If the equation cos^{4} theta + sin^{4} theta | vedclass

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(D) Given the equation $\cos^{4} 	heta + \sin^{4} 	heta + \lambda = 0$,we can write $\lambda = -(\sin^{4} 	heta + \cos^{4} 	heta)$.Using the ident

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

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(D) Given the equation $\cos^{4} 	heta + \sin^{4} 	heta + \lambda = 0$,we can write $\lambda = -(\sin^{4} 	heta + \cos^{4} 	heta)$.Using the identity $\sin^{4} 	heta + \cos^{4} 	heta = (\sin^{2} 	heta + \cos^{2} 	heta)^{2} - 2 \sin^{2} 	heta \cos^{2} 	heta$,we get $\sin^{4} 	heta + \cos^{4} 	heta = 1 - \frac{1}{2} \sin^{2} 2	heta$.Thus,$\lambda = -(1 - \frac{1}{2} \sin^{2} 2	heta) = \frac{1}{2} \sin^{2} 2	heta - 1$.Since $0 \le \sin^{2} 2	heta \le 1$,we have $0 \le \frac{1}{2} \sin^{2} 2	heta \le \frac{1}{2}$.Subtracting $1$ from all parts,we get $-1 \le \frac{1}{2} \sin^{2} 2	heta - 1 \le -\frac{1}{2}$.Therefore,$\lambda \in [-1, -\frac{1}{2}]$.

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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