For a non-zero complex number z , let arg ( z ) denote the principal argument with - π < arg(z) ≤ π then, which of the following statement(s) is (are) FALSE?

Correct answer: A. Arg (- 1 - i) = (π)/(4), wherei = - 1 (A) Arg(- 1 - i) = - (3π)/(4) (B) Arg(- 1 + it) = \ cases π - ^ - 1 (t), & t ≥ 0 - π + ^ - 1 (t), & t Discontinuous at t = 0 ( C ) arg( z _ 1 z _ 2 ) - arg( z _ 1 ) + arg( z _ 2 ) = arg z _ 1 - arg( z _ 2 ) - arg( z _ 1 ) + arg( z _ 2 ) =0 (D) arg( (z - z _ 1 ) ( z _ 2 - z _ 3 ) (z - z _ 3 )( z _ 2 - z _ 1 ) ) = π ⇒ (z - z _ 1 )( z _ 2 - z _ 3 ) (z - z _ 3 )( z _ 2 - z _ 1 ) is real ⇒ z, z _ 1 , z _ 2 , z _ 3 are concyclic

JEE Advanced 2018Paper 1

Exam: JEE Advanced
Year: 2018
Paper: Paper 1
Subject: Mathematics
Chapter: Complex Number

JEE Advanced 2018 · Paper 1 · Mathematics · Complex Number

For a non-zero complex number z , let arg ( z ) denote the...

For a non-zero complex number

z

, let arg

( z )

denote the principal argument with

- \pi < arg\left(z\right) \le \pi

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

Options

AArg $\left(- 1 - i\right) = \dfrac{\pi}{4}, wherei = \sqrt{- 1}$
BThe function $f: \mathbb{R} \to$ \left(- \pi, \pi\right] $defined by$ f\left(t\right) = arg\left(- 1 + it\right)forallt \in \mathbb{R} $, is continuous at all points of$ \mathbb{R}, wherei = \sqrt{- 1}
CFor any two non-zero complex numbers ${z}_{1}and{z}_{2}$ , $arg$ \left(\dfrac{{z}_{1}}{{z}_{2}}\right) - arg\left({z}_{1}\right) + arg\left({z}_{2}\right) $is an integer multiple of$ 2\pi
DFor any three given distinct complex numbers ${z}_{1}, {z}_{2}and{z}_{3}$ , the locus of the point $z$ satisfying the condition $\text{a}\text{r}\text{g}\left(\dfrac{\left(z - {z}_{1}\right)\left({z}_{2} - {z}_{3}\right)}{\left(z - {z}_{3}\right)\left({z}_{2} - {z}_{1}\right)}\right) = \pi$ lies on a straight line

Correct answer

A. Arg $\left(- 1 - i\right) = \dfrac{\pi}{4}, wherei = \sqrt{- 1}$

Step-by-step solution

\left(A\right) Arg\left(- 1 - i\right) = - \dfrac{3\pi}{4}

<br />\left(B\right)

Arg\left(- 1 + it\right) = \left\{

\begin{cases}\pi - \tan^{- 1} \left(t\right), & t \ge 0 \\ - \pi + \tan^{- 1} \left(t\right), & t < 0\end{cases}

\right)
Discontinuous at

t = 0

( C ) arg\left(\dfrac{{z}_{1}}{{z}_{2}}\right) - arg\left({z}_{1}\right) + arg\left({z}_{2}\right)

= arg{z}_{1} - arg\left({z}_{2}\right) - arg\left({z}_{1}\right) + arg\left({z}_{2}\right) =0

\left(D\right) arg\left(\dfrac{\left(z - {z}_{1}\right)

\left({z}_{2} - {z}_{3}\right)}{\left(z - {z}_{3}\right)\left({z}_{2} - {z}_{1}\right)}\right) = \pi

⇒ \dfrac{\left(z - {z}_{1}\right)\left({z}_{2} - {z}_{3}\right)}{\left(z - {z}_{3}\right)\left({z}_{2} - {z}_{1}\right)}

is real

⇒ z, {z}_{1}, {z}_{2}, {z}_{3}

are concyclic

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