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Question

If $$ A=\sin^{2}\theta+\cos^{4}\theta$$, then for all real values of $$\theta$$


A
1A2
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B
34A1
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C
1316A1
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D
34A1316
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Solution

The correct option is B $$\displaystyle \frac{3}{4}\leq A\leq 1$$
Let $$A= \sin ^{2}\theta +\cos ^{4}\theta $$
$$=\cos ^{4}\theta -\cos ^{2}\theta +1$$
$$=\cos ^{2}\theta (1- \sin ^{2}\theta )-\cos^{2}\theta+1$$
$$\displaystyle =1-\frac{(\sin 2 \theta )^{2}}{4}$$
$$\therefore $$ range of $$(\sin 2\theta )^{2}\in [0,1]$$
$$\therefore $$ range of $$\displaystyle A\in \left [1-\frac{1}{4} , 1-\frac{0}{4} \right]$$
$$\displaystyle A\in  \left[\frac{3}{4} , 1 \right]$$

Maths

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