Let be positive valued angles (in radian) such that Define the complex numbers for where . Consider the statements P and Q given below: [2021]

Question

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

$P | z_{2} - z_{1} | + | z_{3} - z_{2} | + \dots + | z_{10} - z_{9} | + | z_{1} - z_{10} | \leq 2 π$

$Q | z_{2}^{2} - z_{1}^{2} | + | z_{3}^{2} - z_{2}^{2} | + \dots + | z_{10}^{2} - z_{9}^{2} | + | z_{1}^{2} - z_{10}^{2} | \leq 4 π$ [2021]

Smart Achievers · Accepted Answer

(3)

$Since, | z_{1} | = | z_{2} | = \dots = | z_{10} | = 1$

$θ_{2} = arc (z_{1}, z_{2})$

$| z_{2} - z_{1} | = length of line A B \leq length of arc A B$

$| z_{3} - z_{2} | = length of line B C \leq length of arc B C$

$∴$ $Sum of length of these 10 lines \leq Sum of length of arcs (i.e. 2 π)$

$[∵ θ_{1} + θ_{2} + θ_{3} + \dots + θ_{10} = 2 π]$

$∴$ $P : | z_{2} - z_{1} | + | z_{3} - z_{2} | + \dots + | z_{1} - z_{10} | \leq 2 π$

$P is true.$

$Now, | z_{2}^{2} - z_{1}^{2} | = | z_{2} - z_{1} | | z_{2} + z_{1} |$

$We know that$

$| z_{2} + z_{1} | \leq | z_{2} | + | z_{1} | \leq 2$

$∴$ $| z_{2}^{2} - z_{1}^{2} | + | z_{3}^{2} - z_{2}^{2} | + \dots + | z_{1}^{2} - z_{10}^{2} | \leq 2 {| z_{2} - z_{1} | + | z_{3} - z_{2} | + \dots + | z_{1} - z_{10} |}$ $\leq 2 (2 π) \Rightarrow Q \leq 4 π$

$Q is also true.$

Solution

1 P is TRUE and Q is FALSE

2 Q is TRUE and P is FALSE

3 both P and Q are TRUE

4 both P and Q are FALSE

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