JEE Advanced 2022 — Paper 1 — PHY-I solution

JEE Advanced2022Paper 1PHY-I

A rod of length 2 cm makes an angle $\frac{2 π}{3}$ rad with the principal axis of a thin convex lens. The lens has a focal length of 10 cm and is placed at a distance of $\frac{40}{3}$ cm from the object as shown in the figure. The height of the image is $\frac{30 3}{13}$ cm and the angle made by it with respect to the principal axis is $α$ rad. The value of $α$ is $\frac{π}{n}$ rad, where $n$ is _______.

Your answer

Solution

Let the rod have endpoints $A$ (on axis, at $40/3$ cm) and $B$ (tilted upward and to the left so the rod makes $2 π /3$ with the axis). Distances of $A$ and $B$ from the lens along the axis:

$u_{A} = \frac{40}{3}$ cm, $u_{B} = \frac{40}{3} + 2 cos (π /3) = \frac{43}{3}$ cm.

Using the lens formula $\frac{1}{v} - \frac{1}{- u} = \frac{1}{f}$ with $f = 10$ cm:

$v_{A} = \frac{( 40/3 ) ( 10 )}{40/3 - 10} = 40$ cm. $v_{B} = \frac{( 43/3 ) ( 10 )}{43/3 - 10} = \frac{430}{13}$ cm.

Horizontal separation of image endpoints: $A^{'} B^{'} = 40 - \frac{430}{13} = \frac{90}{13}$ cm.

Image height (given) = $\frac{30 3}{13}$ cm.

$tan α = \frac{30 3 /13}{90/13} = \frac{1}{3} \Rightarrow α = \frac{π}{6}$ .

$∴ n = 6$ .

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