Question

If $\left|si{n}^{2}x+10-x^2\right|=\left|9-x^2\right|+2si{n}^{2}x+co{s}^{2}x,$ then $x$ lies in

• A. $\left[-8,8\right]$
• B. $\left[-3,3\right]$
• C. $\left[-\sqrt{17},\sqrt{17}\right]$
• D. $\left[-\sqrt{21},\sqrt{21}\right]$

Answer 1

Answer: (B)

$\left|si{n}^{2}x+10-x^2\right|=\left|9-x^2\right|+si{n}^{2}x+\left(si{n}^{2}x+co{s}^{2}x\right)$
$\Rightarrow \left|si{n}^{2}x+1+9-x^2\right|=\left|9-x^2\right|+si{n}^{2}x+1$
$\left|x+y+z\right|=\left|x\right|+\left|y\right|+\left|z\right|$ , if $x,y,z$ have the same sign
$\Rightarrow si{n}^{2}x\left(9-x^2\right)\ge 0$
$\Rightarrow \left(9-x^2\right)\ge 0$
$\Rightarrow x\in \left[-3,3\right]$