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Question: If a, b, g are the roots of the equation x<sup>3</sup> + mx<sup>2</sup> + 3x + m = 0, then the gener...

If a, b, g are the roots of the equation x3 + mx2 + 3x + m = 0, then the general value of tan–1a + tan–1b+ tan–1g is:

A

(2n + 1) π2\frac{\pi}{2}

B

np

C

nπ2\frac{n\pi}{2}

D

Dependent upon the value of p

Answer

np

Explanation

Solution

By tan (A + B + C) formula

tan (tan–1 a + tan–1b + tan–1 g)= tan

tan⁡−1α+tan⁡−1β+tan⁡−1γ−(tan⁡−1α)(tan⁡−1β)tan⁡−1γ1−∑tan⁡−1αtan⁡−1β\frac{\tan^{- 1}\alpha + \tan^{- 1}\beta + \tan^{- 1}\gamma - (\tan^{- 1}\alpha)(\tan^{- 1}\beta)\tan^{- 1}\gamma}{1 - \sum_{}^{}{\tan^{- 1}\alpha\tan^{- 1}\beta}}

= tan (−m)−(−m)1−3\frac{( - m) - ( - m)}{1 - 3}= tan 0 = 0

̃ tan–1 a + tan–1b + tan–1g = np