Question:
If the equation $\cos ^{4} \theta+\sin ^{4} \theta+\lambda=0$ has real solutions for $\theta$, then $\lambda$ lies in the interval :
Correct Option: , 4
Solution:
$\lambda=-\left(\sin ^{4} \theta+\cos ^{4} \theta\right)$
$\lambda=-\left(\sin ^{2} \theta+\cos ^{2} \theta\right)^{2}-2 \sin ^{2} \theta \cos ^{2} \theta$
$\lambda=\frac{\sin ^{2} 2 \theta}{2}-1$
$\frac{\sin ^{2} 2 \theta}{2} \in\left[0, \frac{1}{2}\right]$
$\lambda \in\left[-1,-\frac{1}{2}\right]$
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