Question

Let $\theta_1, \theta_2, …., \theta_{10}$ be positive valued angles (in radian) such that $\theta_1+ \theta_2+ ….+ \theta_{10} = 2\pi$. Define the complex numbers $z_1 = 𝑒^{ 𝑖\theta_1}, 𝑧_𝑘 = 𝑧_{𝑘−1}𝑒^ {𝑖\theta_k} \text{for}\ k = 2, 3, …, 10,$ where $i = \sqrt{-1}$. Consider the statement P and Q given below:
$P: \left|z_2 - z_1\right| + \left|z_3 - z_2\right| + \ldots + \left|z_{10} - z_9\right| + \left|z_1 - z_{10}\right| \leq 2\pi$
$Q: \left|z_{22} - z_{12}\right| + \left|z_{32} - z_{22}\right| + \ldots + \left|z_{102} - z_{92}\right| + \left|z_{12} - z_{102}\right| \leq 4\pi$
Then,

• A. P is TRUE and Q is FALSE
• B. Q is TRUE and P is FALSE
• C. Both P and Q are TRUE
• D. Both P and Q are FALSE

Answer 1

Answer: (C)

Both P and Q are TRUE

Answer 2

The correct answer is Both P and Q are TRUE that is option (C)