For a non-zero complex number $z$, let $\arg (z)$ denote the principal argument with $-\pi<\arg (z) \leq \pi$. Then, which of the following statement(s) is (are) FALSE?

(A) $\arg (-1-i)=\frac{\pi}{4}$, where $i=\sqrt{-1}$; (B) The function $f: \mathbb{R} \rightarrow(-\pi, \pi]$, defined by $f(t)=\arg (-1+i t)$ for all $t \in \mathbb{R}$, is continuous at all points of $\mathbb{R}$, where $i=\sqrt{-1}$; (D) For any three given distinct complex numbers $z_{1}, z_{2}$ and $z_{3}$, the locus of the point $z$ satisfying the condition $\arg \left(\frac{\left(z-z_{1}\right)\left(z_{2}-z_{3}\right)}{\left(z-z_{3}\right)\left(z_{2}-z_{1}\right)}\right)=\pi$, lies on a straight line

(C) For any two non-zero complex numbers $z_{1}$ and $z_{2}$, $\arg \left(\frac{z_{1}}{z_{2}}\right)-\arg \left(z_{1}\right)+\arg \left(z_{2}\right)$ is an integer multiple of $2 \pi$

Complex Numbers — JEE Advanced · 2018 · 20 May 2018 · Paper 1 Question with Solution

For a non-zero complex number

z

, let

ar g (z)

denote the principal argument with

- π < ar g (z) \leq π

. Then, which of the following statement(s) is (are) FALSE?

(A) $ar g (- 1 - i) = \frac{π}{4}$ , where $i = - 1$ ✓ Correct
(B)The function $f : R \to (- π, π]$ , defined by $f (t) = ar g (- 1 + i t)$ for all $t \in R$ , is continuous at all points of $R$ , where $i = - 1$ ✓ Correct
(C)For any two non-zero complex numbers $z_{1}$ and $z_{2}$ , $ar g (\frac{z _{1}}{z _{2}}) - ar g (z_{1}) + ar g (z_{2})$ is an integer multiple of $2 π$
(D)For any three given distinct complex numbers $z_{1}, z_{2}$ and $z_{3}$ , the locus of the point $z$ satisfying the condition $ar g (\frac{( z - z _{1} ) ( z _{2} - z _{3} )}{( z - z _{3} ) ( z _{2} - z _{1} )}) = π$ , lies on a straight line✓ Correct

Answer: (A), (B), (D)

Solution

(A)

ar g (- 1 - i) = \frac{- 3 π}{4}

, not

\frac{π}{4}

. FALSE.

(B)

f (t) = ar g (- 1 + i t)

is discontinuous at

t = 0

. For

t > 0

f (t) = π - tan^{- 1} t

; for

t < 0

f (t) = - π - tan^{- 1} t

. The left and right limits at

t = 0

differ by

2 π

. FALSE.

(C) This is TRUE by the property of arguments:

ar g (z_{1} / z_{2}) = ar g (z_{1}) - ar g (z_{2}) + 2 k π

for some integer

k

.

(D) The condition

ar g (\frac{( z - z _{1} ) ( z _{2} - z _{3} )}{( z - z _{3} ) ( z _{2} - z _{1} )}) = π

means

ar g (\frac{z - z _{1}}{z - z _{3}}) = π - θ

where

θ = ar g (\frac{z _{2} - z _{3}}{z _{2} - z _{1}})

. This describes a circular arc (part of the circumcircle of

△ z_{1} z_{2} z_{3}

), not a straight line. FALSE.

JEE Advanced · 2018 · 20 May 2018 · Paper 1 · One or more correct options

JEE Advanced · 2018 · 20 May 2018 · Paper 1 · Complex Numbers

Solution

More JEE Advanced questions on Complex Numbers