NDA 2019 Maths · Mathematics · Complex Numbers and Quadratic Equations · hard

For every non-zero complex number z, arg z+argbar z=0 (principal arguments). 2. For n>1, the sum of all the nth roots of unity is zero.

Single correct option

Consider the following statements:
1. For every non-zero complex number

z

\arg z+\arg\bar z=0

(principal arguments).
2. For

n>1

, the sum of all the

n

th roots of unity is zero.
Which of the statements given above is/are correct?

Show answer and solution

Answer: (B) 2 only

Statement 1 fails for negative real

z

: with

z=-1

the principal argument is

\pi

for both

z

and

\bar z=-1

, so the sum is

2\pi

, not

0

. Statement 2 is true: the roots are

1,\zeta,\ldots,\zeta^{n-1}

in a geometric series with ratio

\zeta\neq1

, whose sum is

\dfrac{\zeta^n-1}{\zeta-1}=0

. Hence only statement 2 is correct.

Original question written in the NDA 2019 Maths pattern and checked twice for the correct answer. Not an official question paper. How we write and check questions · Corrections policy · Report a mistake