Which of following statement is/are false -<br>(A) t is divisible by (α - β)<br>(B) t is divisible by (β - γ)<br>(C) t is divisible by (γ - α)<br>(D) (γ - α) is divisible by t
(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 in the paragraph is a Vandermonde determinant, which equals (x-y)(y-z)(z-x). Given the equation x^3 - 14x^2 + Px - 36 = 0 has roots \alpha, \beta, \gamma, the determinant t = (\alpha-\beta)(\beta-\gamma)(\gamma-\alpha). Since t is the product of these differences, it is divisible by each of them, but the reverse is not necessarily true.
The determinant is given by |1 1 1; \alpha \beta \gamma; \alpha^2 \beta^2 \gamma^2| = (\alpha-\beta)(\beta-\gamma)(\gamma-\alpha) = t. Since t = (\alpha-\beta)(\beta-\gamma)(\gamma-\alpha), t is divisible by (\alpha-\beta), (\beta-\gamma), and (\gamma-\alpha). However, (\gamma-\alpha) is not necessarily divisible by t, as t is a product of three terms.
Correct Answer: 4