| Exam Board | AQA |
|---|---|
| Module | FP1 (Further Pure Mathematics 1) |
| Year | 2007 |
| Session | January |
| Marks | 8 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Roots of polynomials |
| Type | Sum of powers of roots |
| Difficulty | Moderate -0.3 This is a standard FP1 question on sums and products of roots requiring straightforward application of Vieta's formulas and algebraic manipulation. Part (a) is direct recall, part (b) uses the identity (α+β)²-2αβ, and part (c) extends this by squaring the result from (b). While it requires multiple steps, the techniques are routine for Further Maths students with no novel problem-solving required. |
| Spec | 4.05a Roots and coefficients: symmetric functions |
| Answer | Marks | Guidance |
|---|---|---|
| \(\alpha + \beta = -2\), \(\alpha\beta = \frac{3}{2}\) | B1B1 | 2 marks |
| Answer | Marks | Guidance |
|---|---|---|
| Use of expansion of \((\alpha + \beta)^2\) | M1 | |
| \(\alpha^2 + \beta^2 = (-2)^2 - 2\left(\frac{3}{2}\right) = 1\) | m1A1 | 3 marks |
| Answer | Marks | Guidance |
|---|---|---|
| \(\alpha^3 + \beta^3\) given in terms of \(\alpha + \beta\), \(\alpha\beta\) and/or \(\alpha^2 + \beta^2\) | M1A1 | |
| \(\alpha^3 + \beta^3 = \frac{7}{2}\) | A1 | 3 marks |
### Part (a)
$\alpha + \beta = -2$, $\alpha\beta = \frac{3}{2}$ | B1B1 | 2 marks |
### Part (b)
Use of expansion of $(\alpha + \beta)^2$ | M1 | |
$\alpha^2 + \beta^2 = (-2)^2 - 2\left(\frac{3}{2}\right) = 1$ | m1A1 | 3 marks | convincingly shown (AG); m1A0 if $\alpha + \beta = 2$ used
### Part (c)
$\alpha^3 + \beta^3$ given in terms of $\alpha + \beta$, $\alpha\beta$ and/or $\alpha^2 + \beta^2$ | M1A1 | | M1A0 if num error made
$\alpha^3 + \beta^3 = \frac{7}{2}$ | A1 | 3 marks | OE
### **Total for Question 3: 8 marks**
---3 The quadratic equation
$$2 x ^ { 2 } + 4 x + 3 = 0$$
has roots $\alpha$ and $\beta$.
\begin{enumerate}[label=(\alph*)]
\item Write down the values of $\alpha + \beta$ and $\alpha \beta$.
\item Show that $\alpha ^ { 2 } + \beta ^ { 2 } = 1$.
\item Find the value of $\alpha ^ { 4 } + \beta ^ { 4 }$.
\end{enumerate}
\hfill \mbox{\textit{AQA FP1 2007 Q3 [8]}}