Question

If ${\sin }^{2}x+4\sin x+5\in [k,5k]$, $k=$

• A. $1$
• B. $2$
• C. $3$
• D. $4$

Answer 1

The correct option is B $2$

$f\left(x\right)={\sin }^{2}x+4\sin x+5$
$=(\sin x+2{)}^2+1$
$\therefore$ max. value of $f\left(x\right)=9+1=10\sin x=1$
min value of $f\left(x\right)=2\sin x=-1$
$\therefore$ Range $=[k,5k]=[2,10]$
$k=2$