QJEEAdv21Ma03d8: JEE Advanced 2021 Mathematics - Class 11 JEEAdv21Ma03d8 Solution: Let be positive valued angles (in radian) such that . Define…

Both P and Q are true.

$\because$ Length of direct ditance $\le$ length of arc

i.e. | z₂ $-$ z₁ | = length of line AB $\le$ length of arc AB.


| z₃ $-$ z₂ | = length of line BC $\le$ length of arc BC.

$\therefore$ Sum of length of these 10 lines $\le$ sum of length of arcs (i.e. 2 $\pi$ ) (because $\theta$ ₁ + $\theta$ ₂ + $\theta$ ₃ + .... + $\theta$ ₁₀ = 2 $\pi$ (given)

$\therefore$ | z₂ $-$ z₁ | + | z₃ $-$ z₂ | + ..... + | z₁ $-$ z₁₀ | $\le$ 2 $\pi$ $\to$ P is true.

And | z $_k^2$ $-$ z $_{k - 1}^2$ | = | z_k $-$ z_{k $-$ 1} | | z_k + z_{k $-$ 1} |

As we know that,

| z_k + z_{k $-$ 1} | $\le$ | z_k | + | z_{k $-$ 1} | $\le$ 2

$\therefore$ | z $_2^2$ $-$ z $_1^2$ | + | z $_3^2$ $-$ z $_2^2$ | + .... + | z $_1^2$ $-$ z $_{10}^2$ | $\le$ 2 ( | z₂ $-$ z₁ | + | z₃ $-$ z₂ | + .... + | z₁ $-$ z₁₀ | )

$\le$ 2(2 $\pi$ )

$\le$ 4 $\pi$ $\to$ Q is true.