If one of the roots of the equation $3 x^{3} + 11 x^{2} + 12 x + 4 = 0$ is $(- 1)$ , then all the three roots are in :

Arithmetic progression

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Geometric progression

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Harmonic progression

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None of these

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Solution

The correct option is C Harmonic progression
The roots of the given equation are -2, -1, $\frac{- 2}{3}$ .
These roots are neither in A.P. nor in G.P. These roots are in H.P.
because $- \frac{1}{2}, - 1, - \frac{3}{2}$ are in A.P.
with common difference $- \frac{1}{2}$
$∴ - 2, - 1, - \frac{2}{3}$ are in H.P.

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