Question

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

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

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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