The origin is translated to (1,2). The point (7,5) in the old system undergoes the following transformations successively
(i) Moves to the new point under the given translation of origin
(ii) Translated throgh 2 units along the negative direction of the new X-axis
(iii) Rotated through an angle $\frac{\pi }{4}$about the origin of new system in the positive direction.
The final position of the point (7,5) is

$$\begin{gathered} \mathrm{Given} (\mathrm{h}, \mathrm{k})=(1, 2), \mathrm{P}(\mathrm{x}, \mathrm{y})=(7, 5) \\ \left(\mathrm{i}\right) \mathrm{After} \mathrm{translation} \mathrm{of} \mathrm{origin}, \mathrm{new} \mathrm{position} \\ {\mathrm{P}}_1=(7-1, 5-2)=(6, 3) \\ \left(\mathrm{ii}\right) \mathrm{After} \mathrm{translating} 2 \mathrm{units} \mathrm{in} \mathrm{negative} \mathrm{direction} \mathrm{of} \mathrm{then} \mathrm{new} \mathrm{position} \mathrm{of} \mathrm{the} \mathrm{point} \\ {\mathrm{P}}_2=(6-2, 3)=(4, 3) \\ \left(\mathrm{iii}\right) {\mathrm{OP}}_2={\mathrm{OP}}_3=5 \mathrm{and} \mathrm{tan\theta }=\frac{3}{4} \\ \Rightarrow \mathrm{sin\theta }=\frac{3}{5}, \mathrm{cos\theta }=\frac{4}{5} \\ \mathrm{Now} {\mathrm{P}}_3=\left[{\mathrm{OP}}_3\cos \left(\mathrm{\theta }-\frac{\mathrm{\pi }}{4}\right), {\mathrm{OP}}_3\sin \left(\mathrm{\theta }-\frac{\mathrm{\pi }}{4}\right)\right] \\ =\left[5\left(\mathrm{cos\theta } \cos \frac{\mathrm{\pi }}{4}+\mathrm{sin\theta } \sin \frac{\mathrm{\pi }}{4}\right), 5\left(\mathrm{sin\theta } \cos \frac{\mathrm{\pi }}{4}-\mathrm{cos\theta } \sin \frac{\mathrm{\pi }}{4}\right)\right] \\ =\left[5\left[\frac{4}{5}\left(\frac{1}{\sqrt{2}}\right)+\frac{3}{5}\left(\frac{1}{\sqrt{2}}\right)\right], 5\left[\frac{3}{5}\left(\frac{1}{\sqrt{2}}\right)-\frac{4}{5}\left(\frac{1}{\sqrt{2}}\right)\right]\right] \\ =\left[5\left(\frac{7}{5\sqrt{2}}\right), 5\left(\frac{-1}{5\sqrt{2}}\right)\right]=\left[\frac{7}{\sqrt{2}}, \frac{-1}{\sqrt{2}}\right] \end{gathered}$$