Let z = a + ib , b ≠ 0 be complex numbers satisfying z 2= overline z ⋅ 21-| z |. Then the least value of n ∈ N, such that z n =( z +1) n , is equal to

Question

Let z = a + ib , b ≠ 0 be complex numbers satisfying z 2= overline z ⋅ 21-| z |. Then the least value of n ∈ N, such that z n =( z +1) n , is equal to (A)  (B)  (C)  (D)

Tardigrade Admin · Accepted Answer

Sol. | z 2|=| overline z | ⋅ 21-| z | ⇒| z |=1 z 2= overline z ⇒ z 3=1 ∴ z =ω text or ω2 ω n =(1+ω) n =(-ω2) n Least natural value of n is 6 .

Q. Let $z = a + ib, b \neq = 0$ be complex numbers satisfying $z^{2} = \overline{z} \cdot 2^{1 - ∣ z ∣}$ . Then the least value of $n \in N$ , such that $z^{n} = (z + 1)^{n}$ , is equal to

Sol. $∣ ∣ z^{2} ∣ ∣ = ∣ \overline{z} ∣ \cdot 2^{1 - ∣ z ∣} \Rightarrow ∣ z ∣ = 1$
$z^{2} = \overline{z} \Rightarrow z^{3} = 1$
$∴ z = ω or ω^{2}$
$ω^{n} = (1 + ω)^{n} = (- ω^{2})^{n}$
Least natural value of $n$ is 6 .

Q. Let z=a+ib,b=0 be complex numbers satisfying z2=z⋅21−∣z∣. Then the least value of n∈N, such that zn=(z+1)n, is equal to

Sol. ∣∣​z2∣∣​=∣z∣⋅21−∣z∣⇒∣z∣=1 z2=z⇒z3=1 ∴z=ω or ω2 ωn=(1+ω)n=(−ω2)n Least natural value of n is 6 .

Q. Let $z = a + ib, b \neq = 0$ be complex numbers satisfying $z^{2} = \overline{z} \cdot 2^{1 - ∣ z ∣}$ . Then the least value of $n \in N$ , such that $z^{n} = (z + 1)^{n}$ , is equal to

Sol. $∣ ∣ z^{2} ∣ ∣ = ∣ \overline{z} ∣ \cdot 2^{1 - ∣ z ∣} \Rightarrow ∣ z ∣ = 1$
$z^{2} = \overline{z} \Rightarrow z^{3} = 1$
$∴ z = ω or ω^{2}$
$ω^{n} = (1 + ω)^{n} = (- ω^{2})^{n}$
Least natural value of $n$ is 6 .