Statement I Let z1 and z2 be two complex numbers such that overlinez1+i overlinez2=0 and arg (z1 ⋅ z2)=π, then arg (z1) is (3 π/4). Statement II arg (z1 ⋅ z2)= arg z1+ arg z2.

Question

Statement I Let z1 and z2 be two complex numbers such that overlinez1+i overlinez2=0 and arg (z1 ⋅ z2)=π, then arg (z1) is (3 π/4). Statement II arg (z1 ⋅ z2)= arg z1+ arg z2. (A) Statement I is correct (B) Statement II is correct (C) Both are correct (D) Neither I nor II is correct

Tardigrade Admin · Accepted Answer

Given that, overlinez1+i overlinez2=0 z1=i z2 text , i.e., z2=-i z1 text Thus, arg (z1 z2)= arg z1+ arg (-i z1)=π ∵ arg (z1 z2)= arg z1+ arg z2 ⇒ arg (-i z12)=π ⇒ arg (-i)+2 arg (z1)=π ⇒ (-π/2)+2 arg (z1)=π ⇒ arg (z1)=(3 π/4)

Q. Statement I Let z1​ and z2​ be two complex numbers such that z1​​+iz2​​=0 and arg(z1​⋅z2​)=π, then arg(z1​) is 43π​. Statement II arg(z1​⋅z2​)=argz1​+argz2​.

Statement I is correct

Statement II is correct

Both are correct

Neither I nor II is correct

Given that, z1​​+iz2​​=0 z1​=iz2​, i.e., z2​=−iz1​ Thus, arg(z1​z2​)=argz1​+arg(−iz1​)=π ∵arg(z1​z2​)=argz1​+argz2​ ⇒arg(−iz12​)=π ⇒arg(−i)+2arg(z1​)=π ⇒2−π​+2arg(z1​)=π ⇒arg(z1​)=43π​

Q. Statement I Let $z_{1}$ and $z_{2}$ be two complex numbers such that $\overline{z_{1}} + i \overline{z_{2}} = 0$ and $ar g (z_{1} \cdot z_{2}) = π$ , then $ar g (z_{1})$ is $\frac{3 π}{4}$ .
Statement II $ar g (z_{1} \cdot z_{2}) = ar g z_{1} + ar g z_{2}$ .