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)

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