NDA 1 2024 GAT, question 51

Topic: Physics

Q51

An infinite combination of resistors, each having resistance $R=4\ \Omega$ , is given below. What is the net resistance between the points $A$ and $B$ ? (Each resistance is of equal value, $R=4$ )

Figure for Q51

(a) $0\ \Omega$
(b) $2+2\sqrt{5}\ \Omega$ Correct answer
(c) $2+\sqrt{5}\ \Omega$
(d) $\infty \ \Omega$

Solution.

Answer: (b) $2+2\sqrt{5}\ \Omega$

Let X be the input resistance of the infinite alternating ladder.
After the first series resistor R, the remaining branch is R in parallel with a continuation equivalent to X (interchanging the top and bottom rails changes no resistance).
Hence X = R + RX/(R + X).
This gives X² − RX − R² = 0.
With R = 4 Ω, take the positive root X = 2 + 2√5 Ω.