| Exam Board | AQA |
|---|---|
| Module | Further AS Paper 1 (Further AS Paper 1) |
| Year | 2020 |
| Session | June |
| Marks | 8 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Roots of polynomials |
| Type | Quadratic with transformed roots |
| Difficulty | Standard +0.3 This is a straightforward Further Maths question on roots of quadratics using Vieta's formulas and standard algebraic manipulation. Parts (a) and (b) are direct recall (αβ = 3/2, α+β = -p/2), part (c) requires expanding (α-β)² = (α+β)² - 4αβ, and part (d) uses sum and product of new roots. All techniques are standard textbook exercises with no novel insight required, though the multi-step nature and Further Maths context places it slightly above average A-level difficulty. |
| Spec | 4.05a Roots and coefficients: symmetric functions4.05b Transform equations: substitution for new roots |
| Answer | Marks |
|---|---|
| 9(a)(i) | Obtains the correct value of . |
| Answer | Marks | Guidance |
|---|---|---|
| 𝛼𝛼𝛼𝛼 = 2 | 1.2 | B1 |
| Answer | Marks |
|---|---|
| 9(a)(ii) | Obtains the correct value of . |
| Answer | Marks | Guidance |
|---|---|---|
| 𝛼𝛼+𝛼𝛼 = −2 | 1.2 | B1 |
| Answer | Marks | Guidance |
|---|---|---|
| 9(b) | Expresses in the form of . | |
| 2 2 | 1.1a | M1 |
| Answer | Marks | Guidance |
|---|---|---|
| FT their and | 1.1b | A1F |
| Answer | Marks |
|---|---|
| 9(c) | Selects a method to find the quadratic equation with roots |
| Answer | Marks | Guidance |
|---|---|---|
| 𝛼𝛼−1, 𝛼𝛼+1 | 3.1a | M1 |
| Answer | Marks | Guidance |
|---|---|---|
| Obtains the sum of roots = their . | 1.1b | B1F |
| Answer | Marks | Guidance |
|---|---|---|
| 2 | 1.1b | B1F |
| Answer | Marks | Guidance |
|---|---|---|
| ISW afte(r𝛼𝛼 a− co𝛼𝛼r)rect equation𝛼𝛼 o+r 𝛼𝛼expression. | 1.1b | A1F |
| Total | 8 | |
| Q | Marking instructions | AO |
Question 9:
--- 9(a)(i) ---
9(a)(i) | Obtains the correct value of .
3
𝛼𝛼𝛼𝛼 = 2 | 1.2 | B1 | 3
--- 9(a)(ii) ---
9(a)(ii) | Obtains the correct value of .
𝑝𝑝
𝛼𝛼+𝛼𝛼 = −2 | 1.2 | B1 | 𝛼𝛼𝛼𝛼 =
2
𝑝𝑝
--- 9(b) ---
9(b) | Expresses in the form of .
2 2 | 1.1a | M1 | 𝛼𝛼+𝛼𝛼 = −
2
2 2 2
(𝛼𝛼−𝛼𝛼) = 𝛼𝛼 −2𝛼𝛼𝛼𝛼+𝛼𝛼
2
= (𝛼𝛼+𝛼𝛼) −2𝛼𝛼𝛼𝛼−2𝛼𝛼𝛼𝛼
2
𝑝𝑝 3
= �− � −4 ×
2 2 2
𝑝𝑝
= −6
(𝛼𝛼−𝛼𝛼) (𝛼𝛼+𝛼𝛼) + 𝑚𝑚𝛼𝛼𝛼𝛼
Obtains the correct value of .
2
2 𝑝𝑝
(𝛼𝛼−𝛼𝛼) = 4 −6
(May be unsimplified.)
FT their and | 1.1b | A1F
--- 9(c) ---
9(c) | Selects a method to find the quadratic equation with roots
by expressing the sum and product of roots in
terms of and .
𝛼𝛼−1, 𝛼𝛼+1 | 3.1a | M1 | New sum
𝑝𝑝
= 𝛼𝛼−1+𝛼𝛼+1 = 𝛼𝛼+𝛼𝛼 = −2
2
𝑝𝑝
𝛼𝛼−𝛼𝛼 = � −6
4
positive root only as
New product 𝛼𝛼 > 𝛼𝛼
= (𝛼𝛼−1)(𝛼𝛼+1) = 𝛼𝛼𝛼𝛼+𝛼𝛼−𝛼𝛼−1
2
3 𝑝𝑝
= +�� −6�−1
2 4
2
1 𝑝𝑝
= +� −6
2 4
2
2 𝑝𝑝 1 𝑝𝑝
𝑥𝑥 −𝑥𝑥�− �+ + �� −6� = 0
2 2 4
𝛼𝛼 𝛼𝛼
Obtains the sum of roots = their . | 1.1b | B1F
𝛼𝛼+𝛼𝛼
Finds an expression for in terms of .
FT their . 𝛼𝛼−𝛼𝛼 𝑝𝑝
2 | 1.1b | B1F
(𝛼𝛼−𝛼𝛼)
Obtains a correct quadratic equation with roots .
FT their and their . 𝛼𝛼−1, 𝛼𝛼+1
Condone a quad2ratic expression.
ISW afte(r𝛼𝛼 a− co𝛼𝛼r)rect equation𝛼𝛼 o+r 𝛼𝛼expression. | 1.1b | A1F
Total | 8
Q | Marking instructions | AO | Marks | Typical solutionThe quadratic equation $2x^2 + px + 3 = 0$ has two roots, $\alpha$ and $\beta$, where $\alpha > \beta$.
\begin{enumerate}[label=(\alph*)]
\item \begin{enumerate}[label=(\roman*)]
\item Write down the value of $\alpha\beta$.
[1 mark]
\item Express $\alpha + \beta$ in terms of $p$.
[1 mark]
\end{enumerate}
\item Hence find $(\alpha - \beta)^2$ in terms of $p$.
[2 marks]
\item Hence find, in terms of $p$, a quadratic equation with roots $\alpha - 1$ and $\beta + 1$
[4 marks]
\end{enumerate}
\hfill \mbox{\textit{AQA Further AS Paper 1 2020 Q9 [8]}}