To solve this problem, let's consider a particle projected up an inclined plane. The angle of inclination of the plane is represented by α, and the angle of projection from the inclined plane is represented by θ.

Let's break down the problem step by step:

Step 1: Analyzing the motion

When the particle is projected up the inclined plane, it moves along two directions: one perpendicular to the inclined plane and one parallel to the inclined plane.

The perpendicular direction can be represented by the angle of projection from the inclined plane, θ.

The parallel direction can be represented by the angle of inclination of the plane, α.

Step 2: Resolving the velocities

We can resolve the initial velocity of the particle into two components: one perpendicular to the inclined plane (v⊥) and one parallel to the inclined plane (v||).

The component of velocity perpendicular to the inclined plane is given by v⊥ = v₀ sinθ, where v₀ is the initial velocity of the particle.

The component of velocity parallel to the inclined plane is given by v|| = v₀ cosθ.

Step 3: Analyzing the collision

When the particle strikes the inclined plane at right angles, it means that the perpendicular component of velocity becomes zero. Therefore, v⊥ = 0.

From the equation v⊥ = v₀ sinθ, we can write:

0 = v₀ sinθ

This implies that sinθ = 0.

Step 4: Finding the relation between angles

Since sinθ = 0, the angle θ must be either 0 degrees or 180 degrees.

If θ = 0 degrees, it means that the particle was projected horizontally from the inclined plane. In this case, the angle of inclination of the plane, α, is not defined.

If θ = 180 degrees, it means that the particle was projected vertically downwards onto the inclined plane. In this case, the angle of inclination of the plane, α, is 90 degrees.

Therefore, the relation between α and θ can be written as:

θ = 0 degrees or θ = 180 degrees, for any value of α.

This relation does not match any of the given options (a), (b), or (c).

Option (d) is incomplete as it only states "cot = 2tan alpha". It should be written as cotθ = 2tanα, which is not the correct relation between α and θ.

Hence, none of the given options are the correct relation between the angle of inclination of the plane, α, and the angle of projection from the inclined plane, θ.