Question

Let ${\theta }_1,{\theta }_2,\cdots ,{\theta }_{10}$ 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 $k=2,3\cdots 10,$ where $i=\sqrt{-1}$. Consider the statements $P$ and $Q$ given below:
$P:|z_2-z_1|+|z_3-z_2|+\cdots +|z_{10}-z_9|+|z_1-z_{10}|\le 2\pi$
$Q:|z_2^2-z_1^2|+|z_3^2-z_2^2|+\cdots +|z_{10}^2-z_9^2|+|z_1^2-z_{10}^2|\le 4\pi$

• 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

Answer 1

The correct option is C Both $P$ and $Q$ are TRUE


$|z_2-z_1|=$ length of line $AB\le$ length of arc $AB$
$\Rightarrow |z_2-z_1|\le {\theta }_2$
Similarly, $|z_3-z_2|=$ length of line $BC\le$ length of arc $BC$
$\Rightarrow |z_3-z_2|\le {\theta }_3$
$.$
$.$
$.$
$\Rightarrow |z_1-z_{10}|\le {\theta }_1$
Now,

$\Rightarrow |z_2-z_1|+|z_3-z_2|+\cdots +|z_{10}-z_9|+|z_1-z_{10}|\le {\theta }_1+{\theta }_2+....+{\theta }_{10}$

$\Rightarrow |z_2-z_1|+|z_3-z_2|+\cdots +|z_{10}-z_9|+|z_1-z_{10}|\le 2\pi$
$(\because {\theta }_1+{\theta }_2+....+{\theta }_{10}=2\pi )$
And
$|z_k^2-z_{k-1}^2|=|z_k-z_{k-1}||z_k+z_{k-1}|$

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

$\Rightarrow |z_2^2-z_1^2|+|z_3^2-z_2^2|+\cdots +|z_{10}^2-z_9^2|+|z_1^2-z_{10}^2|\le 2(|z_2-z_1|+|z_3-z_2|+\cdots +|z_{10}-z_9|+|z_1-z_{10}|)$

$\therefore |z_2^2-z_1^2|+|z_3^2-z_2^2|+\cdots +|z_{10}^2-z_9^2|+|z_1^2-z_{10}^2|\le 4\pi$