MCQ
If $\frac{5-\sqrt{3}}{2+\sqrt{3}}=x+y \sqrt{3}$, then
- A$x=-13, y=-7$
- B$x=13, y=-7$
- ✓$x=-13, y=7$
- D$x=13, y=7$
✓
Answer
Correct option: C.
$x=-13, y=7$
$x+y \sqrt{3}=\frac{5-\sqrt{3}}{2+\sqrt{3}}$
$=\frac{5-\sqrt{3}}{2+\sqrt{3}} \times \frac{2-\sqrt{3}}{2-\sqrt{3}}$
$=\frac{(5-\sqrt{3})(2-\sqrt{3})}{(2)^2-(\sqrt{3})^2}$
$=\frac{5(2-\sqrt{3})-\sqrt{3}(2-\sqrt{3})}{4-3}$
$=\frac{10-5 \sqrt{3}-2 \sqrt{3}+3}{1}$
$=13-7 \sqrt{3}$
$\text { Hence, } x+y \sqrt{3}=13-7 \sqrt{3}$
$\Rightarrow x=13, y=-7$
$=\frac{5-\sqrt{3}}{2+\sqrt{3}} \times \frac{2-\sqrt{3}}{2-\sqrt{3}}$
$=\frac{(5-\sqrt{3})(2-\sqrt{3})}{(2)^2-(\sqrt{3})^2}$
$=\frac{5(2-\sqrt{3})-\sqrt{3}(2-\sqrt{3})}{4-3}$
$=\frac{10-5 \sqrt{3}-2 \sqrt{3}+3}{1}$
$=13-7 \sqrt{3}$
$\text { Hence, } x+y \sqrt{3}=13-7 \sqrt{3}$
$\Rightarrow x=13, y=-7$
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