c
Let us consider a spherical shell of radius \(x\) and thickness \(dx\)

Charge on this shell

\(d q=\rho \cdot 4 \pi x^{2} d x=\rho_{0}\left(\frac{5}{4}-\frac{x}{R}\right) .4 \pi x^{2} d x\)

\(\therefore\) Total charge in the spherical region from centre to \(r\) \((r < R)\) is

\(q=\int d q=4 \pi \rho_{0} \int_{0}^{r}\left(\frac{5}{4}-\frac{x}{R}\right) x^{2} d x\)

\(=4 \pi \rho_{0}\left[\frac{5}{4} \cdot \frac{r^{3}}{3}-\frac{1}{R} \cdot \frac{r^{4}}{4}\right]=\pi \rho_{0} r^{3}\left(\frac{5}{3}-\frac{r}{R}\right)\)

Electric field at \(r, E=\frac{1}{4 \pi \epsilon_{0}} \cdot \frac{q}{r^{2}}\)

\(=\frac{1}{4 \pi \epsilon_{0}} \cdot \frac{\pi \rho_{0} r^{3}}{r^{2}}\left(\frac{5}{3}-\frac{r}{R}\right)=\frac{\rho_{0} r}{4 \epsilon_{0}}\left(\frac{5}{3}-\frac{r}{R}\right)\)