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If-A-sin-4-cos-4-then-select-the-correct-option-i-0-lt-A-lt-1-2-ii-1-lt-A-lt-3-2-iii-1-2-A-1-iv-3-2-A-2-




Question Number 206618 by MATHEMATICSAM last updated on 20/Apr/24
If A = sin^4 θ + cos^4 θ then select the   correct option:  i) 0 < A < (1/2)  ii) 1 < A < (3/2)  iii) (1/2) ≤ A ≤ 1  iv) (3/2) ≤ A ≤ 2
$$\mathrm{If}\:\mathrm{A}\:=\:\mathrm{sin}^{\mathrm{4}} \theta\:+\:\mathrm{cos}^{\mathrm{4}} \theta\:\mathrm{then}\:\mathrm{select}\:\mathrm{the}\: \\ $$$$\mathrm{correct}\:\mathrm{option}: \\ $$$$\left.\mathrm{i}\right)\:\mathrm{0}\:<\:\mathrm{A}\:<\:\frac{\mathrm{1}}{\mathrm{2}} \\ $$$$\left.\mathrm{ii}\right)\:\mathrm{1}\:<\:\mathrm{A}\:<\:\frac{\mathrm{3}}{\mathrm{2}} \\ $$$$\left.\mathrm{iii}\right)\:\frac{\mathrm{1}}{\mathrm{2}}\:\leq\:\mathrm{A}\:\leq\:\mathrm{1} \\ $$$$\left.\mathrm{iv}\right)\:\frac{\mathrm{3}}{\mathrm{2}}\:\leq\:\mathrm{A}\:\leq\:\mathrm{2} \\ $$
Answered by MM42 last updated on 20/Apr/24
A=(sin^2 θ+cos^2 θ)^2 −2sin^2 θcos^2 θ  =1−(1/2)sin^2 (2θ)⇒ans:  (1/2)≤A≤1   ✓
$${A}=\left({sin}^{\mathrm{2}} \theta+{cos}^{\mathrm{2}} \theta\right)^{\mathrm{2}} −\mathrm{2}{sin}^{\mathrm{2}} \theta{cos}^{\mathrm{2}} \theta \\ $$$$=\mathrm{1}−\frac{\mathrm{1}}{\mathrm{2}}{sin}^{\mathrm{2}} \left(\mathrm{2}\theta\right)\Rightarrow{ans}:\:\:\frac{\mathrm{1}}{\mathrm{2}}\leqslant{A}\leqslant\mathrm{1}\:\:\:\checkmark \\ $$$$ \\ $$
Answered by Frix last updated on 20/Apr/24
A=(3/4)+((cos 4x)/4)  (1/2)≤A≤1
$${A}=\frac{\mathrm{3}}{\mathrm{4}}+\frac{\mathrm{cos}\:\mathrm{4}{x}}{\mathrm{4}} \\ $$$$\frac{\mathrm{1}}{\mathrm{2}}\leqslant{A}\leqslant\mathrm{1} \\ $$

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