Consider equation (x−sin⁡α)(x−cos⁡α)−2=0. Which of the following is/are true?

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If 0<α<π4, then the equation has both roots in (sin⁡α, cos⁡α)

If π4<α<π2, then the equation has both roots in (sin⁡α, cos⁡α)

If 0<α<π4, then one root lies in (−∞, sin⁡α) and the other in (cos⁡α, ∞)

If π4<α<π2, then one root lies in (−∞, cos⁡α) and the other in (sin⁡α, ∞)

answer is C.

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Detailed Solution

Let f(x)=(x−sin⁡α)(x−cos⁡α)−2.Then, f(sin⁡α)=−2<0 and f(cos⁡α)=−2<0.So, sin⁡α and cos⁡α lie between the roots.For 0<α<π4,sin⁡α

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