| Exam Board | CAIE |
|---|---|
| Module | Further Paper 1 (Further Paper 1) |
| Year | 2023 |
| Session | November |
| Marks | 8 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Roots of polynomials |
| Type | Finding polynomial from root properties |
| Difficulty | Challenging +1.2 This is a standard Further Maths question on symmetric functions and Vieta's formulas requiring systematic application of known identities. Part (a) uses direct relationships between power sums and elementary symmetric functions (e₁=3, e₁²-2e₂=5 gives e₂, and sum of reciprocals gives e₃/e₄). Part (b) applies Newton's identities or recurrence relations. While it requires careful algebraic manipulation and knowledge of multiple techniques, the approach is methodical without requiring novel insight—typical of Further Pure content but more routine than problem-solving intensive questions. |
| Spec | 4.05a Roots and coefficients: symmetric functions |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \(b = -(\alpha+\beta+\gamma+\delta) = -3\) | B1 | |
| \(5 = (-3)^2 - 2(\alpha\beta+\alpha\gamma+\alpha\delta+\beta\gamma+\beta\delta+\gamma\delta)\) | M1 A1 | Uses formula for sum of squares |
| \(c = 2\) | A1 | |
| \(6 = \frac{\alpha\beta\gamma+\beta\gamma\delta+\gamma\delta\alpha+\delta\alpha\beta}{\alpha\beta\gamma\delta} = \frac{-d}{-2}\) | M1 | Uses \(\alpha^{-1}+\beta^{-1}+\gamma^{-1}+\delta^{-1} = \frac{\alpha\beta\gamma+\beta\gamma\delta+\gamma\delta\alpha+\delta\alpha\beta}{\alpha\beta\gamma\delta}\) |
| \(d = 12\) | A1 | Equation is \(x^4-3x^3+2x^2+12x-2=0\) |
| Total: 6 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \(\alpha^4+\beta^4+\gamma^4+\delta^4 = 3(-27)-2(5)-12(3)+2(4)\) | M1 | Uses *their* quartic equation derived in (a) |
| \(-119\) | A1 | |
| Total: 2 |
## Question 3(a):
| Answer | Marks | Guidance |
|--------|-------|----------|
| $b = -(\alpha+\beta+\gamma+\delta) = -3$ | B1 | |
| $5 = (-3)^2 - 2(\alpha\beta+\alpha\gamma+\alpha\delta+\beta\gamma+\beta\delta+\gamma\delta)$ | M1 A1 | Uses formula for sum of squares |
| $c = 2$ | A1 | |
| $6 = \frac{\alpha\beta\gamma+\beta\gamma\delta+\gamma\delta\alpha+\delta\alpha\beta}{\alpha\beta\gamma\delta} = \frac{-d}{-2}$ | M1 | Uses $\alpha^{-1}+\beta^{-1}+\gamma^{-1}+\delta^{-1} = \frac{\alpha\beta\gamma+\beta\gamma\delta+\gamma\delta\alpha+\delta\alpha\beta}{\alpha\beta\gamma\delta}$ |
| $d = 12$ | A1 | Equation is $x^4-3x^3+2x^2+12x-2=0$ |
| **Total: 6** | | |
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## Question 3(b):
| Answer | Marks | Guidance |
|--------|-------|----------|
| $\alpha^4+\beta^4+\gamma^4+\delta^4 = 3(-27)-2(5)-12(3)+2(4)$ | M1 | Uses *their* quartic equation derived in (a) |
| $-119$ | A1 | |
| **Total: 2** | | |
---3 The quartic equation $\mathrm { x } ^ { 4 } + \mathrm { bx } ^ { 3 } + \mathrm { cx } ^ { 2 } + \mathrm { dx } - 2 = 0$ has roots $\alpha , \beta , \gamma , \delta$. It is given that
$$\alpha + \beta + \gamma + \delta = 3 , \quad \alpha ^ { 2 } + \beta ^ { 2 } + \gamma ^ { 2 } + \delta ^ { 2 } = 5 , \quad \alpha ^ { - 1 } + \beta ^ { - 1 } + \gamma ^ { - 1 } + \delta ^ { - 1 } = 6$$
\begin{enumerate}[label=(\alph*)]
\item Find the values of $b , c$ and $d$.
\item Given also that $\alpha ^ { 3 } + \beta ^ { 3 } + \gamma ^ { 3 } + \delta ^ { 3 } = - 27$, find the value of $\alpha ^ { 4 } + \beta ^ { 4 } + \gamma ^ { 4 } + \delta ^ { 4 }$.
\end{enumerate}
\hfill \mbox{\textit{CAIE Further Paper 1 2023 Q3 [8]}}