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Question

Let z1,z2,z3 be three complex numbers such that |z1|=|z2|=|z3|=1 and z21z2z3+z22z1z3+z23z1z2=1. Then the possible value(s) of |z1+z2+z3| is/are

A
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B
1
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C
2
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D
3
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Solution

The correct option is C 2
z21z2z3+z22z1z3+z23z1z2=1
z31+z32+z33+z1z2z3=0

Now, by using the relation a3+b3+c33abc=(a+b+c)(a2+b2+c2abbcca) we can find (z1+z2+z3)

4z1z2z3=z31+z32+z333z1z2z3
4z1z2z3=(z1+z2+z3)(z21+z22+z23z1z2z2z3z3z1)

Let z=z1+z2+z3
z33z(z1z2+z2z3+z3z1)=4z1z2z3
z3=z1z2z3[3z(1z1+1z2+1z3)4]
z3=z1z2z3[3z(¯z1+¯z2+¯z3)4]
z3=z1z2z3(3|z|24)

Taking the absolute values of both sides, we get
|z|3=| 3|z|24 | (|z1|=|z2|=|z3|=1)
If |z|23, then
|z|33|z|2+4=0
|z|=2

If |z|<23, then
|z|2+3|z|24=0
|z|=1

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