The set of all values of for which the equation cos 2 2 x - 2 sin 4 x - 2 cos 2 x = has a real solution x , is [2023]

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Solution

Q.
The set of all values of $λ$ for which the equation $\cos^{2} 2 x - 2 \sin^{4} x - 2 \cos^{2} x = λ$ has a real solution $x,$ is [2023]

1 $[- 2, - 1]$

2 $[- 2, - \frac{3}{2}]$

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

4 $[- \frac{3}{2}, - 1]$

Ans.
(4)

$The equation is given \cos^{2} 2 x - 2 \sin^{4} x - 2 \cos^{2} x = λ$

$then we can write the given equation as$

$λ = \cos^{2} 2 x - 2 \sin^{4} x - 2 \cos^{2} x$

$\Rightarrow λ = {(2 \cos^{2} x - 1)}^{2} - 2 {(1 - \cos^{2} x)}^{2} - 2 \cos^{2} x$

$\Rightarrow λ = 4 \cos^{4} x - 4 \cos^{2} x + 1 - 2 (1 - 2 \cos^{2} x + \cos^{4} x)$ $- 2 \cos^{2} x$

$\Rightarrow λ = 2 \cos^{4} x - 2 \cos^{2} x + 1 - 2 \Rightarrow λ = 2 \cos^{4} x - 2 \cos^{2} x - 1$

$λ = 2 [\cos^{4} x - \cos^{2} x - \frac{1}{2}], λ = 2 [{(\cos^{2} x - \frac{1}{2})}^{2} - \frac{3}{4}]$

$So λ_{\max} = 2 [\frac{1}{4} - \frac{3}{4}] = 2 \times (\frac{- 2}{4}) = - 1 (maximum value)$

$and λ_{\min} = 2 [0 - \frac{3}{4}] = - \frac{3}{2} (minimum value)$

$So range of the value of λ is [\frac{- 3}{2}, - 1] .$

$Hence option (4) is correct answer.$

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