Q30 / 75 · JEE Main 2020 (02 Sep Shift 2) · Mathematics

If the equation $\cos^{4} θ + \sin^{4} θ + λ = 0$ has real solutions for $θ$ then $λ$ lies in interval

Solution

$- λ = \sin^{4} θ + \cos^{4} θ$

$= {(\sin^{2} θ + \cos^{2} θ)}^{2} - 2 \sin^{2} θ \cos^{2} θ$

$= 1 - \frac{4 \sin^{2} θ \cos^{2} θ}{2}$

$= 1 - \frac{\sin^{2} 2 θ}{2}$

$\Rightarrow λ = \frac{\sin^{2} 2 θ}{2} - 1$

$λ \in [- 1, - \frac{1}{2}]$

Hence, the correct answer is option: 'B'.