Complex Numbers
Modulus from Conjugate Equation
nta_pyq_2024_jan
Grade 11
Question:
If $z=x+iy$, $xy\ne 0$, satisfies the equation $z^2+i\bar{z}=0$, then $|z|^2$ is equal to:
Step-by-Step Solution
Key Concept: From $z^2=-i\bar{z}$, take modulus of both sides: $|z|^2=|z|$, giving $|z|(|z|-1)=0$. Since $xy\ne 0$, $|z|\ne 0$, so $|z|=1$ and $|z|^2=1$.
$z^2+i\bar{z}=0\Rightarrow z^2=-i\bar{z}$. Taking modulus: $|z|^2=|-i||z|=|z|$. So $|z|^2-|z|=0\Rightarrow|z|(|z|-1)=0$. Since $xy\ne0$, $|z|\ne0\Rightarrow|z|=1\Rightarrow|z|^2=1$.
Correct Answer: 2