JEE Advanced 2018 · Paper 1 · Mathematics Q1

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MathematicsMultiple correct+4 / -2Question 37 of 54

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

\arg(-1-i) = \dfrac{\pi}{4}

, where

i = \sqrt{-1}

The function

f: \mathbb{R} \to (-\pi, \pi]

, defined by

f(t) = \arg(-1 + it)

for all

t \in \mathbb{R}

, is continuous at all points of

\mathbb{R}

, where

i = \sqrt{-1}

For any two non-zero complex numbers

z_1

and

z_2

\arg\left(\dfrac{z_1}{z_2}\right) - \arg(z_1) + \arg(z_2)

is an integer multiple of

2\pi

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(\dfrac{(z-z_1)(z_2-z_3)}{(z-z_3)(z_2-z_1)}\right) = \pi

, lies on a straight line

Answer and worked solution

Official answer: A, B, D

We want the false statements.

(A) $-1-i$ is in the third quadrant, so $\arg(-1-i)=-\dfrac{3\pi}4$ , not $\dfrac\pi4$ . False.
(B) As $t\to0^+$ , $-1+it$ approaches $-1$ from above and $\arg\to\pi$ . As $t\to0^-$ it approaches from below and $\arg\to-\pi$ . So $f$ jumps at $t=0$ and is not continuous everywhere. False.
(C) $\arg(z_1/z_2)=\arg z_1-\arg z_2+2k\pi$ for some integer $k$ , so the expression is $2k\pi$ . True.
(D) The argument being $\pi$ makes the cross-ratio $\dfrac{(z-z_1)(z_2-z_3)}{(z-z_3)(z_2-z_1)}$ a negative real number. Such $z$ lie on the circle through $z_1,z_2,z_3$ , which is not a straight line in general. False.

(A), (B), (D)

Topics:Modulus and argument Geometry in the complex plane

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JEE Advanced 2018 · Paper 1, Mathematics question 1