The point P (1, 3) undergoes the following transformations successively
i) Reflection with respect to the line y = x
ii) Translation through 3 units along the positive direction of the X-axis
iii) Rotation through an angle of $\frac{π}{6}$ about the origin in the clockwise direction. The final position of the point P is

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$(\frac{7}{\sqrt{2}}, \frac{- 5}{\sqrt{2}})$

$(\frac{6 + \sqrt{3}}{2}, \frac{1 - 6 \sqrt{3}}{2})$

$(\frac{6 \sqrt{3} - 1}{2}, \frac{6 + \sqrt{3}}{2})$

$(\frac{6 \sqrt{3} + 1}{2}, \frac{\sqrt{3} - 6}{2})$

answer is D.

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Detailed Solution

Reflection of (1, 3) w.r.t. y = x is (3, 1)
If the point (3, 1) is translated distance 3 units in the positive direction of x-axis
Then the coordinates of the point is (6, 1)
After the rotation through an angle of $π / 6$ about the origin in the clockwise direction, the point is
$(6 \cos (\frac{- π}{6}) - \sin (\frac{- π}{6}), 6 \sin (\frac{- π}{6}) + \cos (\frac{- π}{6})) = (\frac{6 \sqrt{3} + 1}{2}, \frac{- 6 + \sqrt{3}}{2})$