a
(a) If two resistances are \({R_1}\) and \({R_2}\) then
\(S = {R_1} + {R_2}\) and \(P = \frac{{{R_1}{R_2}}}{{({R_1} + {R_2})}}\)
From given condition \(S = nP\) i.e. \(({R_1} + {R_2}) = n\,\left( {\frac{{{R_1}{R_2}}}{{{R_1} + {R_2}}}} \right)\)
\(==>\) \({({R_1} + {R_2})^2} = n\,\,{R_1}{R_2}\) \(==>\) \({({R_1} - {R_2})^2} + 4{R_1}{R_2} = n{R_1}{R_2}\)
So \(n = 4 + \frac{{{{({R_1} - {R_2})}^2}}}{{{R_1}{R_2}}}.\) Hence minimum value of \(n\) is \(4\).