Question
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?
JEE Advanced
Mathematics
Complex Number
2020
JEE Advanced 2020 (Paper 1)
JEE Advanced Mathematics Question (2020) — Solution
Options
- A. z + 1 2 ≤ 1 2 for all z ∈ S
- B. z ≤ 2 for all z ∈ S
- C. z + 1 2 ≥ 1 2 for all z ∈ S
- D. The set S has exactly four elements
Answer
C. z + 1 2 ≥ 1 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 cos θ ) 1 4 ⇒ 2 Z ≤ 1 + 7 < 4 ⇒ Z ≤ 2 Option (B) is correct.
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