Six equal resistances are connected between points \(P, Q\) and \(R\) as shown in figure. Then net resistance will be maximum between

\(r_{P Q}=r_{11}\left(\frac{r}{3}+\frac{r}{2}\right)=\frac{r \times \frac{5}{6} r}{r+\frac{5}{6} r}=\frac{5}{11} r\)

Resistance between \(Q\) and \(R\)

\(\mathrm{r}_{\mathrm{QR}}=\frac{\mathrm{r}}{2} 11\left(\mathrm{r}+\frac{\mathrm{r}}{3}\right)=\frac{\frac{\mathrm{r}}{2} \times \frac{4}{3} \mathrm{r}}{\frac{\mathrm{r}}{2}+\frac{4}{3} \mathrm{r}}=\frac{4}{11} \mathrm{r}\)

Resistance between \(\mathrm{P}\) and \(\mathrm{R}\)

\({{\rm{r}}_{{\rm{PR}}}} = \frac{{\rm{r}}}{3}11\left( {\frac{{\rm{r}}}{2} + {\rm{r}}} \right) = \frac{{\frac{{\rm{r}}}{3} \times \frac{3}{2}{\rm{r}}}}{{\frac{{\rm{r}}}{3} + \frac{3}{2}{\rm{r}}}} = \frac{3}{{11}}{\rm{r}}\)

Hence, it is clear that \(\mathrm{r}_{\mathrm{PQ}}\) is maximum