Let Z and W be complex numbers such that |Z| = |W| and arg Z denotes the principal argument of Z. Statement 1: If arg Z+ arg W= π, then Z=- overlineW. Statement 2: |Z|=|W|, implies arg Z-arg overlineW=π.

Question

Let Z and W be complex numbers such that |Z| = |W| and arg Z denotes the principal argument of Z. Statement 1: If arg Z+ arg W= π, then Z=- overlineW. Statement 2: |Z|=|W|, implies arg Z-arg overlineW=π. (A) Statement 1 is true, Statement 2 is false (B) Statement 1 is true, Statement 2 is true, Statement 2 is a correct explanation for Statement 1 (C) Statement 1 is true, Statement 2 is true, Statement 2 is not a correct explanation for Statement 1 (D) Statement 1 is false, Statement 2 is true

Tardigrade Admin · Accepted Answer

Let |Z|=|W|=r ⇒ Z=reiθ, W=reiφ where θ+φ=π ∴ =- overlineW=re-iφ Now, Z=rei(π-φ)=reiπ× e-iφ =- overlineW Thus, statement-1 is true but statement-2 is false.

Q. Let $Z$ and $W$ be complex numbers such that $∣ Z ∣ = ∣ W ∣$ and arg $Z$ denotes the principal argument of $Z$ .
Statement 1: If arg $Z +$ arg $W = π$ , then $Z = - \overline{W} .$
Statement 2: $∣ Z ∣ = ∣ W ∣,$ implies arg $Z -$ arg $\overline{W} = π .$

Statement 1 is true, Statement 2 is false

Statement 1 is true, Statement 2 is true, Statement 2 is a correct explanation for Statement 1

Statement 1 is true, Statement 2 is true, Statement 2 is not a correct explanation for Statement 1

Statement 1 is false, Statement 2 is true

Let $∣ Z ∣ = ∣ W ∣ = r$
$\Rightarrow Z = r e^{i θ}, W = r e^{i ϕ}$
where $θ + ϕ = π$
$∴ = - \overline{W} = r e^{- i ϕ}$
Now, $Z = r e^{i (π - ϕ)} = r e^{iπ} \times e^{- i ϕ}$
$= - \overline{W}$
Thus, statement-1 is true but statement-2 is false.

Q. Let Z and W be complex numbers such that ∣Z∣=∣W∣ and arg Z denotes the principal argument of Z. Statement 1: If arg Z+ arg W=π, then Z=−W. Statement 2: ∣Z∣=∣W∣, implies arg Z−arg W=π.