સમીકરણ cos ^2 2 x-2 sin ^4 x-2 cos ^2 x=lambd | vedclass

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Question

$\lambda=\cos ^2 2 x-2 \sin ^4 x-2 \cos ^2 x$

$	ext { convert all in to } \cos x 	ext {. }$

$\lambda=\left(2 \cos ^2 x-1ight)^2-2\left(1-\co

Vedclass · Accepted Answer

$\left[-\frac{3}{2},-1\right]$

Vedclass · Answer

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

સમીકરણ $\cos ^2 2 x-2 \sin ^4 x-2 \cos ^2 x=\lambda$ ને વાસ્તવિક ઉકેલ $x$ હોય તેવી $\lambda$ ની તમામ કિંમતોનો ગણ $...........$ છે.

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