Question

If the equation $\cos ^{4} \theta+\sin ^{4} \theta+\lambda=0$ has real solutions for $\theta,$ then $\lambda$ lies in the interval :

• A. $\left[-\frac{3}{2},-\frac{5}{4}\right]$
• B. $\left(-\frac{1}{2},-\frac{1}{4}\right]$
• C. $\left(-\frac{5}{4},-1\right)$
• D. $\left[-1,-\frac{1}{2}\right]$

Answer 1

Answer: (D)

$\lambda=-\left(\sin ^{4} \theta+\cos ^{4} \theta\right)$
$\lambda=-\left[\left(\sin ^{2} \theta+\cos ^{2} \theta\right)^{2}-2 \sin ^{2} \theta \cos ^{2} \theta\right]$
$\lambda=\frac{\sin ^{2} 2 \theta}{2}-1$
$\frac{\sin ^{2} 2 \theta}{2} \in\left[0, \frac{1}{2}\right]$
$\lambda \in\left[-1,-\frac{1}{2}\right]$