Question

Which of above combinations have equivalent resistance $1\Omega$ between X and Y?

• A. (a)
• B. (b)
• C. (c)
• D. (d)

Answer 1

Answer: (A), (D)

The correct options are
A (a)
D (d)

Answer is option A and D.
Components connected in parallel are connected so the same voltage is applied to each component. In a parallel circuit, the voltage across each of the components is the same, and the total current is the sum of the currents through each component.
To find the total resistance of all components, the reciprocals of the resistances of each component is added and the reciprocal of the sum is taken. The total resistance will always be less than the value of the smallest resistance. That is, $\frac{1}{R_{total}}=\frac{1}{R_1}+\frac{1}{R_2}thatis,R={R_{total}}^{-1}$. The voltage is given as $V=V_1=V_2$. The total resistance is always lesser than the least resistance.
In the figure (a), the resistances two 2 ohms resistors are connected in parallel and the total resistance is $\frac{1}{R}=\frac{1}{2}+\frac{1}{2}=\frac{1}{1}.ThatisR=1ohm.$
In the figure (b), the resistances two 1 ohm and a 2 ohms resistors are connected in parallel and the total resistance is $\frac{1}{R}=\frac{1}{1}+\frac{1}{1}+\frac{1}{2}=\frac{1}{2.5}.ThatisR=0.4ohm.$
In the figure (c), the resistances two 1 ohm resistors are connected in parallel and the total resistance is $\frac{1}{R}=\frac{1}{1}+\frac{1}{1}=\frac{1}{2}.ThatisR=0.5ohm.$
In the figure (d), the resistances two pairs 1 ohm resistors are connected in parallel. a pair of 1 ohm resistors form 2 ohms resistors as they are serially connected. Hence, the total resistance is $\frac{1}{R}=\frac{1}{2}+\frac{1}{2}=\frac{1}{1}.ThatisR=1ohm.$
Hence, the figures a and d have a total resistance of 1 ohm each.