The height \({ }^{\prime} h ^{\prime}\) at which the weight of a body will be the same as that at the same depth \('h'\) from the surface of the earth is (Radius of the earth is \(R\) and effect of the rotation of the earth is neglected):

The temperature at which the kinetic energy of oxygen molecules becomes double than its value at \(27^{\circ}\,C\) is \(............^{\circ}\,C\)

The maximum percentage error in the measurment of density of a wire is [Given, mass of wire \(=(0.60 \pm 0.003) \mathrm{g}\)
radius of wire \(=(0.50 \pm 0.01) \mathrm{cm}\)
length of wire \(=(10.00 \pm 0.05) \mathrm{cm}]\)

A letter \('A'\) is constructed of a uniform wire with resistance \(1.0\,\Omega \) per \(cm\) , The sides of the letter are \(20\, cm\) and the cross piece in the middle is \(10\, cm\) long. The apex angle is \(60\) . The resistance between the ends of the legs is close to ................ \(\Omega\)

If mass is written as \(\mathrm{m}=\mathrm{kc}^{\mathrm{p}} \mathrm{G}^{-1 / 2} \mathrm{~h}^{1 / 2}\) then the value of \(P\) will be _______. (Constants have their usual meaning with \(\mathrm{k}\) a dimensionless constant)

A cubical block of side \(0.5\,m\) floats on water with \(30\%\) of its volume under water. ....... \(kg\) is the maximum weight that can be put on the block without fully submerging it under water ? [Take density of water \(= 10^3\,kg/m^3\) ]

A square loop of sides \(a=1 \mathrm{~m}\) is held normally in front of a point charge \(\mathrm{q}=1 \mathrm{C}\). The flux of the electric field through the shaded region is \(\frac{5}{\mathrm{p}} \times \frac{1}{\varepsilon_0} \frac{\mathrm{Nm}^2}{\mathrm{C}}\), where the value of p is _____.

A leak proof cylinder of length \(1 \;\mathrm{m},\) made of a metal which has very low coefficient of expansion is floating vertically in water at \(0^{\circ} \mathrm{C}\) such that its height above the water surface is \(20\; \mathrm{cm} .\) When the temperature of water is increased to \(4^{\circ} \mathrm{C},\) the height of the cylinder above the water surface becomes \(21 \;\mathrm{cm} .\) The density of water at \(\mathrm{T}=4^{\circ} \mathrm{C},\) relative to the density at \(\mathrm{T}=0^{\circ} \mathrm{C}\) is close to

Let \(M\) be a \(3 \times 3\) matrix such that \(M \begin{pmatrix} 1 \\ 0 \\ 0 \end{pmatrix} = \begin{pmatrix} 1 \\ 2 \\ 3 \end{pmatrix}\), \(M \begin{pmatrix} 0 \\ 1 \\ 0 \end{pmatrix} = \begin{pmatrix} 0 \\ 1 \\ 2 \end{pmatrix}\) and \(M \begin{pmatrix} 0 \\ 0 \\ 1 \end{pmatrix} = \begin{pmatrix} -1 \\ 1 \\ 1 \end{pmatrix}\). If \(M \begin{pmatrix} x \\ y \\ z \end{pmatrix} = \begin{pmatrix} 1 \\ 7 \\ 11 \end{pmatrix}\), then \(x + y + z\) equals :

A software company sets up \(m\) number of computer systems to finish an assignment in \(17\) days. If \(4\) computer systems crashed on the start of the second day, \(4\) more computer systems crashed on the start of the third day and so on, then it took \(8\) more days to finish the assignment. The value of \(m\) is equal to :

For a train engine moving with speed of \(20 \;ms ^{-1}\). the driver must apply brakes at a distance of \(500 \;m\) before the station for the train to come to rest at the station. If the brakes were applied at half of this distance, the train engine would cross the station with speed \(\sqrt{ x }\; ms ^{-1}\). The value of \(x\) is \(..............\) (Assuming same retardation is produced by brakes)

A piston of mass \(M\) is hung from a massless spring whose restoring force law goes as \(\mathrm{F}=-\mathrm{kx}^3\), where k is the spring constant of appropriate dimension. The piston separates the vertical chamber into two parts, where the bottom part is filled with ' \(n\) ' moles of an ideal gas. An external work is done on the gas isothermally (at a constant temperature T) with the help of a heating filament (with negligible volume) mounted in lower part of the chamber, so that the piston goes up from a height \(\mathrm{L}_0\) to \(\mathrm{L}_1\), the total energy delivered by the filament is (Assume spring to be in its natural length before heating)

A cylindrical vessel of cross-section \(A\) contains water to a height \(h\) . There is a hole in the bottom of radius \('a'\) . The time in which it will be emptied is