A) \[(2,\,\,3)\]
B) \[(2,\,\,\sqrt{3})\]
C) \[(\sqrt{3},\,\,2)\]
D) \[(\sqrt{3},\,\,5)\]
Correct Answer: D
Solution :
We have, \[x=4,\,\,y=2\sqrt{3}\] and\[\theta =-{{30}^{o}}\] \[\therefore \] \[X=x\cos \theta +y\sin \theta \] and \[Y=-x\sin \theta +y\cos \theta \] \[\Rightarrow \] \[X=4\cos {{30}^{o}}-2\sqrt{3}\sin {{30}^{o}}\] \[=4\times \frac{\sqrt{3}}{2}-2\sqrt{3}\times \frac{1}{2}\] and \[Y=4\sin {{30}^{o}}+2\sqrt{3}\cos {{30}^{o}}\] \[=4\times \frac{1}{2}+2\sqrt{3}\times \frac{\sqrt{3}}{2}\] \[\Rightarrow \] \[X=\sqrt{3}\]and\[Y=5\] Hence, the given point in the new system is\[(\sqrt{3},\,5)\].
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