Challenging +1.2 This is a Further Maths question requiring students to find the roots of a cubic (which factor nicely), plot them on an Argand diagram, and verify they lie on a circle by calculating their moduli. While it involves multiple steps and Further Maths content, the polynomial factors straightforwardly and the verification is computational rather than requiring deep insight. It's moderately above average difficulty due to the Further Maths context and multi-step nature, but not exceptionally challenging.
7 In this question you must show detailed reasoning.
It is given that \(\mathrm { f } ( \mathrm { z } ) = \mathrm { z } ^ { 3 } - 13 z ^ { 2 } + 65 z - 125\).
The points representing the three roots of the equation \(\mathrm { f } ( z ) = 0\) 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.
7 In this question you must show detailed reasoning.\\
It is given that $\mathrm { f } ( \mathrm { z } ) = \mathrm { z } ^ { 3 } - 13 z ^ { 2 } + 65 z - 125$.\\
The points representing the three roots of the equation $\mathrm { f } ( z ) = 0$ 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.
\hfill \mbox{\textit{OCR Further Pure Core AS Q7 [9]}}