If the equation ${sin}^{4} x - (k + 2) {sin}^{2} x - (k + 3) = 0$ has a solution then k must lie in the interval

(- 4, - 2)

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[- 3, - 2)

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(- 4, - 3)

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[- 3, - 2]

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Solution

The correct option is C $[- 3, - 2]$
$G i v e n - {s i n}^{4} x - {(k + 2) s i n}^{2} x - (k + 3) = 0. T o f i n d o u t - t h e i n t e r v a l, i n w h i c h k i s l y i n g S o l u t i o n - {s i n}^{4} x - {(k + 2) s i n}^{2} x - (k + 3) = 0 ⟹ {s i n}^{4} x - {(k + 3) s i n}^{2} x + s i n x - (k + 3) = 0 ⟹ {{s i n}^{2} x - (k + 3)} ({s i n}^{2} x + 1) = 0 ∴ {s i n}^{2} x = (k + 3) . N o w t h e v a l u e o f {s i n}^{2} x l i e b e t w e e n 0 & 1. S o k + 3 = 0 ⟹ k = - 3 a n d k + 3 = 1 ⟹ k = - 2. ∴ T h e i n t e r v a l, i n w h i c h k i s l y i n g, i s [- 3, - 2] . A n s - O p t i o n D .$

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