Let $\theta_1,\theta_1,……..,\theta_{10}$ be positive valued angles (in radian) such that $\theta_1+\theta_2+……..+\theta_{10}=2\pi$. Define the complex numbers $z_1=e^{i\theta_1},z_k=z_{k-1}e^{i\theta_k}$ for k=2, 3, …….., 10 where $i=\sqrt {-1}$. Consider the statements P and Q given below:

P: $|z_2-z_1|+|z_3-z_2|+………..+|z_{10}-z_9|+|z_1-z_{10}|\leq 2\pi $

Q: $|z_2^2-z_1^2|+|z_3^2-z_2^2|+………..+|z_{10}^2-z_9^2|+|z_1^2-z_{10}^2|\leq 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

Solution

We have, $z_2=z_1e^{i\theta_2}=e^{i(\theta_1+\theta_2)}$

$z_3=z_2e^{i\theta_3}=e^{i(\theta_1+\theta_2+\theta_3)}$

& so on till $z_{10}=e^{i(\theta_1+\theta_2+…….+\theta_{10})}=e^{i.2\pi}$

In Argand plane, all these $z_k$s are posited on unit circle $z_{10}$ being on the real axis having angle $2\pi$ as shown below.

Regarding Statement P:

$|z_2-z_1|$ is the chord joining points $z_1$ and $z_2$.

So, P involves summation of all the 10 chords. Summation of 10 chords is the perimeter of decagon inscribed in unit circle having circumference $2\pi$.

Perimeter of decagon cannot exceed circumference of circle.

Hence, P is TRUE.

Regarding Statement Q:

$z_1^2=e^{i.2\theta_1}=Z_1$ (say)

Likewise, $z_2^2=e^{i.2(\theta_1+\theta_2)}=Z_2$

$z_3^2=e^{i.2(\theta_1+\theta_2+\theta_3)}=Z_3$

& so on till $z_{10}^2=e^{i.4\pi}=Z_{10}$

$|z_2^2-z_1^2|+|z_3^2-z_2^2|+………..+|z_{10}^2-z_9^2|+|z_1^2-z_{10}^2|$ = $|Z_2-Z_1|+|Z_3-Z_2|+………..+|Z_{10}-Z_9|+|Z_1-Z_{10}|$

Here, summation of 10 chords is spanned over $4\pi$ angle or two circles. Thus, Q is also TRUE.

Answer: Option (C).