Correct Answer: $-2$

Solution : Given: $\left(5 \frac{1}{4} \div \frac{3}{7}\right.$ of $\left.\frac{1}{2}\right) \div\left(5 \frac{1}{9}-7 \frac{7}{8} \div 9 \frac{9}{20}\right) \times \frac{11}{21}-\left(5 \div 2\right.$ of $\left.\frac{1}{2}\right)$
= $\left( \frac{21}{4} \div (\frac{3}{7}\right.\times \left.\frac{1}{2}\right)) \div\left( \frac{46}{9}-\frac{63}{8} \div  \frac{189}{20}\right) \times \frac{11}{21}-\left(5 \div (2\right. \times \left.\frac{1}{2})\right)$
= $\left( \frac{21}{4} \times \frac{14}{3}\right.) \div\left( \frac{46}{9}-\frac{63}{8} \times  \frac{20}{189}\right) \times \frac{11}{21}-\left(5 \times1\right)$
= $\left( \frac{49}{2}\right) \div\left( \frac{46}{9}-\frac{5}{6}\right) \times \frac{11}{21}-5$
= $\left( \frac{49}{2}\right) \div\left( \frac{77}{18}\right) \times \frac{11}{21}-5$
= $\left( \frac{49}{2}\right) \times \left( \frac{18}{77}\right)\times \frac{11}{21}-5$
= $\left( \frac{7}{1}\right) \times \left( \frac{9}{11}\right)\times \frac{11}{21}-5$
= $3-5$
= $- 2$
Hence the correct answer is $-2$.