Let $\theta_{1}, \theta_{2}, \ldots, \theta_{10}$ be positive valued angles (in radian) such that $\theta_{1} + \theta_{2} + \ldots + \theta_{10} = 2\pi$. Define the complex numbers $z_{1} = e^{i\theta_{1}}$, $z_{k} = z_{k-1}e^{i\theta_{k}}$ for $k = 2, 3, \ldots, 10$, where $i = \sqrt{-1}$. Consider the statements $P$ and $Q$ given below : $P \quad |z_{2} - z_{1}| + |z_{3} - z_{2}| + \ldots + |z_{10} - z_{9}| + |z_{1} - z_{10}| \leq 2\pi$ $Q \quad |z_{2}^{2} - z_{1}^{2}| + |z_{3}^{2} - z_{2}^{2}| + \ldots + |z_{10}^{2} - z_{9}^{2}| + |z_{1}^{2} - z_{10}^{2}| \leq 4\pi$

Let θ₁, θ₂, …, θ₁₀ be positive valued angles (in radian) such that θ₁ + θ₂ + … + θ₁₀ = 2π. Define the complex numbers z₁ = e^(iθ₁), zₖ = zₖ₋₁e^(iθₖ) for k = 2, 3, …, 10, where i = √-1. Consider the statements P and Q given below : P |z₂ - z₁| + |z₃ - z₂| + … + |z₁₀ - z₉| + |z₁ - z₁₀| ≤ 2π Q |z₂² - z₁²| + |z₃² - z₂²| + … + |z₁₀² - z₉²| + |z₁² - z₁₀²| ≤ 4π $\theta_1+\theta_2+\cdots+\theta_{10}=2\pi,\quad \theta_k>0$ $z_1=e^{i\theta_1},\quad z_k=z_{k-1}\,e^{i\theta_k}\ (k=2,\ldots,10)$ $P:\ \sum_{k=1}^{10}|z_{k+1}-z_k|\le 2\pi\quad (z_{11}:=z_1)$ $Q:\ \sum_{k=1}^{10}|z_{k+1}^{2}-z_k^{2}|\le 4\pi$ $\text{Which option (1)--(4) is correct?}$ $|z_1|=|e^{i\theta_1}|=1$ $|z_k|=|z_{k-1}|\cdot|e^{i\theta_k}|=|z_{k-1}|=1$ $\Rightarrow\ |z_1|=\cdots=|z_{10}|=1$ $\arg z_k=\theta_1+\cdots+\theta_k,\qquad \arg z_{10}=2\pi$ $z_{10}=e^{i2\pi}=1$ $P:\ \sum_{k=1}^{10}|z_{k+1}-z_k|\le 2\pi$ $\text{perimeter of the inscribed polygon} \le \text{circumference } 2\pi$ $|z_{k+1}-z_k|=2\sin\dfrac{\theta_{k+1}}{2}$ $2\sin\dfrac{\theta}{2}\le \theta\quad (\theta\ge 0)$ $\sum_{k=1}^{10}|z_{k+1}-z_k| = 2\sum_{k=1}^{10}\sin\dfrac{\theta_k}{2} \le \sum_{k=1}^{10}\theta_k = 2\pi$ $\therefore\ P\ \text{is TRUE}$ $Q:\ \sum_{k=1}^{10}|z_{k+1}^2-z_k^2|\le 4\pi$ $|z_k^2|=|z_k|^2=1,\qquad \arg(z_k^2)=2\arg z_k$ $z_{k+1}^2-z_k^2=z_k^2\left(e^{2i\theta_{k+1}}-1\right)$ $\left|e^{2i\theta}-1\right|=2|\sin\theta|$ $\Rightarrow\ |z_{k+1}^2-z_k^2|=2|\sin\theta_{k+1}|$ $\sum_{k=1}^{10}|z_{k+1}^2-z_k^2| = 2\sum_{k=1}^{10}|\sin\theta_k| \le 2\sum_{k=1}^{10}\theta_k = 4\pi$ $\therefore\ Q\ \text{is TRUE}$ $\theta_k=\dfrac{\pi}{5}:\quad \sum|z_{k+1}-z_k|=10\cdot 2\sin\dfrac{\pi}{10}\approx 6.18<2\pi\approx 6.28$ $\sum|z_{k+1}^2-z_k^2|=10\cdot 2\sin\dfrac{\pi}{5}\approx 11.76<4\pi\approx 12.57$ $P\ \text{TRUE},\qquad Q\ \text{TRUE}$ $\boxed{\text{Option (3): both } P \text{ and } Q \text{ are TRUE}}$