Question

If \log_{\sin \frac{\pi}{6}} \left\\{\frac{\left|z-2\right| + 3 }{3\left|z - 2\right| - 1 }\right\\}>1 , then

• A. $|z - 2| > 7$
• B. $|z - 2| < 7$
• C. $|z - 2| < 3$
• D. $|z - 2| < 6$

Answer 1

Answer: (A)

$|z - 2| > 7$

Answer 2

We have, \log _{\sin \frac{\pi}{6}}\left\\{\frac{|z-2|+3}{3|z-2|-1}\right\\}>\,1 \Rightarrow \, \log _{1 / 2}\left\\{\frac{|z-2|+3}{3|z-2|-1}\right\\}>\,1 $\therefore$ Here, base of log is less than $1$ i.e., $\frac{1}{2}\,7$