| Exam Board | CAIE |
|---|---|
| Module | FP1 (Further Pure Mathematics 1) |
| Year | 2005 |
| Session | November |
| Marks | 7 |
| Paper | Download PDF ↗ |
| Topic | Roots of polynomials |
| Type | Sum of powers of roots |
| Difficulty | Challenging +1.8 This is a Further Maths question requiring systematic application of Vieta's formulas combined with inequality manipulation across three related parts. While the techniques (AM-GM, sum of powers identities) are standard for FP1, the multi-part proof structure with progressively complex expressions and the need to carefully handle inequalities with roots constrained to be >1 elevates this above routine exercises. It requires mathematical maturity but follows a clear logical progression. |
| Spec | 4.05a Roots and coefficients: symmetric functions |
In the equation
$$x^3 + ax^2 + bx + c = 0,$$
the coefficients $a$, $b$ and $c$ are real. It is given that all the roots are real and greater than $1$.
\begin{enumerate}[label=(\roman*)]
\item Prove that $a < -3$. [1]
\item By considering the sum of the squares of the roots, prove that $a^2 > 2b + 3$. [2]
\item By considering the sum of the cubes of the roots, prove that $a^3 < -9b - 3c - 3$. [4]
\end{enumerate}
\hfill \mbox{\textit{CAIE FP1 2005 Q5 [7]}}