If $u, v$ and $w$ are the three roots of the equation $z^{3} - 1 = 0$ . Calculate $u . v + v . w + w . u$ .

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Solution

The correct option is B $0$
$Z^{3} - 1 = 0 ⟹ (Z - 1) (Z^{2} + 1) = 0 ⟹ Z^{2} + Z + 1 = 0$
roots $= \frac{- 1 \pm \sqrt{3} i}{2}$
$u = 1$
$v = \frac{- 1 + \sqrt{3} i}{2} = w$
$w = \frac{- 1 - \sqrt{3} i}{2} = w^{2}$
$u . v + v . w + w . u = w + w^{2} + w^{3}$
$⟹ 1 - \frac{1}{2}$ $+ \frac{\sqrt{3} i}{2}$ $- \frac{1}{2}$ $- \frac{\sqrt{3} i}{2}$
$⟹ 0$

Suggest Corrections

\to u, \to v

\to w

(\to u + \to v - \to w) . (\to u - \to v) \times (\to v - \to w)

a, b, c, d

u, v, w

u + 2 v + 3 w = 6

4 u + 5 v + 6 w = 12

6 u + 9 v = 4

(\frac{1}{u} + \frac{1}{v} + \frac{1}{w}) x^{2} + [{(b - c)}^{2} + {(c - a)}^{2} + {(d - b)}^{2}] x + u + v + w = 0

{20 x}^{2} + 10 {(a - d)}^{2} x - 9 = 0

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