JEE MAIN - Physics (2004 - No. 16)
The resistance of the series combination of two resistances is $$S.$$ When they are jointed in parallel the total resistance is $$P.$$ If $$S = nP$$ then the Minimum possible value of $$n$$ is
$$2$$
$$3$$
$$4$$
$$1$$
Explanation
$$S = {R_1} + {R_2}$$ and $$P = {{{R_1}{R_2}} \over {{R_1} + {R_2}}}$$
$$S = nP \Rightarrow {R_1} + {R_2} = {{n\left( {{R_1}{R_2}} \right)} \over {\left( {{R_1} + {R_2}} \right)}}$$
$$ \Rightarrow {\left( {{R_1} + {R_2}} \right)^2} = n{R_1}{R_2}$$
$$ \Rightarrow n = {{R_1^1 + R_2^2 + {R_1}{R_2}} \over {{R_1}{R_2}}}$$
$$n = {{{R_1}} \over {{R_2}}} + {{{R_2}} \over {{R_1}}} + 2$$
Arithmetic mean $$ > $$ Geometric mean
Minimum value of $$n$$ is $$4$$
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