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For a non-zero complex number z, let \( \arg (\mathrm{z}) \) denote the principal argument with \( -\pi\arg (\mathrm{z}) \leq \pi \). Then, which of the following statement(s) is(are) FALSE?
\( \mathrm{P} \)
(A) \( \arg (-1-\mathrm{i})=\frac{\pi}{4} \), where \( \mathrm{i}=\sqrt{-1} \).
W
(B) The function \( \mathrm{f}: \mathrm{R} \rightarrow(-\pi, \pi] \), defined by \( \mathrm{f}(\mathrm{t})=\arg (-1+\mathrm{it}) \) for all \( \mathrm{t} \in \mathrm{R} \), is continuous at all points of \( R \), where \( i=\sqrt{-1} \).
(C) For any two non-zero complex numbers \( z_{1} \) and \( z_{2}, \arg \left(\frac{z_{1}}{z_{2}}\right)-\arg \left(z_{1}\right)+\arg \left(z_{2}\right) \) is an integer multiple of \( 2 \pi \).
(D) For any three given distinct complex numbers \( z_{1}, z_{2} \) and \( z_{3} \), the locus of the point \( z \) satisfying the condition \( \arg \left(\frac{\left(\mathrm{z}-\mathrm{z}_{1}\right)\left(\mathrm{z}_{2}-\mathrm{z}_{3}\right)}{\left(\mathrm{z}-\mathrm{z}_{3}\right)\left(\mathrm{z}_{2}-\mathrm{z}_{1}\right)}\right)=\pi \) lies on a straight line.
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