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Question Number 206618 by MATHEMATICSAM last updated on 20/Apr/24

$$\mathrm{If}\mathrm{A}={\sin }^4\theta +{\cos }^4\theta \mathrm{then}\mathrm{select}\mathrm{the}$$$$\mathrm{correct}\mathrm{option}:$$$$\mathrm{i})0<\mathrm{A}<\frac{1}{2}$$$$\mathrm{ii})1<\mathrm{A}<\frac{3}{2}$$$$\mathrm{iii})\frac{1}{2}\le \mathrm{A}\le 1$$$$\mathrm{iv})\frac{3}{2}\le \mathrm{A}\le 2$$

Answered by MM42 last updated on 20/Apr/24

$$A={({sin}^2\theta +{cos}^2\theta )}^2-2{sin}^2\theta {cos}^2\theta$$$$=1-\frac{1}{2}{sin}^2\left(2\theta \right)\Rightarrow ans:\frac{1}{2}⩽A⩽1✓$$$$$$

Answered by Frix last updated on 20/Apr/24

$$A=\frac{3}{4}+\frac{\cos 4x}{4}$$$$\frac{1}{2}⩽A⩽1$$

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