Step 1: Given Information
We are given the following conditions:
- The prism has an angle of \( \theta = 60^\circ \).
- The refractive indices of the left and right halves of the prism are \( n_1 \) and \( n_2 \), respectively, where \( n_2 \geq n_1 \).
- The angle of incidence \( i \) is chosen such that the incident light rays will have minimum deviation when \( n_1 = n_2 = n \).
- For the case of unequal refractive indices, \( n_1 = n \) and \( n_2 = n + \Delta n \) (where \( \Delta n \ll n \)), the angle of emergence is \( e = i + \Delta e \).
We are asked to determine which of the following statements is/are correct.
Step 2: Minimum Deviation and Relation between \( \Delta e \) and \( \Delta n \)
At minimum deviation, the incident light ray undergoes the least bending. For a prism with a refractive index \( n_1 = n_2 = n \), the angle of emergence \( e \) and the angle of incidence \( i \) are related by the prism's geometry.
When the refractive indices of the left and right halves are unequal, with \( n_2 = n + \Delta n \), the angle of emergence \( e \) will shift. Specifically, the shift in the angle of emergence, \( \Delta e \), will depend on the change in the refractive index, \( \Delta n \). This change is proportional to \( \Delta n \). Hence, the shift in the angle of emergence is directly proportional to the change in the refractive index.
Therefore, statement (B) is correct: \( \Delta e \) is proportional to \( \Delta n \).
Step 3: Estimating \( \Delta e \) for \( \Delta n = 2.8 \times 10^{-3} \)
We are given that \( \Delta n = 2.8 \times 10^{-3} \). We need to estimate the value of \( \Delta e \), which lies between 2.0 and 3.0 milliradians. The value of \( \Delta e \) is small because \( \Delta n \) is small. The linear relationship between \( \Delta e \) and \( \Delta n \) means that if \( \Delta n = 2.8 \times 10^{-3} \), the value of \( \Delta e \) will indeed lie between 2.0 and 3.0 milliradians.
Therefore, statement (C) is correct: \( \Delta e \) lies between 2.0 and 3.0 milliradians if \( \Delta n = 2.8 \times 10^{-3} \).
Final Answer:
The correct options are:
- (B) \( \Delta e \) is proportional to \( \Delta n \)
- (C) \( \Delta e \) lies between 2.0 and 3.0 milliradians if \( \Delta n = 2.8 \times 10^{-3} \)
Let $ S $ denote the locus of the point of intersection of the pair of lines $$ 4x - 3y = 12\alpha,\quad 4\alpha x + 3\alpha y = 12, $$ where $ \alpha $ varies over the set of non-zero real numbers. Let $ T $ be the tangent to $ S $ passing through the points $ (p, 0) $ and $ (0, q) $, $ q > 0 $, and parallel to the line $ 4x - \frac{3}{\sqrt{2}} y = 0 $.
Then the value of $ pq $ is
Let $ y(x) $ be the solution of the differential equation $$ x^2 \frac{dy}{dx} + xy = x^2 + y^2, \quad x > \frac{1}{e}, $$ satisfying $ y(1) = 0 $. Then the value of $ 2 \cdot \frac{(y(e))^2}{y(e^2)} $ is ________.