Skip to main content
Physics Guru is an online education and competitive examinations preparation portal founded by Manish Verma, an IIT Madras alumnus, providing learning resources for competitive examinations.

The emphasis is on conceptual understanding, analytical thinking and problem solving rather than simply memorising formulas and standard methods.

The objective is to help students understand what they are learning and develop the ability to solve problems they have not seen before.

Let $\theta_1,\theta_1,........,\theta_{10}$ be ...

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).