Question

If a, b, g are the roots of equation x³ – 3x² + 3x + 7 = 0 and w is cube roots of unity then the value of $\frac{\alpha - 1}{\beta - 1}$ + $\frac{\beta - 1}{\gamma - 1}$+ $\frac{\gamma - 1}{\alpha - 1}$ is-

• A. w²
• B. 2w²
• C. 3w²
• D. –3w²

Answer 1

Answer: (C)

3w²

Answer 2

Sol. x³ – 3x² + 3x + 7 = 0

Ž (x – 1)³ = –8

Ž $\frac{x - 1}{- 2}$ = (1)^1/3 Ž 1, w, w²

a = –1, b = 1– 2w, g = 1 – 2w²

\ E = $\frac{- 2}{- 2\omega}$+ $\frac{- 2}{- 2\omega^{2}}$+ $\frac{- 2\omega^{2}}{- 2}$

= $\frac{1}{\omega}$ + $\frac{1}{\omega}$ + $\frac{1}{\omega}$ = $\frac{3}{\omega}$ = 3w² .