(A) t is divisible by (α - β)
(B) t is divisible by (β - γ)
(C) t is divisible by (γ - α)
(D) (γ - α) is divisible by t
Step-by-Step Solution
Key Concept: The determinant given is a Vandermonde determinant. The value t = (\alpha-\beta)(\beta-\gamma)(\gamma-\alpha). Since t is the product of these factors, it is divisible by each of them. However, (\gamma-\alpha) is not necessarily divisible by t.
The determinant is a Vandermonde determinant: |1 1 1; \alpha \beta \gamma; \alpha^2 \beta^2 \gamma^2| = (\beta-\alpha)(\gamma-\beta)(\gamma-\alpha). Given the determinant is t, t = -(\alpha-\beta)(\beta-\gamma)(\gamma-\alpha). Thus, t is divisible by (\alpha-\beta), (\beta-\gamma), and (\gamma-\alpha). Statement (D) is false.
Correct Answer: D