2025 TYS

Do not use a calculator in answering this question.

It is given that $1+2\mathrm{i}$ is a complex root of the equation ${{z}^{3}}+p{{z}^{2}}+qz-10=0$, where $p$ and $q$ are real numbers.

1. State another complex root of the equation in the form $a+\mathrm{i}b$, where $a$ and $b$ are real numbers and $b\ne 0$. Justify your answer.

   [2]

2. Find the values of $p$ and $q$, and find the third root.

   [6]

3. Hence find the roots of the equation $1+pw+q{{w}^{2}}-10{{w}^{3}}=0$.

   [2]