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Complex Numbers question

2021 · Shift 1 · Q23

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PYQ Library
JEE Advanced
Mathematics
Complex Numbers
2021 · Shift 1 · Q23

Complex Numbers question

2021 · Shift 1 · Q23

JEE AdvancedMathematicsComplex NumbersMCQ+3 / −1

Let

\theta_1, \theta_2, \ldots, \theta_{10}

be positive valued angles (in radian) such that

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

. Define the complex numbers

z_1=e^{i \theta_1}, z_k=z_{k-1} e^{i \theta_k}

for

k=2,3, \ldots, 10

, where

i=\sqrt{-1}

. Consider the statements

P

and

Q

given below:

P:\left| {{z_2} - {z_1}} \right| + \left| {{z_3} - {z_2}} \right| + ..... + \left| {{z_{10}} - {z_9}} \right| + \left| {{z_1} - {z_{10}}} \right| \le 2\pi Q:\left| {z_2^2 - z_1^2} \right| + \left| {z_3^2 - z_2^2} \right| + .... + \left| {z_{10}^2 - z_9^2} \right| + \left| {z_1^2 - z_{10}^2} \right| \le 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

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