If $α, β (α < β)$ are the roots of the equation $6 x^{2} + 11 x + 3 = 0$ then which of the following are real?

{cos}^{- 1} α

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{sin}^{- 1} β

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c o s e c^{- 1} α

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Both

{cot}^{- 1} α

and

{cot}^{- 1} β

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Solution

The correct options are
B $c o s e c^{- 1} α$
C ${sin}^{- 1} β$
D Both ${cot}^{- 1} α$ and ${cot}^{- 1} β$
Given, $6 x^{2} + 11 x + 3 = 0$
$6 x^{2} + 9 x + 2 x + 3 = 0$
$3 x (2 x + 3) + 1 (2 x + 3) = 0$
$(3 x + 1) (2 x + 3) = 0$
$x = \frac{- 1}{3} = β$ and $x = \frac{- 3}{2} = α$
Hence
${sin}^{- 1} (β)$ and ${cos}^{- 1} (β)$ is real.
$c o s e c^{- 1} (α)$ is real.
${tan}^{- 1} (α)$ ${tan}^{- 1} (β)$ ${cot}^{- 1} (α)$ and ${cot}^{- 1} (β)$ are real.

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