JEE Advanced 2021 Paper 1Question 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:
$P: |z_2-z_1|+|z_3-z_2|+\cdots+|z_{10}-z_9|+|z_1-z_{10}|\le 2\pi$
$Q: |z_2^2-z_1^2|+|z_3^2-z_2^2|+\cdots+|z_{10}^2-z_9^2|+|z_1^2-z_{10}^2|\le 4\pi$
Then,
Complex Numbers
Solution
$|z_k-z_{k-1}|=|e^{i\theta_k}-1|=2\sin\frac{\theta_k}{2}\le\theta_k$ (chord $\le$ arc), so the sum is $\le\sum\theta_k=2\pi$ (the last term is $|z_1-1|=2\sin\frac{\theta_1}2$ since $z_{10}=1$), P true. Similarly $|z_k^2-z_{k-1}^2|=2|\sin\theta_k|\le2\theta_k$, sum $\le4\pi$, Q true.