Question

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

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

Answer 1

The correct option is A $[-1,-\frac{1}{2}]$

${\cos }^4\theta +{\sin }^4\theta +\lambda =0$
$\lambda =-\{1-\frac{1}{2}{\sin }^{2}2\theta \}$
$2(\lambda +1)={\sin }^{2}2\theta$
$0\le 2(\lambda +1)\le 1$
$0\le \lambda +1\le \frac{1}{2}$
$\Rightarrow -1\le \lambda \le -\frac{1}{2}$