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Complex Numbers question

2018 · Shift 1 · Q19

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PYQ Library
JEE Advanced
Mathematics
Complex Numbers
2018 · Shift 1 · Q19

Complex Numbers question

2018 · Shift 1 · Q19

JEE AdvancedMathematicsComplex NumbersMultiple correct+4 / −1

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?

A
arg( $-$ 1 $-$ i) = ${\pi \over 4}$ , where i = $\sqrt { - 1}$
B
The function f : R $\to$ ( $-\pi$ , $\pi$ ), defined by f(t) = arg ( $-$ 1 + it) for all t $\in$ R, is continuous at all points of R, where i = $\sqrt { - 1}$ .
C
For any two non-zero complex numbers z1 and z2, arg $\left( {{{{z_1}} \over {{z_2}}}} \right)$ $-$ arg (z1) + arg(z2) is an integer multiple of 2 $\pi$ .
D
For any three given distinct complex numbers z1, z2 and z3, the locus of the point z satisfying the condition arg $\left( {{{(z - {z_1})({z_2} - {z_3})} \over {(z - {z_3})({z_2} - {z_1})}}} \right) = \pi$ , lies on a straight line.

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