Question

If $A=2si{n}^2\theta -cos2\theta$,then A lies in the interval

• A. [-1, 3]
• B. [1, 2]
• C. [-2, 4]
• D. none of these

Answer 1

The correct option is A

[-1, 3]

$A=2si{n}^2\theta -cos2\theta =2si{n}^2\theta -(1-2si{n}^2\theta )=4si{n}^2\theta -1\because 0\le si{n}^2\theta \le 1\Rightarrow 4\times 0\le 4\times si{n}^2\theta \le 4\times 1\Rightarrow 0\le 4si{n}^2\theta \le 4\Rightarrow 0-1\le 4si{n}^2\theta \le 4\Rightarrow 0-1\le 4si{n}^2\theta -1\le 4-1\Rightarrow -1\le 4si{n}^2\theta -1\le 3\Rightarrow Aϵ[-1,3]$