question_answer

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 \[{{30}^{\text{o}}}\] with the positive direction of the x-axis, then the sum of the x-coordinates of the vertices of the square is: [JEE Online 09-04-2017]

A)  \[2\sqrt{3}-2\]                 

B)  \[\sqrt{3}-2\]

C)  \[2\sqrt{3}-1\]                 

D)  \[\sqrt{3}-1\]

Correct Answer: A

Solution :

                \[\frac{x}{\cos {{30}^{\text{o}}}}\,=\frac{y}{\sin {{30}^{\text{o}}}}=2\]                 \[x=\frac{2\sqrt{3}}{2}\,=\sqrt{3}\] \[y=1\] \[\frac{x}{\cos {{120}^{\text{o}}}}\,=\frac{y}{\sin {{120}^{\text{o}}}}\,=2\] \[x=-1,\,\,y=\sqrt{3}\] \[\frac{x}{\cos {{75}^{\text{o}}}}\,=\frac{y}{\sin {{75}^{\text{o}}}}=2\sqrt{2}\] \[x=\sqrt{3}-1\] \[y=\sqrt{3}+1\] \[sum\,=0+\sqrt{3}\,+\sqrt{3}-1+(-1)\] \[=2\sqrt{3}-2\]