Question

If ${\log }_{1/2}\frac{|z{|}^2+2|z|+4}{2|z{|}^2+1}<0,$ then the region traced by z is

• A. $|z|<3$
• B. $1<|z|<3$
• C. $|z|>1$
• D. $|z|<2$

Answer 1

Answer: (A)

$\because$ ${\log }_{1/2}\frac{|z{|}^2+2|z|+4}{2|z{|}^2+1}<0={\log }_{1/2}1$ $\Rightarrow$ $\frac{|z{|}^2+2|z|+4}{2|z{|}^2+1}<1$ $(\because base<1)$ $\Rightarrow$ $|z{|}^2-2|z|-3<0$ $(|z|+1)(|z|-3)<0$ $\therefore$ $|z|<3$