If x, y, z are the roots of t^3 - 21t^2 + bt - 343 = 0, b ∈ R, then D is equal to-
Step-by-Step Solution
Key Concept: The determinant D is a Vandermonde-like determinant. If x, y, z are roots of a cubic equation, their product xyz = 343. The determinant D = (x-y)(y-z)(z-x)(xy+yz+zx-1) or similar structure. Given the roots of t^3 - 21t^2 + bt - 343 = 0, the determinant evaluates to 0.
The determinant D = |x x^3 x^4-1; y y^3 y^4-1; z z^3 z^4-1|. This can be split into |x x^3 x^4; y y^3 y^4; z z^3 z^4| - |x x^3 1; y y^3 1; z z^3 1|. The first part is xyz|1 x^2 x^3; 1 y^2 y^3; 1 z^2 z^3|. Given the roots of t^3 - 21t^2 + bt - 343 = 0, we have xyz = 343. The determinant evaluates to 0.
Correct Answer: B