Challenging +1.2 This is a Further Maths FP1 complex numbers question requiring students to find the roots (one real, two complex conjugates by inspection or factor theorem), verify they all have the same modulus, and identify k=5. While it involves multiple steps including factorization, using conjugate root theorem, and calculating moduli, the techniques are standard for FP1 and the structure is guided by the 'show that' format. The 9 marks reflect length rather than exceptional difficulty—it's above average due to being Further Maths content requiring synthesis of several ideas, but not requiring deep insight.
**In this question you must show detailed reasoning.**
It is given that \(f(z) = z^3 - 13z^2 + 65z - 125\).
The points representing the three roots of the equation \(f(z) = 0\) are plotted on an Argand diagram.
Show that these points lie on the circle \(|z| = k\), where \(k\) is a real number to be determined. [9]
**In this question you must show detailed reasoning.**
It is given that $f(z) = z^3 - 13z^2 + 65z - 125$.
The points representing the three roots of the equation $f(z) = 0$ are plotted on an Argand diagram.
Show that these points lie on the circle $|z| = k$, where $k$ is a real number to be determined. [9]
\hfill \mbox{\textit{OCR FP1 AS 2017 Q7 [9]}}