Question

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

• A. $1\le A \le 2$
• B. $\frac{3}{4}1\le A \le 1$
• C. $\frac{13}{16}1\le A \le 1$
• D. $\frac{3}{4}1\le A \le \frac{13}{16}$

Answer 1

Answer: (B)

$Given, A = sin^2 \theta + (1 - sin^2 \theta)^2 $
$\Rightarrow A=sin^4 \theta -sin^2 \theta +1$
$\Rightarrow A=\Bigg(sin^2 \theta-\frac{1}{2}\Bigg)^2+\frac{3}{4}$
$\Rightarrow 0 \le\Bigg(sin^2 \theta-\frac{1}{2}\Bigg)^2\le \frac{1}{4} [\because 0 \le sin^2 \theta \le 1]$
$\therefore \frac{3}{4} \le A \le 1$