Introduction:
When a ray of light enters a prism, it undergoes refraction and deviation due to the change in medium. However, if one of the surfaces of the prism is silvered, the ray of light can be reflected back and retrace its path. In this scenario, we need to determine the angle of incidence at which the ray should be incident on the prism.

Given:
- Angle of the prism = 30 degrees
- Refractive index of the prism = √2

Procedure:

Step 1: Finding the angle of refraction:
To determine the angle of incidence, we first need to find the angle of refraction. The angle of refraction can be calculated using Snell's law:

n₁sin(i) = n₂sin(r)

where n₁ is the refractive index of the medium the light is coming from, i is the angle of incidence, n₂ is the refractive index of the medium the light is entering, and r is the angle of refraction.

In this case, the light is passing from air (refractive index ≈ 1) to the prism (refractive index = √2). Therefore, the equation becomes:

1sin(i) = √2sin(r)

Step 2: Finding the angle of deviation:
The angle of deviation of a prism can be calculated using the formula:

A = (μ - 1) × A₁

where A is the angle of deviation, μ is the refractive index of the prism, and A₁ is the angle of the prism.

In this case, A₁ is given as 30 degrees, and μ is √2. Therefore, the angle of deviation becomes:

A = (√2 - 1) × 30

Step 3: Finding the angle of incidence:
To find the angle of incidence at which the ray retraces its path after reflection, we need to consider the law of reflection. According to the law of reflection, the angle of incidence and the angle of reflection are equal.

Therefore, the angle of incidence should be equal to the angle of deviation (A) calculated in Step 2.

Conclusion:
To make a ray of light retrace its path after reflection from the silvered surface of the prism, the angle of incidence should be equal to the angle of deviation, which can be calculated using the formula A = (√2 - 1) × 30.

In this question angle of prism given =30 and we know that A=r1+r2 so second surface coated then r2=0 so A=r1=30 and refractive index =root 2 then according to snell law (u) =sin(I) /sin(r) so =root 2=sin i/sin 30 so i=45