A charge Q is located at the centre of an initially uncharged,...

When a charge $Q$ is placed at the centre of the uncharged hollow metallic sphere, the electric field inside the conducting material must be zero.

By Gauss's law, a charge $-Q$ is induced on the inner surface of the sphere. Since the sphere is initially uncharged, a charge $+Q$ appears on the outer surface.

Assuming the sphere is a thin shell of radius $a$, the surface charge density on the inner surface is $-\dfrac{Q}{4\pi a^2}$ and on the outer surface is $\dfrac{Q}{4\pi a^2}$.

When an additional charge $q$ is given to the sphere, it resides entirely on the outer surface. The charge on the inner surface remains $-Q$ to maintain zero electric field inside the metal.

The total charge on the outer surface becomes $Q + q$.

The new surface charge densities are $-\dfrac{Q}{4\pi a^2}$ on the inner surface and $\dfrac{Q+q}{4\pi a^2}$ on the outer surface.

For a point at a distance $x$ from the centre inside the cavity ($x < a$), a spherical Gaussian surface encloses only the central charge $Q$. The uniform charge distributions on the spherical shell do not contribute to the electric field inside the cavity.

Applying Gauss's law, $E (4\pi x^2) = \dfrac{Q}{\varepsilon_0}$, which gives $E = \dfrac{Q}{4\pi\varepsilon_0 x^2}$. This electric field remains the same in both situations.