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Q.

For a non-zero complex number 𝑧, let arg (𝑧) denote the principal argument with .−π

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a

arg(−1−i)= π4

b

The functionf:ℝ→−π,π , defined by ft=arg−1+it for all t∈ℝ is continuous at all points of ℝ

c

For any two non-zero complex numbers  z1and z2 , argz1z2−argZ1+argZ2 is an integer multiple of 2π

d

For any three given distinct complex numbers,  z1,z2 and z3, the locus of the point 𝑧 satisfying the condition argz−z1z2−z3z−z3z2−z1=π, lies on a straight line

answer is A.

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Detailed Solution

a) arg(-1-i)=-3π4 b) f(t)=arg(-1+it)=π-tan-1(t),t≥0-π+tan-1(t)t<0 Discontinuous at t=0 c) argz1z2-argz1+argz2=argz1-argz2+2nπ-argz1+argz2=2nπ d) argz-z1z2-z3z-z3z2-z1=π⇒z-z1z2-z3z-z3z2-z1 is real. ⇒z,z1,z2,z3 are concyclic
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