2026 NYJC P1 Q4

Do not use a calculator in answering this question.

It is given that \(1-3i\) is a complex root of the equation \(z^3+pz^2+(p+12)z+q=0\), where \(p\) and \(q\) are real numbers.

1. State another complex root of the equation in the form \(a+ib\), where \(a\) and \(b\) are real numbers and \(b\ne0\).[1]
2. Find the values of \(p\) and \(q\), and find the third root.[4]
3. The point \(A\) represents the complex number \(1-3i\) on an Argand diagram with respect to the origin \(O\). The point \(C\) represents the complex number \(x+iy\), where \(x\) and \(y\) are real numbers and \(x>0\). Find the values of \(x\) and \(y\) such that \(OA=AC\) and angle \(OAC\) is \(90^\circ\).[2]