Evaluate Truth Value of Complex Number Inequalities

We analyze the statements $P$ and $Q$ using the geometric properties of complex numbers on the unit circle.

For statement $P$, each term $|z_k - z_{k-1}|$ represents the chord length on the unit circle corresponding to the angle $\theta_k$. Using the inequality $\sin x < x$ for $x > 0$, we have: $|z_k - z_{k-1}| = 2\sin\left(\frac{\theta_k}{2}\right) < \theta_k$ Summing these chord lengths over all 10 segments gives: $\sum_{k=2}^{10} |z_k - z_{k-1}| + |z_1 - z_{10}| < \sum_{k=1}^{10} \theta_k = 2\pi$ Thus, statement $P$ is TRUE.

For statement $Q$, using the identity $|z_k^2 - z_{k-1}^2| = 2\sin\theta_k$ along with $\sin\theta_k \le \theta_k$ for $\theta_k > 0$, the total sum yields: $\sum_{k=2}^{10} |z_k^2 - z_{k-1}^2| + |z_1^2 - z_{10}^2| = \sum_{k=1}^{10} 2\sin\theta_k \le 2\sum_{k=1}^{10} \theta_k = 4\pi$ Thus, statement $Q$ is TRUE.

Hence, both $P$ and $Q$ are TRUE, which corresponds to Option (C).