If A=2 ^2x- 2x, then A lies in the interval:

MCQ

If $\text{A}=2\sin^2\text{x}-\cos2\text{x},$ then $A$ lies in the interval:

✓
$[-1,3]$
B
$[1,2]$
C
$[-2,4]$
D
None of these

✓

Answer

Correct option: A.

$[-1,3]$

$\text{A}=2\sin^2\text{x}-\cos2\text{x}$
$=2\sin^2\text{x}-(1-2\sin^2\text{x})$
$=4\sin^2\text{x}-1$
$\therefore0\leq\sin^2\text{x}\leq1$
$\Rightarrow4\times0\leq4\times\sin^2\text{x}\leq4\times1$
$\Rightarrow0\leq4\sin^2\text{x}\leq4$
$\Rightarrow0-1\leq4\sin^2\text{x}-1\leq4-1$
$\Rightarrow-1\leq4\sin^2\text{x}-1\leq3$
$\Rightarrow-1\leq\text{A}\leq3$
$\Rightarrow\text{A}\in[-1,3]$

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