Two identical condensers \(M\) and \(N\) are connected in series with a battery. The space between the plates of \(M\) is completely filled with a dielectric medium of dielectric constant 8 and a copper plate of thickness \(\frac{d}{2}\) is introduced between the plates of \(N\). ( \(d\) is the distance between the plates). Then potential differences across \(M\) and \(N\) are, respectively, in the ratio

An electric dipole with dipole moment \(p\) is placed with its axis at \(30^{\circ}\) to a uniform electric field. The work done in rotating the dipole to a position where its axis is perpendicular to the field is

A wire of length 0.5 m and area of cross-section \(4 \times 10^{-6} \mathrm{~m}^2\) at a temperature of \(100^{\circ} \mathrm{C}\) is suspended vertically by fixing its upper end to the ceiling. The wire is then cooled to \(0^{\circ} \mathrm{C}\), but is prevented from contracting, by attaching a mass at the lower end. If the mass of the wire is negligible, then the value of the mass attached to the wire is
[Young's modulus of material of the wire \(=10^{11} \mathrm{~N} \mathrm{~m}^{-2}\); coefficient of linear expansion of the material of the wire \(=10^{-5} \mathrm{~K}^{-1}\) and acceleration due to gravity \(\left.=10 \mathrm{~m} \mathrm{~s}^{-2}\right]\)

In the given circuit values of \(I_1, I_2, I_3\) are respectively

A proton, a deuteron (nucleus of \({ }_1 \mathrm{H}^2\) ) and an \(\alpha\)-particle with same kinetic energy enter a region of uniform magnetic field moving at right angles to the field. The ratio of the radii of their circular paths is :

A thin converging lens of focal length \(25 \mathrm{~cm}\) forms a sharp image of an object on a screen placed at a distance of \(75 \mathrm{~cm}\) from the lens. Later the screen is moved closer to the lens by a distance \(25 \mathrm{~cm}\). The distance through which the object is to be shifted so that its image on the screen is sharp again is

The energy of a parallel plate capacitor when connected to a battery is \(E\). With the battery still in connection, if the plates of the capacitor are separated, so that the distance between them is twice the original distance, then the electrostatic energy becomes

Polymers that can be softened on heating and hardened on cooling are called

The slopes of the lines represented by \(x^2+2 h x y+2 y^2=0\) are in the ratio \((1: 2)\), then \(h\) is equal

The \(\mathrm{pH}\) of \(10^{-8} \mathrm{M} \mathrm{HCl}\) solution is

If \(\int \frac{d x}{(x-1)^{3 / 2}(x-3)^{1 / 2}}=\sqrt{f(x)}+c\) then \(f(-1)-f(0)=\)

If \(\sqrt{3} \cos \theta+\sin \theta>0\) then

\(\cos \frac{\pi}{12}=\)