Challenging +1.2 This is a Further Maths question requiring knowledge that complex roots come in conjugate pairs, use of Vieta's formulas, and geometric interpretation on an Argand diagram. While it involves multiple concepts (polynomial roots, complex conjugates, area calculation), the solution path is relatively standard once you recognize that β and γ must be conjugate pairs of form 2±bi. The area condition gives b=4, then Vieta's formulas yield c and d directly. More routine than typical Further Maths proof questions but requires synthesis of several ideas.
11 The equation \(x ^ { 3 } - 8 x ^ { 2 } + c x + d = 0\) where \(c\) and \(d\) are real numbers, has roots \(\alpha , \beta , \gamma\).
When plotted on an Argand diagram, the triangle with vertices at \(\alpha , \beta , \gamma\) has an area of 8 .
Given \(\alpha = 2\), find the values of \(c\) and \(d\).
Fully justify your solution. [0pt]
[5 marks]
11 The equation $x ^ { 3 } - 8 x ^ { 2 } + c x + d = 0$ where $c$ and $d$ are real numbers, has roots $\alpha , \beta , \gamma$.\\
When plotted on an Argand diagram, the triangle with vertices at $\alpha , \beta , \gamma$ has an area of 8 .
Given $\alpha = 2$, find the values of $c$ and $d$.
Fully justify your solution.\\[0pt]
[5 marks]
\hfill \mbox{\textit{AQA Further AS Paper 1 Q11 [5]}}