In the following circuit the value of total resistance between $X$ and $Y$ in $o h m$ is:

(1 + \sqrt{3}) r

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(\sqrt{3} - 1) r

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\infty

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50 r

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Solution

The correct option is D $(1 + \sqrt{3}) r$
Equivalent resistance
$R_{e} = r + r + \frac{1}{⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ \frac{1}{{(\frac{1}{\frac{1}{r} + \frac{1}{3 r}}) + r + r}} + \frac{1}{r} ⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦}$

$R_{e} = r + r + \frac{1}{⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ \frac{1}{{(\frac{1}{\frac{4}{3 r}}) + r + r}} + \frac{1}{r} ⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦}$

$R_{e} = r + r + \frac{1}{⎡ ⎢ ⎢ ⎢ ⎣ \frac{1}{{(\frac{3 r}{4}) + r + r}} + \frac{1}{r} ⎤ ⎥ ⎥ ⎥ ⎦}$

$R_{e} = r + r + \frac{1}{⎡ ⎢ ⎢ ⎢ ⎣ \frac{1}{{(\frac{11 r}{4})}} + \frac{1}{r} ⎤ ⎥ ⎥ ⎥ ⎦}$

$R_{e} = r + r + \frac{1}{[\frac{4}{11 r} + \frac{1}{r}]} = r + r + \frac{11 r}{15} = \frac{41 r}{15} = 2.72 r$

Answer: (A) $(1 + \sqrt{3}) r$

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