JEE PYQ: Trigonometric Ratios And Identities Question 3
Question 3 - 2020 (02 Sep Shift 2)
If the equation $\cos^4 \theta + \sin^4 \theta + \lambda = 0$ has real solutions for $\theta$, then $\lambda$ lies in the interval:
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(1) $\left(-\frac{5}{4}, -1\right)$
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(2) $\left[-1, -\frac{1}{2}\right]$
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(3) $\left(-\frac{1}{2}, -\frac{1}{4}\right]$
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(4) $\left(-\frac{3}{2}, -\frac{5}{4}\right)$
Show Answer
Answer: (2) $\left[-1, -\frac{1}{2}\right]$
Solution
$\sin^4\theta + \cos^4\theta = -\lambda \Rightarrow 1 - 2\sin^2\theta\cos^2\theta = -\lambda \Rightarrow \lambda = \frac{\sin^2 2\theta}{2} - 1$. Since $\sin^2 2\theta \in [0,1]$, $\lambda \in [-1, -\frac{1}{2}]$.
BETA
