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

Moderate

A

If $0 < α < \frac{π}{4}$ , then the equation has both roots in $(\sin α, \cos α)$
B

If $\frac{π}{4} < α < \frac{π}{2}$ , then the equation has both roots in $(\sin α, \cos α)$
C

If $0 < α < \frac{π}{4}$ , then one root lies in $(- \infty, \sin α)$ and the other in $(\cos α, \infty)$
D

If $\frac{π}{4} < α < \frac{π}{2}$ , then one root lies in $(- \infty, \cos α)$ and the other in $(\sin α, \infty)$

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 $0 < α < \frac{π}{4}, \sin α < \cos α$
Therefore, equation f (x) = 0 has one root in $(- \infty, \sin α)$ and other in $(\cos α, \infty)$ .
Also, for $\frac{π}{4} < α < \frac{π}{2}$ , $\cos α < \sin α$
Therefore, equation f(x) = 0 has one root in $(- \infty, \cos α)$ and other in $(\sin α, \infty)$ .

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