JEE Advanced 2024 Paper 1 Q07 Physics Waves & Optics Geometrical Optics Hard

JEE Advanced 2024 Paper 1 · Q07 · Geometrical Optics

A glass beaker has a solid, plano-convex base of refractive index 1.60, as shown in the figure. The radius of curvature of the convex surface (SPU) is 9 cm, while the planar surface (STU) acts as a mirror. This beaker is filled with a liquid of refractive index $n$ up to the level QPR. If the image of a point object O at a height of $h$ (OT in the figure) is formed onto itself, then, which of the following option(s) is(are) correct?

[Figure: cross-section showing point O above the beaker on axis at height $h$ from T (the planar bottom surface), liquid surface QPR at the level of P (top of the convex glass), and the plano-convex glass base STU.]

  1. A. For $n = 1.42$, $h = 50\,\text{cm}$.
  2. B. For $n = 1.35$, $h = 36\,\text{cm}$.
  3. C. For $n = 1.45$, $h = 65\,\text{cm}$.
  4. D. For $n = 1.48$, $h = 85\,\text{cm}$.
JEE Advanced multi-correct — pick every correct option, then check.
Reveal answer + step-by-step solution

Correct answer:A, B

Solution

For self-imaging through a plane mirror at the back, rays inside the glass must hit the plane mirror normally, i.e. travel vertically inside the glass. Two refractions: (i) flat liquid surface QPR (parallel slab — shifts object virtually), (ii) spherical liquid-glass interface SPU of radius 9 cm.

Applying refraction at the spherical surface (object distance from P after the apparent-depth shift through liquid): for the rays to be parallel inside the glass, $\dfrac{n_{\text{glass}}}{v} - \dfrac{n}{u} = \dfrac{n_{\text{glass}} - n}{R}$ with $v \to \infty$ giving the self-consistency equation involving $n$, $h$, and $R = 9$ cm.

Working through the lens-mirror compound (FIITJEE solution): the pairs $(n, h) = (1.42, 50\,\text{cm})$ and $(1.35, 36\,\text{cm})$ satisfy the self-imaging condition. The pairs $(1.45, 65)$ and $(1.48, 85)$ violate it. Hence (A) and (B) are correct.

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