JEE Advanced 2026 — Paper 1 — PHY-IV solution

JEE Advanced2026Paper 1PHY-IV

List-I shows four configurations made of straight and semi-circular narrow tubes containing air. A sound wave of wavelength $λ = 0.29$ m enters these structures at the point $S$ and a sound detector is placed at $D$ . Between the points $S$ and $D$ , the sound travels only through the tubes. List-II contains the possible smallest values of $l$ (refer to the figures) for which the detector $D$ records maximum amplitude. Ignore effects of sharp corners. [Given $cos (1 5^{\circ}) = 0.97$ ]

Choose the option that best describes the match between the entries in List-I to those in List-II.

A
P $\to$ 4; Q $\to$ 3; R $\to$ 5; S $\to$ 1
B
P $\to$ 4; Q $\to$ 3; R $\to$ 1; S $\to$ 5
C
P $\to$ 3; Q $\to$ 4; R $\to$ 1; S $\to$ 2
D
P $\to$ 3; Q $\to$ 4; R $\to$ 5; S $\to$ 2

Solution

General principle. Maximum amplitude at $D$ requires constructive interference, i.e. path difference $Δ x = nλ$ with $n = 1$ giving the smallest non-zero $l$ .

(P) Semicircular detour of diameter $0.5 l$ over a straight base $l$ . Arc length $= π (0.25 l) \cdot 2 = π \cdot 0.5 l /2 \cdot$ ... more directly, the semicircle of diameter $0.5 l$ has length $π (0.25 l) = π l /4$ ... reworking: path difference $Δ x = π R - 2 R = (π - 2) \cdot \frac{l}{2}$ where $R = l /4$ , giving $Δ x = (π - 2) \frac{l}{2}$ . Setting equal to $λ$ : $l_{m i n} = \frac{2 λ}{π - 2} \approx \frac{0.58}{1.14} \approx 0.51$ m. P $\to$ 3.

(Q) Rectangular detour: three sides of length $0.5 l + l + 0.5 l = 2 l$ versus straight $l$ . $Δ x = 2 l - l = l$ . Setting $l = λ$ : $l_{m i n} = 0.29$ m. Q $\to$ 4.

(R) Quarter-arc of radius $l$ plus a straight segment $l$ around the corner; the other path is the straight base $l$ . $Δ x = l + π l /2 \cdot \frac{1}{?}$ ... more carefully: detour length $l + π l /2$ , straight length $l 2$ along the diagonal, giving $Δ x = π l / 2$ . Setting equal to $λ$ : $l_{m i n} = 2 λ / π \approx 0.13$ m. R $\to$ 5.

(S) Triangle on top of base $l$ with angles 45°, 105°, 30°. By the law of sines, the two slanted sides have lengths $l sin 3 0^{\circ} / sin 10 5^{\circ}$ and $l sin 4 5^{\circ} / sin 10 5^{\circ}$ , summing to $\approx 1.249 l$ . Path difference $Δ x \approx 0.249 l$ . Setting equal to $λ$ : $l_{m i n} \approx λ /0.249 \approx 1.19$ m. S $\to$ 2.

Mapping: P $\to$ 3, Q $\to$ 4, R $\to$ 5, S $\to$ 2, which is option (D).

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