| Exam Board | CAIE |
|---|---|
| Module | Further Paper 1 (Further Paper 1) |
| Year | 2024 |
| Session | June |
| Marks | 6 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Roots of polynomials |
| Type | Symmetric functions of roots |
| Difficulty | Standard +0.8 This is a multi-part Further Maths question on symmetric functions of roots requiring systematic application of Vieta's formulas, algebraic manipulation to find symmetric functions of new roots (αβ, βγ, αγ), and forming a new cubic equation. Part (d) requires using the identity α²+β²+γ² = (α+β+γ)² - 2(αβ+βγ+γα). While methodical, it demands fluency with multiple techniques and careful algebraic work across several steps, placing it moderately above average difficulty. |
| Spec | 4.05a Roots and coefficients: symmetric functions |
| Answer | Marks |
|---|---|
| \(-\frac{1}{2}p\) | B1 |
| Answer | Marks | Guidance |
|---|---|---|
| \(\alpha\beta^2\gamma + \alpha^2\beta\gamma + \alpha\beta\gamma^2 = \alpha\beta\gamma(\beta + \gamma + \alpha) = \frac{5}{2}\left(-\frac{1}{2}\right)\) | M1 | Factorises |
| \(-\frac{5}{4}\) | A1 |
| Answer | Marks | Guidance |
|---|---|---|
| \(4z^3 + 2pz^2 - 5z - 25 = 0\) | B1 FT | OE; FT on *their* value of \(\alpha\beta + \beta\gamma + \gamma\alpha\) (in terms of \(p\)) |
| Answer | Marks | Guidance |
|---|---|---|
| \(\frac{1}{3} = \frac{1}{4} + p\) | M1 | Uses \(\alpha^2 + \beta^2 + \gamma^2 = (\alpha + \beta + \gamma)^2 - 2(\alpha\beta + \beta\gamma + \gamma\alpha)\) |
| \(p = \frac{1}{12}\) | A1 | CAO |
## Question 1:
**Part (a):**
$-\frac{1}{2}p$ | B1 |
---
**Part (b):**
$\alpha\beta^2\gamma + \alpha^2\beta\gamma + \alpha\beta\gamma^2 = \alpha\beta\gamma(\beta + \gamma + \alpha) = \frac{5}{2}\left(-\frac{1}{2}\right)$ | M1 | Factorises
$-\frac{5}{4}$ | A1 |
---
**Part (c):**
$4z^3 + 2pz^2 - 5z - 25 = 0$ | B1 FT | OE; FT on *their* value of $\alpha\beta + \beta\gamma + \gamma\alpha$ (in terms of $p$)
---
**Part (d):**
$\frac{1}{3} = \frac{1}{4} + p$ | M1 | Uses $\alpha^2 + \beta^2 + \gamma^2 = (\alpha + \beta + \gamma)^2 - 2(\alpha\beta + \beta\gamma + \gamma\alpha)$
$p = \frac{1}{12}$ | A1 | CAO1 The cubic equation $2 x ^ { 3 } + x ^ { 2 } - p x - 5 = 0$, where $p$ is a positive constant, has roots $\alpha , \beta , \gamma$.
\begin{enumerate}[label=(\alph*)]
\item State, in terms of $p$, the value of $\alpha \beta + \beta \gamma + \gamma \alpha$.
\item Find the value of $\alpha ^ { 2 } \beta \gamma + \alpha \beta ^ { 2 } \gamma + \alpha \beta \gamma ^ { 2 }$.
\item Deduce a cubic equation whose roots are $\alpha \beta , \beta \gamma , \alpha \gamma$.
\item Given that $\alpha ^ { 2 } + \beta ^ { 2 } + \gamma ^ { 2 } = \frac { 1 } { 3 }$, find the value of $p$.
\end{enumerate}
\hfill \mbox{\textit{CAIE Further Paper 1 2024 Q1 [6]}}