JEE Advanced Physics: Geometrical Optics / Ray Optics (NCERT Physics Part II, Chapter 9) questions with solutions

Q1. Two candles of equal length L = 20 cm are lit simultaneously. Candle 1 is at distance d1 = 10 cm from the left wall; candle 2 is at distance d3 = 30 cm from the right wall; the two candles are d2 = 20 cm apart. Each candle acts as the light source for the other candle's shadow. The shadow of candle 1 on the left wall does not change in length, while the shadow of candle 2 on the right wall shortens at v = 1 mm/s. Which of the following statements is INCORRECT?

1. (A) The left candle will burn for a longer time than the right candle.
2. (B) The candle that extinguishes first does so at t = 400 s.
3. (C) The candle that extinguishes second does so at t = 800 s.
4. (D) The right candle is decreasing in length at a faster rate than the left candle.

Answer: (C) The candle that extinguishes second does so at t = 800 s.

Using similar triangles with d1=10, d2=20, d3=30: shadow of candle 1 on left wall has length S1 = (h1*30 - h2*10)/20, and shadow of candle 2 on right wall has length S2 = (h2*50 - h1*30)/20. Setting dS1/dt = 0 gives dh2/dt = 3*dh1/dt, and substituting into dS2/dt = -1 mm/s gives dh1/dt = -1/6 mm/s and dh2/dt = -1/2 mm/s. Right candle burns out at t = 200/(0.5) = 400 s; left candle burns out at t = 200/(1/6) = 1200 s. Statement C claims 800 s, which is wrong; A, B, D are correct.