Step-by-step Derivations:

1. Understanding the Argand Plane Representation In the Argand plane, we can represent movements (displacements) as complex numbers. We'll set the origin as the starting point, $z_0 = 0$. The positive real axis corresponds to the East direction, and the positive imaginary axis corresponds to the North direction.

2. Representing the First Movement The man walks a distance of 3 units from the origin towards the North-East (N 45° E) direction.

   • The direction N 45° E means 45° East of North. In the standard angle measurement (counter-clockwise from the positive real axis/East), this corresponds to an angle of $90^\circ - 45^\circ = 45^\circ$ or $\theta_1 = \frac{\pi}{4}$ radians.
   • The magnitude (distance) is $r_1 = 3$.
   • The first displacement can be represented by the complex number $z_1$ in polar form: $z_1 = r_1 e^{i\theta_1} = 3e^{i\pi/4}$.
   • After this movement, his position is $A = z_0 + z_1 = 3e^{i\pi/4}$.

3. Representing the Second Movement From his new position A, he walks a distance of 4 units towards the North-West (N 45° W) direction.

   • The direction N 45° W means 45° West of North. This corresponds to an angle of $90^\circ + 45^\circ = 135^\circ$ or $\theta_2 = \frac{3\pi}{4}$ radians from the positive real axis.
   • The magnitude (distance) is $r_2 = 4$.
   • This second displacement is represented by the complex number $z_2 = r_2 e^{i\theta_2} = 4e^{i3\pi/4}$.

4. Calculating the Final Position P The final position P is the sum of the initial position and all subsequent displacements. The final position vector is the sum of the displacement vectors. The complex number representing the position P is $z_P = z_1 + z_2$. $z_P = 3e^{i\pi/4} + 4e^{i3\pi/4}$

5. Simplifying the Expression We need to simplify this expression to match one of the given options. Notice that the options have a common factor of $e^{i\pi/4}$. We can express $e^{i3\pi/4}$ in terms of $e^{i\pi/4}$. We can write the angle $\frac{3\pi}{4}$ as $\frac{\pi}{2} + \frac{\pi}{4}$. Using the property $e^{i(a+b)} = e^{ia}e^{ib}$, we have: $e^{i3\pi/4} = e^{i(\pi/2 + \pi/4)} = e^{i\pi/2} \cdot e^{i\pi/4}$ We know that $e^{i\pi/2} = \cos(\frac{\pi}{2}) + i\sin(\frac{\pi}{2}) = 0 + i(1) = i$. So, $e^{i3\pi/4} = i \cdot e^{i\pi/4}$.

6. Substituting and Finalizing the Answer Substitute this back into the expression for $z_P$: $z_P = 3e^{i\pi/4} + 4(i e^{i\pi/4})$ Now, we can factor out $e^{i\pi/4}$: $z_P = (3 + 4i)e^{i\pi/4}$ This matches option D.

Conclusion

The position of P in the Argand plane is $(3 + 4i)e^{i\pi/4}$.