Two circuits (shown below) are called ‘Circuit $A$ ’and ‘Circuit $B$’. The equivalent resistance of ‘Circuit $a$’ is $x$ and that of ‘Circuit $B$’ is $y$ between $1$ and $2.$
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Circuit $A:$ $x=\frac{R(2 R+x)}{3 R+x}$ or $3 R x+x^{2}=2 R^{2}+R x$
or $x^{2}+2 R x-2 R^{2}=0$ or $x=\frac{-2 R+\sqrt{4 R^{2}+8 R^{2}}}{2}$
or $x=-\frac{2 R+2 \sqrt{3 R}}{2}=(\sqrt{3}-1) R$
Circuit $B:$ $y=\frac{y R}{y+R}+2 R$ or $y^{2}+R y=y R+2 R y+2 R^{2}$
or $y^{2}-2 Y R-2 R^{2}=0$ or $y=(\sqrt{3}+1) R$
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