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Question

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?

A
arg(1i)=π4, where i=1
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B
The function f:R(π,π], defined by f(t)=arg(1+it) for all tR, is continuous at all points of R, where i=1
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C
For any two non-zero complex numbers z1 and z2, arg(z1z2)arg(z1)+arg(z2) is an integer multiple of 2π.
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D
For any three given distinct complex numbers z1,z2 and z3, the locus of the point z satisfying the condition arg((zz1)(z2z3)(zz3)(z2z1))=π,
lies on a straight line.
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Solution

The correct options are
A arg(1i)=π4, where i=1
B The function f:R(π,π], defined by f(t)=arg(1+it) for all tR, is continuous at all points of R, where i=1
D For any three given distinct complex numbers z1,z2 and z3, the locus of the point z satisfying the condition arg((zz1)(z2z3)(zz3)(z2z1))=π,
lies on a straight line.
arg(1i)=π4π=3π4
f(t)=arg(1+it)={πtan1t, t0tan1tπ, t<0
Clearly, f is discontinuous at t=0.

arg(z1z2)arg(z1)+arg(z2)
=arg(z1)arg(z2)+2nπarg(z1)+arg(z2)
=2nπ

arg((zz1)(z2z3)(zz3)(z2z1))=π
arg(zz1z2z1)+arg(z2z3zz3)=π
z, z1, z2, z3 are concyclic.
Hence, locus of z is a circle.

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