Let be positive valued angles (in radian) such that

MATHEMATICS · Single correct · IB Diploma Programme · Mathematics: Analysis and Approaches · Number and Algebra · Number and Algebra

Problem

Let

\theta_1, \theta_2, \theta_3, \dots, \theta_{10}

be positive valued angles (in radian) such that

\theta_1 + \theta_2 + \dots + \theta_{10} = 2\pi

. Define the complex numbers

z_k = e^{i\theta_k}

z_k = \cos \theta_k + i \sin \theta_k

, for

k = 1, 2, 3, \dots, 10

, where

i = \sqrt{-1}

. Consider the statement P and Q given below: P:

|z_2 - z_1| + |z_3 - z_2| + \dots + |z_{10} - z_9| \leq 2\pi

. Q:

|z_2^2 - z_1^2| + |z_3^2 - z_2^2| + \dots + |z_{10}^2 - z_9^2| \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.