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Given $A = \sin^{2} θ + \cos^{4} θ$ , then for all real $θ$

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a

1≤A≤2

b

34≤A≤1

c

1316≤A≤1

d

34≤A≤1316

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detailed solution

Correct option is B

We have,A=sin2⁡θ+cos4⁡θ⇒ A=1−cos⁡2θ2+1+cos⁡2θ22⇒ A=12−12cos⁡2θ+14+12cos⁡2θ+14cos2⁡2θ⇒ A=34+14cos⁡4θ+12=34+18+18cos⁡4θ Now, −1≤cos⁡4θ≤1⇒ −18≤cos⁡4θ8≤18⇒ 34+18−18≤34+141+cos⁡4θ2≤34+18+18⇒ 34≤A≤1

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