1. Initial point

   The particle starts at $z_0 = 1+2i$ so its coordinates are $(1,2)$.

2. Moves horizontally away from origin by 5 units

   Since the point is on the right of the origin and it moves horizontally away from origin, the real part increases by $5$.

   So, $(1,2) \to (1+5,2) = (6,2)$

3. Then vertically away from origin by 3 units

   Since the point is above the origin and it moves vertically away from origin, the imaginary part increases by $3$.

   Thus, $(6,2) \to (6,2+3) = (6,5)$

   Therefore, $z_1 = 6+5i$

4. Moves $\sqrt{2}$ units in the direction of vector $\hat i + \hat j$

   The direction vector is $(1,1)$, whose magnitude is $\sqrt{1^2+1^2} = \sqrt{2}$

   So the unit vector in this direction is $\frac{1}{\sqrt{2}}(\hat i + \hat j)$

   Moving a distance $\sqrt{2}$ in this direction gives displacement $\sqrt{2}\cdot \frac{1}{\sqrt{2}}(\hat i + \hat j)= \hat i + \hat j$

   Hence, $(6,5) \to (7,6)$

   So the new point is $7+6i$

5. Then rotates through an angle $\pi/2$ anticlockwise about the origin

   Rotation by $\pi/2$ anticlockwise transforms $(x,y) \to (-y,x)$

   Applying this to $(7,6)$: $(7,6) \to (-6,7)$

   Therefore, $z_2 = -6+7i$

6. Check options

   • A: $6+7i$ ❌
   • B: $-7+6i$ ❌
   • C: $7+6i$ ❌
   • D: $-6+7i$ ✅

Hence, the correct answer is Option D.