question_answer

Six resistances each of value \[r=5\,\,\Omega \]. are connected between points A, B and C as shown in the figure. If \[{{R}_{1}},{{R}_{2}}\] and \[{{R}_{3}}\] are the net resistances between A and B, between B and C and between A and C respectively, then\[{{R}_{1}}:{{R}_{2}}:{{R}_{3}}\] will be equal to

A)  6 : 3 : 2                                

B)  1 : 2 : 3

C)  5 : 4 : 3                                

D)  4 : 3 : 2

Correct Answer: C

Solution :

                         Resistance between points A and B                 \[{{R}_{AB}}=r|\,|\left( \frac{r}{3}+\frac{r}{2} \right)\]                 \[=\frac{r\times 5/6\,\,r}{r+5/6r}=\frac{5}{11}r\] Resistance between points B and C                 \[{{R}_{BC}}=\frac{r}{2}\,|\,|\left( r+\frac{r}{3} \right)\]                 \[=\frac{\frac{r}{2}\times \frac{4r}{3}}{\frac{r}{2}+\frac{4r}{3}}=\frac{4}{11}r\] Resistance between points C and D                 \[{{R}_{CA}}=\frac{r}{3}|\,|\left( \frac{r}{2}+r \right)\]                 \[=\frac{\left( \frac{r}{3}\times \frac{3r}{2} \right)}{\frac{r}{3}+\frac{3r}{2}}\]                 \[=\frac{3}{11}r\] \[{{R}_{AB}}:{{R}_{BC}}:{{R}_{CA}}=5:4:3\]