Question

The set of all values of $\lambda$ for which the equation $\cos ^2 2 x-2 \sin ^4 x-2 \cos ^2 x=\lambda$

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

Answer 1

Answer: (D)

$\lambda=\cos ^2 2 x-2 \sin ^4 x-2 \cos ^2 x$
convert all in to $\cos x$.
$\lambda=\left(2 \cos ^2 x-1\right)^2-2\left(1-\cos ^2 x\right)^2-2 \cos ^2 x$
$= 4 \cos ^4 x-4 \cos ^2 x+1-2\left(1-2 \cos ^2 x+\cos ^4 x\right)- 2 \cos ^2 x $
$= 2 \cos ^4 x-2 \cos ^2 x+1-2 $
$= 2 \cos ^4 x-2 \cos ^2 x-1 $
$= 2\left[\cos ^4 x-\cos ^2 x-\frac{1}{2}\right] $
$= 2\left[\left(\cos ^2 x-\frac{1}{2}\right)^2-\frac{3}{4}\right] $
$ \lambda_{\max }=\left.2\left[\frac{1}{4}-\frac{3}{4}\right]=2 \times\left(-\frac{2}{4}\right)=-1 \text { (max Value }\right) $
$ \lambda_{\min }= 2\left[0-\frac{3}{4}\right]=-\frac{3}{2}(\text { MinimumValue }) $
$ \text { So, Range }=\left[-\frac{3}{2},-1\right]$