JEE Advanced2020MathematicsComplex NumberActual
Let S be the set of all complex numbers z satisfying z 2 + z + 1 = 1 . Then which of the following statements is/are TRUE?
Options
- Az + 1 2 ≤ 1 2 for all z ∈ S
- Bz ≤ 2 for all z ∈ S
- Cz + 1 2 ≥ 1 2 for all z ∈ S
- DThe set S has exactly four elements
Correct answer
B. z ≤ 2 for all z ∈ S
Step-by-step solution
z 2 + z + 1 = e i θ   θ ∈ ( − π ,   π ] ⇒ z 2 + z + 1 − e i θ = 0 By quadratic formula, we get z = − 1 ± 4 e i θ − 3 2   . . . . . . . . . . . . . 1 ⇒ z + 1 2 = ± 4 cos θ − 3 + i 4 sin θ ⇒ z + 1 2 = 4 cos θ − 3 2 + 4 sin θ 2 1 4 ⇒ z + 1 2 ∈ 1 ,   7 Option (C) is correct By equation 1 , 2 z ≤ 1 + 4 e i θ − 3 ⇒ 2 Z ≤ 1 + ( 25 − 24 c