1) Reflection about x=y
2)Transformation through a distance 2 units along +ve x axis
3) Rotation through an angle $\pi/4$ about the origin in the counterclockwise direction
Find the final coordinates
The point becomes (1,4)
Then after shifting origin $$X=x-h$$ $$X=-1$$ So (-1,4)
After rotating the axes $$X=x\cos \pi/4+y\sin \pi/4$$ $$X=\frac{-1}{\sqrt 2} +\frac{4}{\sqrt 2}$$ $$X=\frac{3}{\sqrt 2}$$ But the x coordinate given in the answer is $\frac{-1}{\sqrt 2}$. What’s wrong?
On
Another way is by complex numbers $$ $$ Let $Z_1$=(3+i4) is rotated about origin by π/4 and complex number after rotation is $Z_2$ $$\frac{Z_2}{Z_1}=e^{\frac{iπ}{4}}$$ $$Z_2=(3+i4)(\frac{1}{\sqrt{2}}+i\frac{1}{\sqrt{2}})$$ $$Z_2=-\frac{1}{\sqrt{2}}+i\frac{7}{\sqrt{2}}$$ Hence point after rotation is $(-\frac{1}{\sqrt{2}},\frac{7}{\sqrt{2}})$. For more on rotation of complex numbers check this link https://www.mathsdiscussion.com/best-iitjee-maths-for-mains-and-advance/
Step 2 transformation by 2 unit along +X axis point after this is (3,4) $$ $$ Now rotation by π/4 $$ X=5(\frac{3}{5}.\frac{1}{\sqrt{2}}-\frac{4}{5}.\frac{1}{\sqrt{2}})$$ $$X=\frac{-1}{\sqrt{2}}$$