Given that $$x = 4\sqrt{12} + 5\sqrt{27 }– 3\sqrt{75 }+ \sqrt{300}$$​ and y = $$(2 + \sqrt3)(2 – \sqrt3).$$​ If $$\frac x y = a + b\sqrt3,$$​ then what is the value of (a + 2b)?

Correct option is B

​$$\textbf{Given:} \\x = 4\sqrt{12} + 5\sqrt{27} - 3\sqrt{75} + \sqrt{300} \\y = (2 + \sqrt{3})(2 - \sqrt{3}) \\\text{and } \frac{x}{y} = a + b\sqrt{3} \\\text{Required value } = a + 2b \\\textbf{Concept Used:} \\\text{Simplification of surds and rationalization} \\\textbf{Formula Used:} \\\sqrt{mn} = \sqrt{m}\sqrt{n} \\(a + b)(a - b) = a^2 - b^2 \\\textbf{Solution:} \\x = 4\sqrt{12} + 5\sqrt{27} - 3\sqrt{75} + \sqrt{300} \\= 4(2\sqrt{3}) + 5(3\sqrt{3}) - 3(5\sqrt{3}) + 10\sqrt{3} \\= 8\sqrt{3} + 15\sqrt{3} - 15\sqrt{3} + 10\sqrt{3} \\= 18\sqrt{3} \\y = (2 + \sqrt{3})(2 - \sqrt{3}) \\= 4 - 3 \\= 1 \\\frac{x}{y} = \frac{18\sqrt{3}}{1} \\= 18\sqrt{3} \\\text{Comparing with } a + b\sqrt{3}: \\a = 0,\quad b = 18 \\a + 2b = 0 + 2(18) \\= 36 \\\textbf{Final Answer:} \\36$$​​