| Exam Board | CAIE |
|---|---|
| Module | FP1 (Further Pure Mathematics 1) |
| Year | 2012 |
| Session | June |
| Marks | 5 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Roots of polynomials |
| Type | Sum of powers of roots |
| Difficulty | Standard +0.8 This is a Further Maths question requiring knowledge of Newton's sums and symmetric functions beyond basic Vieta's formulas. Part (i) uses the standard identity (Σα)² - 2Σαβ, while part (ii) requires either Newton's identity or substituting roots into the original equation. These are established techniques in FP1, making this moderately above average difficulty but not requiring novel insight. |
| Spec | 4.05a Roots and coefficients: symmetric functions |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| \(\sum\alpha = 7\), \(\sum\alpha\beta = 2\) | B1 | |
| \(\sum\alpha^2 = 7^2 - 2\times2 = 45\) | B1 | Uses formula correctly |
| \(\sum\alpha^3 = 7\sum\alpha^2 - 2\sum\alpha + 9\) | M1 | Uses formula for \(\sum\alpha^3\) |
| \(= 315 - 14 + 9 = 310\) | A1A1 | To obtain result |
## Question 1:
| Answer/Working | Marks | Guidance |
|---|---|---|
| $\sum\alpha = 7$, $\sum\alpha\beta = 2$ | B1 | |
| $\sum\alpha^2 = 7^2 - 2\times2 = 45$ | B1 | Uses formula correctly |
| $\sum\alpha^3 = 7\sum\alpha^2 - 2\sum\alpha + 9$ | M1 | Uses formula for $\sum\alpha^3$ |
| $= 315 - 14 + 9 = 310$ | A1A1 | To obtain result |
**Total: [5]**
---1 The roots of the cubic equation $x ^ { 3 } - 7 x ^ { 2 } + 2 x - 3 = 0$ are $\alpha , \beta , \gamma$. Find the values of\\
(i) $\alpha ^ { 2 } + \beta ^ { 2 } + \gamma ^ { 2 }$,\\
(ii) $\alpha ^ { 3 } + \beta ^ { 3 } + \gamma ^ { 3 }$.
\hfill \mbox{\textit{CAIE FP1 2012 Q1 [5]}}