| Exam Board | CAIE |
|---|---|
| Module | FP1 (Further Pure Mathematics 1) |
| Year | 2008 |
| Session | November |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Roots of polynomials |
| Type | Sum of powers of roots |
| Difficulty | Challenging +1.2 This is a structured Further Maths question on Newton's sums with clear scaffolding. Part (i) is a standard proof that roots satisfy the recurrence relation derived from the polynomial. Parts (ii)-(iii) involve routine application of Newton's identities and the recurrence. Part (iv) requires recognizing the expression as S₂·S₄ - S₆, which is a moderate algebraic manipulation insight but well within expected FM techniques. The multi-part structure guides students through the solution, making it easier than it initially appears. |
| Spec | 4.05a Roots and coefficients: symmetric functions4.05b Transform equations: substitution for new roots |
The roots of the equation
$$x ^ { 4 } - 5 x ^ { 2 } + 2 x - 1 = 0$$
are $\alpha , \beta , \gamma , \delta$. Let $S _ { n } = \alpha ^ { n } + \beta ^ { n } + \gamma ^ { n } + \delta ^ { n }$.\\
(i) Show that
$$S _ { n + 4 } - 5 S _ { n + 2 } + 2 S _ { n + 1 } - S _ { n } = 0 .$$
(ii) Find the values of $S _ { 2 }$ and $S _ { 4 }$.\\
(iii) Find the value of $S _ { 3 }$ and hence find the value of $S _ { 6 }$.\\
(iv) Hence find the value of
$$\alpha ^ { 2 } \left( \beta ^ { 4 } + \gamma ^ { 4 } + \delta ^ { 4 } \right) + \beta ^ { 2 } \left( \gamma ^ { 4 } + \delta ^ { 4 } + \alpha ^ { 4 } \right) + \gamma ^ { 2 } \left( \delta ^ { 4 } + \alpha ^ { 4 } + \beta ^ { 4 } \right) + \delta ^ { 2 } \left( \alpha ^ { 4 } + \beta ^ { 4 } + \gamma ^ { 4 } \right) .$$
\hfill \mbox{\textit{CAIE FP1 2008 Q12 OR}}