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: , 2
Solution:
$\sin ^{4} \theta+\cos ^{4} \theta=-\lambda$
$\Rightarrow\left(\sin ^{2} \theta+\cos ^{2} \theta\right)^{2}-2 \sin ^{2} \theta \cdot \cos ^{2} \theta=-\lambda$
$\Rightarrow 1-2 \sin ^{2} \theta \cos ^{2} \theta=-\lambda$
$\Rightarrow \lambda=\frac{(\sin 2 \theta)^{2}}{2}-1$
$\Rightarrow$ as $\sin ^{2} 2 \theta \in[0,1] \Rightarrow \lambda \in\left[-1, \frac{-1}{2}\right]$
Click here to get exam-ready with eSaral
For making your preparation journey smoother of JEE, NEET and Class 8 to 10, grab our app now.