If z - αz + α (α ∈ R) is a purely imaginary number and |z| = 2, then a value of α is:

Let the given expression be denoted by w. Then

w = z - αz + α = z(1 - α) + α

Since w is purely imaginary, its real part is zero. Let z = x + iy, where x and y are real numbers. Then

Re(w) = x(1 - α) + α = 0

Also, we are given that |z| = 2, which means

|z|² = x² + y² = 4

From the equation x(1 - α) + α = 0, we get

x = -α / (1 - α)

Substituting this into x² + y² = 4:

(-α / (1 - α))² + y² = 4

y² = 4 - α² / (1 - α)²

Since y² ≥ 0, we must have

4 - α² / (1 - α)² ≥ 0

4(1 - α)² ≥ α²

4(1 - 2α + α²) ≥ α²

4 - 8α + 4α² ≥ α²

3α² - 8α + 4 ≥ 0

(3α - 2)(α - 2) ≥ 0

This inequality holds when α ≤ 2/3 or α ≥ 2.

Let's check the options:

If α = √2, then α is approximately 1.414, which satisfies α ≥ 2/3
If α = 1, then α is not greater than or equal to 2.
If α = 2, then α satisfies α ≥ 2.
If α = 12, then α satisfies α ≥ 2.

Therefore, possible values of α are √2, 2, and 12. The question asks for a value of α, so any of these values is a valid solution. The options provided suggest 2 or √2 are the intended answers. Choosing α = 2 as an example, we would have:

x = -2/(1-2) = 2

y² = 4 - 4/(1-2)² = 0

y = 0

Then z = 2, and w = 2(1-2) + 2 = 0, which is not purely imaginary. There must be a mistake in the problem statement or options.