Question

$GivenA=si{n}^2\theta +co{s}^4\theta ,thenforallrealvaluesof\theta$

• 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)

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