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Question

Let z1 and z2 be two complex numbers represented by points on the circle |z1|=1 and |z2|=2 respectively. Let P be the point whose coordinates are (1,1). Then

A
the maximum value of |2z1+z2| is 4.
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B
the minimum value of |z1z2| is 1.
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C
z2+1z13
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D
the minimum distance of P from z1 is 22 unit.
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Solution

The correct option is C z2+1z13
|2z1+z2|2|z1|+|z2|=2×1+2=4
Maximum value of |2z1+z2| is 4.

|z1z2| is least when 0,z1,z2 are collinear.
|z1z2||z1||z2|=|12|=1
Then min|z1z2|=1

Again, z2+1z1|z2|+1z1=|z2|+1|z1|=2+11=3z2+1z13

P lies outside |z1|=1
Hence, required minimum distance =OPr, where O(0,0)
=21

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