Step-by-step Solution

1. Understanding the Complex Numbers $z_k$

First, let's express the complex numbers $z_k$ in a more explicit form. We are given:

• $z_1 = e^{i\theta_1}$
• $z_k = z_{k-1} e^{i\theta_k}$ for $k=2, 3, \ldots, 10$.

Let's find the general form for $z_k$:

• $z_2 = z_1 e^{i\theta_2} = (e^{i\theta_1}) e^{i\theta_2} = e^{i(\theta_1+\theta_2)}$
• $z_3 = z_2 e^{i\theta_3} = (e^{i(\theta_1+\theta_2)}) e^{i\theta_3} = e^{i(\theta_1+\theta_2+\theta_3)}$

By induction, we can see that $z_k = e^{i(\theta_1+\theta_2+\cdots+\theta_k)}$. Let's define the cumulative angle $\phi_k = \sum_{j=1}^k \theta_j$. Then, $z_k = e^{i\phi_k}$.

The modulus of each $z_k$ is $|z_k| = |e^{i\phi_k}| = 1$. This means all the complex numbers $z_k$ lie on the unit circle in the complex plane.

We are given that $\sum_{k=1}^{10} \theta_k = 2\pi$. This means $\phi_{10} = 2\pi$. So, $z_{10} = e^{i\phi_{10}} = e^{i(2\pi)} = \cos(2\pi) + i\sin(2\pi) = 1$.

2. Analyzing Statement P

Statement P is: $\left| {{z_2} - {z_1}} \right| + \left| {{z_3} - {z_2}} \right| + \dots + \left| {{z_{10}} - {z_9}} \right| + \left| {{z_1} - {z_{10}}} \right| \le 2\pi$.

The expression represents the sum of the lengths of the chords connecting the points $z_1, z_2, \dots, z_{10}$ in a cycle. This is the perimeter of the 10-sided polygon with vertices $z_1, z_2, \dots, z_{10}$. These vertices all lie on the unit circle.

A fundamental geometric principle is that the length of a chord connecting two points on a circle is always less than or equal to the length of the arc connecting them.

Let's consider a generic chord length $|z_k - z_{k-1}|$. The points are $z_k = e^{i\phi_k}$ and $z_{k-1} = e^{i\phi_{k-1}}$. The angle subtended by the arc between them at the center is $\Delta\phi = \phi_k - \phi_{k-1} = \theta_k$. Since the circle has radius $r=1$, the arc length is $r \cdot \Delta\phi = \theta_k$. So, we have the inequality: $|z_k - z_{k-1}| \le \text{arc length between } z_{k-1} \text{ and } z_k = \theta_k$ for $k=2, 3, \ldots, 10$. Equality holds only if $\theta_k=0$, but we are given that $\theta_k$ are positive.

For the last term, $|z_1 - z_{10}|$, the points are $z_1 = e^{i\theta_1}$ and $z_{10} = 1 = e^{i0}$. The angle between their position vectors is $\theta_1$. The arc length is $\theta_1$. So, $|z_1 - z_{10}| \le \theta_1$

Summing all these inequalities, we get: $\left| {{z_2} - {z_1}} \right| + \dots + \left| {{z_{10}} - {z_9}} \right| + \left| {{z_1} - {z_{10}}} \right| \le \theta_2 + \dots + \theta_{10} + \theta_1$

The sum on the right side is $\sum_{k=1}^{10} \theta_k$, which is given to be $2\pi$. Therefore, we have: $\left| {{z_2} - {z_1}} \right| + \dots + \left| {{z_1} - {z_{10}}} \right| \le 2\pi$ Thus, statement P is TRUE.

3. Analyzing Statement Q

Statement Q is: $\left| {z_2^2 - z_1^2} \right| + \left| {z_3^2 - z_2^2} \right| + \dots + \left| {z_{10}^2 - z_9^2} \right| + \left| {z_1^2 - z_{10}^2} \right| \le 4\pi$.

Let's define a new set of complex numbers $w_k = z_k^2$. Since $|z_k|=1$, we have $|w_k| = |z_k^2| = |z_k|^2 = 1^2 = 1$. So, the points $w_k$ also lie on the unit circle.

We have $z_k = e^{i\phi_k}$, so $w_k = (e^{i\phi_k})^2 = e^{i(2\phi_k)}$. The expression in Q is the perimeter of the polygonal chain with vertices $w_1, w_2, \dots, w_{10}$.

Let's apply the same arc length argument. The chord length is $|w_k - w_{k-1}|$. The points are $w_k = e^{i(2\phi_k)}$ and $w_{k-1} = e^{i(2\phi_{k-1})}$. The angle subtended by the arc between them is $2\phi_k - 2\phi_{k-1} = 2(\phi_k - \phi_{k-1}) = 2\theta_k$. The arc length is $2\theta_k$. So, we have the inequality: $|w_k - w_{k-1}| \le \text{arc length between } w_{k-1} \text{ and } w_k = 2\theta_k$ for $k=2, 3, \ldots, 10$.

For the last term, $|w_1^2 - w_{10}^2|$, the points are $w_1 = z_1^2 = e^{i(2\theta_1)}$ and $w_{10} = z_{10}^2 = 1^2 = 1 = e^{i0}$. The angle between their position vectors is $2\theta_1$. The arc length is $2\theta_1$. So, $|w_1 - w_{10}| \le 2\theta_1$

Summing these inequalities: $\left| {z_2^2 - z_1^2} \right| + \dots + \left| {z_{10}^2 - z_9^2} \right| + \left| {z_1^2 - z_{10}^2} \right| \le 2\theta_2 + \dots + 2\theta_{10} + 2\theta_1$

The sum on the right side is $2(\theta_1 + \theta_2 + \dots + \theta_{10}) = 2 \sum_{k=1}^{10} \theta_k = 2(2\pi) = 4\pi$. Therefore, we have: $\left| {z_2^2 - z_1^2} \right| + \dots + \left| {z_1^2 - z_{10}^2} \right| \le 4\pi$ Thus, statement Q is TRUE.

4. Conclusion

Both statements P and Q are TRUE. This corresponds to option C.