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Question

If z is a complex number lying in the fourth quadrant of Argand plane and [kzk+1]+2i>2 for all real values of k (k1), then range of arg(z) is
  1. None of these
  2. (π4,0)
  3. (π8,0)
  4. (π6,0)


Solution

The correct option is B (π4,0)
 z1=kz(k+1) represents any point lying on the line joining origin and z.
Given,
[kzk+1]+2i>2
Hence, kzk+1 will lie outside the circle |z1+2i|=2.


So, z should lie in the shaded region.
Now OPA is a rightangled triangle
OP2=OA2AP2=42=2OP=APPOA=π4
π4<arg(z)<0

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