| Exam Board | AQA |
|---|---|
| Module | Further Paper 2 (Further Paper 2) |
| Year | 2023 |
| Session | June |
| Marks | 11 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Roots of polynomials |
| Type | Quadratic with transformed roots |
| Difficulty | Challenging +1.8 This is a Further Maths question requiring systematic manipulation of symmetric functions of roots across multiple parts. Part (a) is routine Vieta's formulas, but parts (b) and (c) demand non-trivial algebraic manipulation—expressing α⁴+β⁴ using recurrence relations and constructing a new quadratic from complex expressions of the roots. The 11-mark allocation and multi-step reasoning place this well above average difficulty, though it follows established Further Maths techniques rather than requiring novel insight. |
| Spec | 4.02i Quadratic equations: with complex roots4.05a Roots and coefficients: symmetric functions |
| Answer | Marks | Guidance |
|---|---|---|
| 13(a) | Obtains correct sum of roots. | 1.1b |
| Answer | Marks | Guidance |
|---|---|---|
| Obtains correct product of roots. | 1.1b | B1 |
| Subtotal | 2 | |
| Q | Marking Instructions | AO |
| Answer | Marks | Guidance |
|---|---|---|
| 13(b) | Expresses α2 +β2 in terms of | |
| α+βand αβ | 1.1a | M1 |
| Answer | Marks | Guidance |
|---|---|---|
| α2 +β2 | 1.1b | A1 |
| Answer | Marks | Guidance |
|---|---|---|
| sums and/or products of α, β, | 2.2a | M1 |
| Answer | Marks | Guidance |
|---|---|---|
| CSO | 2.1 | R1 |
| Subtotal | 4 | |
| Q | Marking Instructions | AO |
| Answer | Marks |
|---|---|
| 13(c) | Expresses the sum of roots of |
| Answer | Marks | Guidance |
|---|---|---|
| 𝛼𝛼 𝛽𝛽 | 3.1a | M1 |
| Answer | Marks | Guidance |
|---|---|---|
| Obtains correct sum of roots. | 1.1b | A1 |
| Answer | Marks | Guidance |
|---|---|---|
| =α3β3 +α4 +β4 +αβ | 1.1a | M1 |
| Obtains correct product of roots. | 1.1b | A1 |
| Answer | Marks | Guidance |
|---|---|---|
| Allow any variable. | 2.2a | A1 |
| Subtotal | 5 | |
| Question total | 11 | |
| Q | Marking Instructions | AO |
Question 13:
--- 13(a) ---
13(a) | Obtains correct sum of roots. | 1.1b | B1 | α+β=5
αβ=8
Obtains correct product of roots. | 1.1b | B1
Subtotal | 2
Q | Marking Instructions | AO | Marks | Typical solution
--- 13(b) ---
13(b) | Expresses α2 +β2 in terms of
α+βand αβ | 1.1a | M1 | α2 +β2 =(α+β)2 −2αβ
=25-16=9
( )2
α 4 + β 4 = α2+β2 −2α2β2
2 2
9 -2×8
= −47
=
Obtains correct value of
α2 +β2 | 1.1b | A1
Expresses α4 +β4 in terms of
sums and/or products of α, β, | 2.2a | M1
2 2
C𝛼𝛼o,m𝛽𝛽pletes a reasoned
argument to obtain the required
result.
CSO | 2.1 | R1
Subtotal | 4
Q | Marking Instructions | AO | Marks | Typical solution
--- 13(c) ---
13(c) | Expresses the sum of roots of
the new equation in terms of
sums and/or products of α and β
or and
2 2
𝛼𝛼 𝛽𝛽 | 3.1a | M1 | Sum of roots = α 3 + β3 + α + β
3 3
=125-3×8×5+5
= (𝛼𝛼+𝛽𝛽) − 𝛼𝛼𝛽𝛽(𝛼𝛼+𝛽𝛽)+𝛼𝛼+𝛽𝛽
=10
P roduct of roots = ( α3 +β )( β3 +α )
=α3β3 +α4 +β4 +αβ
3
=8 -47+8
=473
z2 - 10z + 473 = 0
Obtains correct sum of roots. | 1.1b | A1
Expresses the product of roots
of the new equation as
=α3β3 +α4 +β4 +αβ | 1.1a | M1
Obtains correct product of roots. | 1.1b | A1
Deduces a correct quadratic
equation with integer
coefficients.
Allow any variable. | 2.2a | A1
Subtotal | 5
Question total | 11
Q | Marking Instructions | AO | Marks | Typical solutionThe quadratic equation $z^2 - 5z + 8 = 0$ has roots $\alpha$ and $\beta$
\begin{enumerate}[label=(\alph*)]
\item Write down the value of $\alpha + \beta$ and the value of $\alpha\beta$
[2 marks]
\item Without finding the value of $\alpha$ or the value of $\beta$, show that $\alpha^4 + \beta^4 = -47$
[4 marks]
\item Find a quadratic equation, with integer coefficients, which has roots $\alpha^3 + \beta$ and $\beta^3 + \alpha$
[5 marks]
\end{enumerate}
\hfill \mbox{\textit{AQA Further Paper 2 2023 Q13 [11]}}