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A square. of each side 2, lies above the x-axis and has one vertex at the origin. If one of the sides passing through the origin makes an angle 30° with the positive direction of the x-axis, then the sum of the x-coordinates of the vertices of the square is

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$\sqrt{3} - 2$

$2 \sqrt{3} - 1$

$\sqrt{3} - 1$

$2 \sqrt{3} - 2$

answer is D.

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Detailed Solution

Side of square = 2

$\begin{array}{l} \frac{x}{\cos 30^{\circ}} = \frac{y}{\sin 30^{\circ}} = 2 \\ \Rightarrow x = \frac{2 \sqrt{3}}{2} = \sqrt{3} \end{array}$

and $y = 1$

$\begin{aligned} \frac{x}{\cos 120^{\circ}} = \frac{y}{\sin 120^{\circ}} = 2 \\ \Rightarrow x = - 1, y = \sqrt{3} \\ \frac{x}{\cos 75^{\circ}} = \frac{y}{\sin 75^{\circ}} = 2 \sqrt{2} \end{aligned}$

$\Rightarrow x = \sqrt{3} - 1$ and $y = \sqrt{3} + 1$

$∴$ Required sum $= 0 + \sqrt{3} + \sqrt{3} - 1 + (- 1) = 2 \sqrt{3} - 2$

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