Evaluate \(\frac{2}{6-5\sqrt{3}}\)
A
\(-\left(\frac{12}{39}-\frac{10\sqrt{3}}{39}\right)\)
B
\(\frac{12}{39}-\frac{10\sqrt{3}}{39}\)
C
\(-\left(\frac{12}{39}+\frac{10\sqrt{3}}{39}\right)\)
D
\(\frac{12}{39}+\frac{10\sqrt{3}}{39}\)
Verified Correct Answer: C.
Step-by-Step Mathematical Solution
\(\frac{2}{6-5\sqrt{3}} = \frac{2}{6-5\sqrt{3}} \times \frac{6+5\sqrt{3}}{6+5\sqrt{3}}\\
=\frac{2(6+5\sqrt{3})}{(6-5\sqrt{3})(6+5\sqrt{3})}\\
=\frac{12+10\sqrt{3}}{36-25(3)}\\
=\frac{12+10\sqrt{3}}{36-75}\\
=\frac{12+10\sqrt{3}}{-39}\\
=-\left(\frac{12}{39}+\frac{10\sqrt{3}}{39}\right)\)
Source: Myschool Past Questions Archive (Record #8829)