For a non-zero complex number z ,let arg (z) denote the principal argument with -\pi < \text{arg(z)} \leq \pi Then, which of the following statement(s) is (are) FALSE? [JEE Adv 2018 P1]

arg(-1-i)=\pi/4, where i=\sqrt{-1}

the function f:\mathbb{R}\to(-\pi,\pi], defined by f(t)=arg(-1+it) for all t∊\mathbb{R}, is continuous at allpoints of ℝ,where i=\sqrt{-1}

For any two non-zero complex numbers z_1 and z_2,\text{arg} (z_1/z_2)-\text{arg}(Z_1)+\text{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 \text{arg}(\frac{(z-z_1)(z_2-z_3)}{(z-z_3)(z_2-z_1)})=\pi, lies on a straight line

Complex Numbers

Section: Mathematics

Question : 17 of 36

Marks: +1, -0

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?

[JEE Adv 2018 P1]

arg(−1−i)=
π
4
, where i=√−1
the function f:ℝ→(−π,π], defined by f(t)=arg(−1+it) for all t∊ℝ, is continuous at allpoints of ℝ,where i=√−1
For any two non-zero complex numbers z1 and
z2,arg(
z1
z2
)−arg(Z1)+arg(Z2)
is an integer multiple of 2π
For any three given distinct complex numbers z1,z2 and z3, the locus of the point z satisfying the condition arg(
(z−z1)(z2−z3)
(z−z3)(z2−z1)
)=π, lies on a straight line

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π−tan−1(t),	t≥0
−π+tan−1	t<0