Given that z1,z2 and z3 are complex numbers with |z1|=|z2|=|z3|=1,z1+z2+z3=1 and z1z2z3=1 , then |(z1 + 2) (z2 + 2) (z3 + 2)| is equal to

Question

Given that z1,z2 and z3 are complex numbers with |z1|=|z2|=|z3|=1,z1+z2+z3=1 and z1z2z3=1 , then |(z1 + 2) (z2 + 2) (z3 + 2)| is equal to (A) 14 (B) 15 (C) 20 (D) 9

Tardigrade Admin · Accepted Answer

(z1 + 2)(z2 + 2)(z3 + 2)=z1z2z3+2(z1 z2 + z2 z3 = z3 z1)+4(z1 + z2 + z3)+8 =1+2(z1 z2 + z2 z3 + z3 z1)+4+8=13+2(z1 z2 + z2 z3 + z3 z1) Now z1z2+z2z3+z3z1=z1z2z3((1/z1) + (1/z2) + (1/z3))=(1/z1)+(1/z2)+(1/z3) =( barz1/|z1|2)+( barz2/|z2|2)+( barz3/|z3|2)= barz1+ barz2+ barz3=z1+z2+z3=1

Q. Given that $z_{1}, z_{2}$ and $z_{3}$ are complex numbers with $∣ z_{1} ∣ = ∣ z_{2} ∣ = ∣ z_{3} ∣ = 1, z_{1} + z_{2} + z_{3} = 1$ and $z_{1} z_{2} z_{3} = 1$ , then $∣ (z_{1} + 2) (z_{2} + 2) (z_{3} + 2) ∣$ is equal to

$14$

$15$

$20$

$9$

Q. Given that z1​,z2​ and z3​ are complex numbers with ∣z1​∣=∣z2​∣=∣z3​∣=1,z1​+z2​+z3​=1 and z1​z2​z3​=1 , then ∣(z1​+2)(z2​+2)(z3​+2)∣ is equal to

14

15

20

9

Q. Given that $z_{1}, z_{2}$ and $z_{3}$ are complex numbers with $∣ z_{1} ∣ = ∣ z_{2} ∣ = ∣ z_{3} ∣ = 1, z_{1} + z_{2} + z_{3} = 1$ and $z_{1} z_{2} z_{3} = 1$ , then $∣ (z_{1} + 2) (z_{2} + 2) (z_{3} + 2) ∣$ is equal to

$14$

$15$

$20$

$9$