The parameter that remains the same for molecules of all gases at a given temperature is _______.

In normal adjustment, for a refracting telescope, the distance between objective and eye piece is \(30\,cm\). The focal length of the objective, when the angular magnification of the telescope is \(2\) , will be \(.....cm\) 

Consider two separate ideal gases of electrons and protons having same number of particles. The temperature of both the gases are same. The ratio of the uncertainty in determining the position of an electron to that of a proton is proportional to :-

The area of cross section of the rope used to lift a load by a crane is \(2.5 \times 10^{-4} m ^{2}\). The maximum lifting capacity of the crane is \(10\) metric tons. To increase the lifting capacity of the crane to \(25\) metric tons, the required area of cross section of the rope should be.\(.........\times 10^{-4} \,m ^{2}\) (take \(g =10\, ms ^{-2}\) )

The equation of state for a gas is given by \(PV = nRT + \alpha V\), where \(n\) is the number of moles and \(\alpha \) is a positive constant. The initial temperature and pressure of one mole of the gas contained in a cylinder are \(T_o\) and \(P_o\) respectively. The work done by the gas when its temperature doubles isobarically will be

The escape velocities of two planets \(A\) and \(B\) are in the ratio \(1: 2\). If the ratio of their radii respectively is \(1: 3\), then the ratio of acceleration due to gravity of planet \(A\) to the acceleration of gravity of planet \(B\) will be:

Three infinitely long charged thin sheets are placed as shown in figure. The magnitude of electric field at the point \(P\) is \(\frac{x \sigma}{\epsilon_0}\). The value of \(x\) is _______. (all quantities are measured in \(SI\) units).

Three parallel plate capacitors \(\mathrm{C}_1, \mathrm{C}_2\) and \(\mathrm{C}_3\) each of capacitance \(5 \mu \mathrm{~F}\) are connected as shown in figure. The effective capacitance between points A and B, when the space between the parallel plates of \(C_1\) capacitor is filled with a dielectric medium having dielectric constant of 4, is :

The equilateral triangle \(ABC\) is cut from a thin solid sheet of wood. (See figure) \(D, E\) and \(F\) are the mid points of its sides as shown and \(G\) is the centre of the triangle. The moment of inertia of the triangle about an axis passing through \(G\) and perpendicular to the plane of the triangle is \(I_0.\) If the smaller triangle \(DEF\) is removed from \(ABC,\) the moment of inertia of the remaining figure about the same axis is \(I.\) Then

On a linear temperature scale \(Y\), water freezes at \(- 160^o\, Y\) and boils at \(- 50^o\, Y\). On this \(Y\) scale , a temperature of \(340\, K\) would be read as........ \(^oY\)  ( water freezes at \(273\, K\) and boils at \(373\, K\))

A body of mass ' \(m\) ' connected to a massless and unstretchable string goes in verticle circle of radius ' \(R\) 'under gravity \(g\). The other end of the string is fixed at the center of circle. If velocity at top of circular path is \(n \sqrt{g R}\), where, \(n \geqslant 1\), then ratio of kinetic energy of the body at bottom to that at top of the circle is