JEE Advanced 2022 · Paper 1 · Physics Q12

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PhysicsMultiple correct+4 / -2Question 30 of 54

Three plane mirrors form an equilateral triangle with each side of length $L$ . There is a small hole at a distance $l > 0$ from one of the corners as shown in the figure. A ray of light is passed through the hole at an angle $\theta$ and can only come out through the same hole. The cross section of the mirror configuration and the ray of light lie on the same plane.

Which of the following statement(s) is(are) correct?

An equilateral triangle of three plane mirrors (side L), mirrored faces inward. A small hole in the base at distance l from the bottom-right corner. A ray enters through the hole from below, making angle θ with the base (measured from the base towards the right corner).

The ray of light will come out for

\theta = 30^\circ

, for

0 < l < L

There is an angle for

l = \dfrac L2

at which the ray of light will come out after two reflections.

The ray of light will NEVER come out for

\theta = 60^\circ

, and

l = \dfrac L3

The ray of light will come out for

\theta = 60^\circ

, and

0 < l < \dfrac L2

after six reflections.

Answer and worked solution

Official answer: A, B

Put the base along the $x$ -axis from the left corner $(0,0)$ to the right corner $(L,0)$ . The hole is at $(L-l,0)$ , and the ray enters going up and to the left at angle $\theta$ to the base.

(A) True. For $\theta=30^\circ$ the ray travels at $150^\circ$ to the $+x$ axis. The left mirror lies along $60^\circ$ , so the ray meets it at exactly $90^\circ$ . Moving left, the ray cannot reach the right mirror first. It is reflected straight back along its own path and leaves through the hole, for every $0<l<L$ .

(B) True. For $l=\frac L2$ and $\theta=60^\circ$ , the ray goes from the midpoint of the base to the midpoint of the left side, reflects to the midpoint of the right side, and reflects back to the hole. That is two reflections.

The $\theta=60^\circ$ path in general.

The ray runs parallel to the right side.
It reflects off the left side and runs parallel to the base.
It reflects off the right side and runs parallel to the left side, reaching the base at distance $l$ from the left corner.
After three more reflections (base, left, right) it returns to the hole.

That is five reflections in all (two when $l=\frac L2$ ).

(C) False: for $l=\frac L3$ the ray does come out.
(D) False: it comes out after five reflections, not six.

Topics:Reflection and plane mirrors

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JEE Advanced 2022 · Paper 1, Physics question 12