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,
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,
\(P: \left|z_2 - z_1\right| + \left|z_3 - z_2\right| + \ldots + \left|z_{10} - z_9\right| + \left|z_1 - z_{10}\right| \leq 2\pi\)
\(Q: \left|z_{22} - z_{12}\right| + \left|z_{32} - z_{22}\right| + \ldots + \left|z_{102} - z_{92}\right| + \left|z_{12} - z_{102}\right| \leq 4\pi\)
Then,
Updated On: Aug 19, 2024
P is TRUE and Q is FALSE
Q is TRUE and P is FALSE
Both P and Q are TRUE
Both P and Q are FALSE
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The Correct Option is C
Solution and Explanation
The correct answer is Both P and Q are TRUE that is option (C)
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