| Exam Board | CAIE |
|---|---|
| Module | FP1 (Further Pure Mathematics 1) |
| Year | 2018 |
| Session | November |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Roots of polynomials |
| Type | Roots with special relationships |
| Difficulty | Standard +0.8 This is a Further Maths question requiring systematic application of Vieta's formulas to roots in geometric progression, followed by algebraic manipulation to eliminate α and derive relationships between coefficients. While the techniques are standard for FP1, the multi-step algebraic manipulation in part (ii) to eliminate α and derive p³r = q³ requires careful work and is more demanding than typical A-level pure questions. |
| Spec | 4.05a Roots and coefficients: symmetric functions |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \(\alpha + 2\alpha + 4\alpha = -p\) | B1 | Sum of roots |
| \(2\alpha^2 + 4\alpha^2 + 8\alpha^2 = q\) | B1 | Sum of products in pairs |
| \(\dfrac{14\alpha^2}{7\alpha} = -\dfrac{q}{p}\) | M1 | Combines equations |
| \(\Rightarrow 2p\alpha + q = 0\) | A1 | Verifies result (AG) |
| Total: 4 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \(8\alpha^3 = -r\) | B1 | Product of roots |
| \(\Rightarrow r = \dfrac{q^3}{p^3} \Rightarrow p^3r - q^3 = 0\) | B1 | Verifies result (AG) |
| Total: 2 |
## Question 2:
**Part (i)**
| Answer | Marks | Guidance |
|--------|-------|----------|
| $\alpha + 2\alpha + 4\alpha = -p$ | B1 | Sum of roots |
| $2\alpha^2 + 4\alpha^2 + 8\alpha^2 = q$ | B1 | Sum of products in pairs |
| $\dfrac{14\alpha^2}{7\alpha} = -\dfrac{q}{p}$ | M1 | Combines equations |
| $\Rightarrow 2p\alpha + q = 0$ | A1 | Verifies result (AG) |
| **Total: 4** | | |
**Part (ii)**
| Answer | Marks | Guidance |
|--------|-------|----------|
| $8\alpha^3 = -r$ | B1 | Product of roots |
| $\Rightarrow r = \dfrac{q^3}{p^3} \Rightarrow p^3r - q^3 = 0$ | B1 | Verifies result (AG) |
| **Total: 2** | | |2 The roots of the equation
$$x ^ { 3 } + p x ^ { 2 } + q x + r = 0$$
are $\alpha , 2 \alpha , 4 \alpha$, where $p , q , r$ and $\alpha$ are non-zero real constants.\\
(i) Show that
$$2 p \alpha + q = 0$$
(ii) Show that
$$p ^ { 3 } r - q ^ { 3 } = 0$$
\hfill \mbox{\textit{CAIE FP1 2018 Q2}}