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 given by the product of (x-y)(y-z)(z-x). Since x, y, z are roots of t^3 - 21t^2 + bt - 343 = 0, we have xyz = 343 = 7^3. If x, y, z are in G.P. (as implied by the context of the paragraph), then y^3 = xyz = 343, so y = 7. The roots are 7/r, 7, 7r. The sum of roots is 7(1/r + 1 + r) = 21, so 1/r + 1 + r = 3, which means r^2 - 2r + 1 = 0, so r = 1. Thus x = y = z = 7. The determinant D = |x x^3 x^4-1; y y^3 y^4-1; z z^3 z^4-1|. If x=y=z, the rows are identical, so D = 0.
The determinant is D = |x x^3 x^4-1; y y^3 y^4-1; z z^3 z^4-1|. This can be written as |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. For the specific case where x, y, z are roots, if they are equal, the determinant is 0.
Correct Answer: 2