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

The Correct Option is C

Step 1: Understanding the Problem
We are given a sequence of complex numbers \( z_1, z_2, \dots, z_{10} \) where each \( z_k \) is defined as: \[ z_1 = e^{i\theta_1}, \quad z_k = z_{k-1} e^{i\theta_k} \text{ for } k = 2, 3, \dots, 10 \] The angles \( \theta_1, \theta_2, \dots, \theta_{10} \) are positive and satisfy: \[ \theta_1 + \theta_2 + \dots + \theta_{10} = 2\pi \] We are tasked with evaluating two statements \( P \) and \( Q \) based on these complex numbers, specifically the sums of distances between consecutive points on the unit circle.

Step 2: Interpretation of \( z_1, z_2, \dots, z_{10} \)
Each \( z_k \) represents a point on the unit circle in the complex plane. The angles \( \theta_1, \theta_2, \dots, \theta_{10} \) define the angular separations between these consecutive points. Since \( \theta_1 + \theta_2 + \dots + \theta_{10} = 2\pi \), the total angle around the unit circle is \( 2\pi \), so the points \( z_1, z_2, \dots, z_{10} \) represent a partition of the unit circle. Each complex number \( z_k \) is related to its predecessor by an angular rotation determined by \( \theta_k \), and the points on the unit circle are connected in a polygonal manner.

Step 3: Analyzing Statement \( P \)
The statement \( P \) says: \[ P: \left|z_2 - z_1\right| + \left|z_3 - z_2\right| + \dots + \left|z_{10} - z_9\right| + \left|z_1 - z_{10}\right| \leq 2\pi \] This represents the total length of the polygon formed by connecting the points \( z_1, z_2, \dots, z_{10} \). The length of each side of the polygon corresponds to the Euclidean distance between consecutive points on the unit circle, which is determined by the angle between them. For two points \( z_k \) and \( z_{k+1} \), the distance is given by the chord length: \[ |z_{k+1} - z_k| = 2 \sin\left(\frac{\theta_k}{2}\right) \] The sum of the distances \( \sum_{k=1}^{9} |z_{k+1} - z_k| + |z_1 - z_{10}| \) is the perimeter of the polygon. Since the total angle \( \theta_1 + \theta_2 + \dots + \theta_{10} = 2\pi \), the total perimeter is bounded by \( 2\pi \), so the statement \( P \) is true.

Step 4: Analyzing Statement \( Q \)
The statement \( Q \) says: \[ Q: \left|z_{22} - z_{12}\right| + \left|z_{32} - z_{22}\right| + \dots + \left|z_{102} - z_{92}\right| + \left|z_{12} - z_{102}\right| \leq 4\pi \] This is similar to statement \( P \), but it involves the points \( z_{12}, z_{22}, \dots, z_{102} \), which correspond to multiples of the points \( z_1, z_2, \dots, z_{10} \) arranged in a similar fashion but with additional cycles around the unit circle. These points correspond to repeating patterns of the unit circle, and the sum of distances between these points also forms a polygon whose perimeter is proportional to the total angle \( 4\pi \). Thus, the sum of distances is bounded by \( 4\pi \), making statement \( Q \) true.

Step 5: Conclusion
Both statements \( P \) and \( Q \) are true, as the sums of distances are bounded by \( 2\pi \) and \( 4\pi \), respectively, due to the geometric properties of points on the unit circle and the conditions given in the problem. Thus, the correct answer is:
Both P and Q are TRUE