In the circuit shown here, what is the value of the unknown resistor $R$ so that the total resistance of the circuit between points $P$ and $Q$ is also equal to $R$
Medium
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The given circuit can be simplified as follows
$R = 3 + \frac{{10 \times (3 + R)}}{{10 + 3 + R}}$ $ = 3 + \frac{{30 + 10R}}{{13 + R}}$
$R = \frac{{39 + 3R + 30 + 10R}}{{13 + R}} = \frac{{69 + 13R}}{{13 + R}}$
$13R + {R^2} = 69 + 13R$ $ \Rightarrow $ $R = \sqrt {69}\, \Omega $.
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