Challenging +1.3 Part (a) requires understanding Vieta's formulas and executing a substitution to transform roots, which is a standard Further Maths technique but requires careful algebraic manipulation. Part (b) involves properties of complex conjugate roots and modulus calculations using Vieta's formulas—conceptually straightforward for Further Maths students but requiring multiple connected steps. The numerical verification is routine. Overall, this is a solid Further Maths question requiring several techniques but no exceptional insight.
The cubic equation \(x^3 + 2x^2 + 3x - 4 = 0\) has roots \(p\), \(q\) and \(r\). A second cubic equation has roots \(qr\), \(rp\) and \(pq\). Show how the substitution \(y = \frac{4}{x}\) can be used to determine this second equation. Hence, or otherwise, find this equation in the form \(y^3 + ay^2 + by + c = 0\). [6]
The cubic equation \(x^3 - 4x^2 + 5x - 4 = 0\) has roots \(\alpha\), \(\beta\) and \(\gamma\). You are given that \(\alpha\) is real and positive, and that \(\beta\) and \(\gamma\) are complex.
Describe the relationship between \(\beta\) and \(\gamma\). [1]
\begin{enumerate}[label=(\alph*)]
\item The cubic equation $x^3 + 2x^2 + 3x - 4 = 0$ has roots $p$, $q$ and $r$. A second cubic equation has roots $qr$, $rp$ and $pq$. Show how the substitution $y = \frac{4}{x}$ can be used to determine this second equation. Hence, or otherwise, find this equation in the form $y^3 + ay^2 + by + c = 0$. [6]
\item The cubic equation $x^3 - 4x^2 + 5x - 4 = 0$ has roots $\alpha$, $\beta$ and $\gamma$. You are given that $\alpha$ is real and positive, and that $\beta$ and $\gamma$ are complex.
\begin{enumerate}[label=(\roman*)]
\item Describe the relationship between $\beta$ and $\gamma$. [1]
\item Explain why $|\beta| = \frac{2}{\sqrt{\alpha}}$. [2]
\item Verify that $\alpha = 2.70$ correct to 3 significant figures, and deduce that $\text{Re}(\beta) = 0.65$ correct to 2 significant figures. [4]
\end{enumerate}
\end{enumerate}
\hfill \mbox{\textit{Pre-U Pre-U 9795/1 2015 Q11 [13]}}