What will be the projection of vector \(A=\hat{i}+\hat{j}+\hat{k}\) on vector \(\vec{B}=\hat{i}+\hat{j}\).

A black coloured solid sphere of radius \(R\) and mass \(M\) is inside a cavity with vacuum inside. The walls of the cavity are maintained at temperature \(T_0\). The initial temperature of the sphere is \(3T_0\). If the specific heat of the material of the sphere varies as \(\alpha T^3\) per unit mass with the temperature \(T\) of the sphere, where \(\alpha \) is a constant, then the time taken for the sphere to cool down to temperature \(2T_0\) will be ( \(\sigma \) is Stefan Boltzmann constant)

In the electric network shown, when no current flows through the \(4\, \Omega \) resistor in the arm \(EB\), the potential difference between the points \(A\) and \(D\) will be ............... \(V\)

During the propagation of electromagnetic waves in a medium

A double convex lens has power \(P\) and same radii of curvature \(R\) of both the surfaces. The radius of curvature of a surface of a plano-convex lens made of the same material with power \(1.5P\) is ......\(R\)

A capacitor with capacitance \(5\,\mu F\) is charged to \(5\,\mu C.\) If the plates are pulled apart to reduce the capacitance to \(2\,\mu F,\) how much work is done?

A rigid diatomic ideal gas undergoes an adiabatic process at room temperature. The rational between temperature and volume for the process is \(TV^x =\) constant, then \(x\) is

\(n\) mole a perfect gas undergoes a cyclic process \(ABCA\) (see figure) consisting of the following processes. \(A \rightarrow B :\) Isothermal expansion at temperature \(T\) so that the volume is doubled from \(V _{1}\) to \(V _{2}=2 V _{1}\) and pressure changes from \(P _{1}\) to \(P _{2}\) \(B \rightarrow C :\) Isobaric compression at pressure \(P _{2}\) to initial volume \(V _{1}\) \(C \rightarrow A\) : Isochoric change leading to change of pressure from \(P _{2}\) to \(P _{1}\) Total workdone in the complete cycle \(ABCA\) is

Preeti reached the metro station and found that the escalator was not working. She walked up the stationary escalator in time \(t_1\) . On other days, if she remains stationary on the moving escalator, then the escalator takes her up in time \(t_2\). The time taken by her to walk up on the moving escalator will be 

A ball will a speed of \(9\, m / s\) collides with another identical ball at rest. After the collision, the direction of each ball makes an angle of \(30^{\circ}\) with the original direction. The ratio of velocities of the balls after collision is \(x : y\) where \(x\) is........

A capillary tube made of glass of radius \(0.15\, mm\) is dipped vertically in a beaker filled with methylene iodide (surface tension \(=0.05\, Nm ^{-1},\) density \(\left.=667\, kg m ^{-3}\right)\) which rises to height \(h\) in the tube. It is observed that the two tangents drawn from liquid-glass interfaces (from opp. sides of the capillary) make an angle of \(60^{\circ}\) with one another. Then \(h\) is close to \(......m\) \(\left(g=10\, ms ^{-2}\right)\)