If the projection of \(2 \hat{i}+4 \hat{j}-2 \hat{k}\) on \(\hat{i}+2 \hat{j}+\alpha \hat{k}\) is zero. Then, the value of \(\alpha\) will be.

A Zener diode is connected to a battery and a load as show below: The currents, \(I, I_Z\) and \(I_L\) are respectively .

Four closed surfaces and corresponding charge distributions are shown below Let the respective electric fluxes through the surfaces be \({\phi _1},{\phi _2},{\phi _3}\) and \({\phi _4}\) . Then

A light ray enters a solid glass sphere of refractive index \(\mu=\sqrt{3}\) at an angle of incidence \(60^{\circ}\). The ray is both reflected and refracted at the farther surface of the sphere. The angle (in degrees) between the reflected and refracted rays at this surface is......

On heating water, bubbles being formed at the bottom of the vessel detach and rise. Take the bubbles to be spheres of radius \(R\) and making a circular contact of radius \(r\) with the bottom of the vessel. If \(r < < R\), and the surface tension of water is \(T\), value of \(r\) just before bubbles detach is (density of water is \(\rho_{w}\))

A body is projected vertically upwards from the surface of earth with a velocity sufficient enough to carry it to infinity. The time taken by it to reach height \(h\) is \(....\,S.\)

A steel wire of length 2 m and Young's modulus \(2.0 \times 10^{11} \mathrm{Nm}^{-2}\) is stretched by a force. If Poisson ratio and transverse strain for the wire are 0.2 and \(10^{-3}\) respectively, then the elastic potential energy density of the wire is _______ \(\times 10^5\) (in SI units)

An object of mass \(m _{1}\) collides with another object of mass \(m _{2}\), which is at rest. After the collision the objects move with equal speeds in opposite direction. The ratio of the masses \(m _{2}: m _{1}\) is

If a body of mass \(1\) kg falls on the earth from infinity, it attains velocity \((v)\) and kinetic energy \((k)\) on reaching the surface of earth. The values of \(v\) and \(k\) respectively are _______. (Take radius of earth to be \(6400\) km and \(g = 9.8\) m/s\(^2\))

Two different adiabatic paths for the same gas intersect two isothermal curves as shown in\(P-V\) diagram. The relation between the ratio \(\frac{V_a}{V_d}\) and the ratio \(\frac{V_b}{V_c}\) is _______.

move in opposite direction with speed ratio \(1: 2\). The following figure represents two biconvex lenses \(L_1\) and \(L_2\) having focal length \(10 \mathrm{~cm}\) and \(15 \mathrm{~cm}\) respectively. The distance between \(\mathrm{L}_1 \& \mathrm{~L}_2\) is _______.