The sum of the distinct real values of μ, for which the vectors μi + j + k, i + μj + k, i + j + μk are coplanar, is:

The correct option is C
Let the given vectors be

→a = μi + j + k
→b = i + μj + k
→c = i + j + μk
Since →a, →b, and →c are coplanar, their scalar triple product is 0.
[→a →b →c] = 0
∣∣∣μ11∣∣∣=0
1μ1∣∣∣
11μ∣∣∣
Expanding the determinant, we get:
μ(μ² - 1) - 1(μ - 1) + 1(1 - μ) = 0
μ³ - μ - μ + 1 + 1 - μ = 0
μ³ - 3μ + 2 = 0
This is a cubic equation in μ. We can find its roots by observation or by using numerical methods.
By observation, we can see that μ = 1 is a root:
1³ - 3(1) + 2 = 0
Therefore, (μ - 1) is a factor. Performing polynomial division, we get:
(μ - 1)(μ² + μ - 2) = 0
(μ - 1)(μ - 1)(μ + 2) = 0
(μ - 1)²(μ + 2) = 0
The distinct real values of μ are 1 and -2.
The sum of the distinct real values of μ is 1 + (-2) = -1
Therefore, the sum of the distinct real values of μ is -1.