In the figure, the value of resistors to be connected between $C$ and $D$ so that the resistance of the entire circuit between $A$ and $B$ does not change with the number of elementary sets used is
Diffcult
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(b) Cut the series from $XY$ and let the resistance towards right of $XY$ be ${R_0}$ whose value should be such that when connected across $AB$ does not change the entire resistance. The combination is reduced to as shown below.
The resistance across $EF, = {R_{EF}} = ({R_0} + 2R)$
Thus ${R_{AB}} = \frac{{({R_0} + 2R)R}}{{{R_0} + 2R + R}} = \frac{{{R_0}R + 2{R^2}}}{{{R_0} + 3R}} = {R_0}$
$ \Rightarrow R_0^2 + 2R{R_0} - 2{R^2} + 0 \Rightarrow {R_0} = R(\sqrt 3 - 1)$
The resistance across $EF, = {R_{EF}} = ({R_0} + 2R)$
Thus ${R_{AB}} = \frac{{({R_0} + 2R)R}}{{{R_0} + 2R + R}} = \frac{{{R_0}R + 2{R^2}}}{{{R_0} + 3R}} = {R_0}$
$ \Rightarrow R_0^2 + 2R{R_0} - 2{R^2} + 0 \Rightarrow {R_0} = R(\sqrt 3 - 1)$
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