in isobaric expansion to volume $ V $ and $ W_2 $ is the work done in isothermal expansion to volume $ V $, then:

Two soap bubbles of radii $R_1$ and $R_2$ are kept in vacuum at constant temperature, the ratio of masses of air inside them are:

Three point masses $m_1, m_2, m_3$ are located at the vertices of an equilateral triangle of side 'a'. The moment of inertia of the system about an axis along the altitude of the triangle passing through $m_1$ is:

In a hydroelectric power station, the water is flowing at $2 \, {ms}^{-1}$ in the river which is $100 \, {m}$ wide and $5 \, {m}$ deep. The maximum power output from the river is:

A body of mass $2 \, {kg}$ is placed on a smooth horizontal surface. Two forces $F_1 = 20 \, {N}$ and $F_2 = 10\sqrt{3} \, {N}$ are acting on the body in the directions making angles of $30^\circ$ and $60^\circ$ to the surface. The reaction of the surface on the body is:

In the wave equation $ y = 3 \cos \pi (100t - x) \, \text{cm}$, the wavelength is

Young’s modulus of steel is $ 2 \times 10^{11} \, \text{N/m}^2 $ and strain at elastic limit is 0.15. The value of limiting stress will be