| Exam Board | OCR MEI |
|---|---|
| Module | FP1 (Further Pure Mathematics 1) |
| Year | 2008 |
| Session | January |
| Marks | 5 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Roots of polynomials |
| Type | Finding polynomial from root properties |
| Difficulty | Standard +0.3 This is a straightforward application of standard relationships between roots and coefficients (Vieta's formulas) combined with the algebraic identity (α+β+γ)² = α²+β²+γ² + 2(αβ+βγ+γα). While it requires knowing these formulas and one algebraic manipulation, it's a routine Further Maths question with no novel insight needed—slightly easier than average even for FP1 standards. |
| Spec | 4.05a Roots and coefficients: symmetric functions |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| \(p = -3,\ r = 7\) | B2 [2] | One mark for each; s.c. B1 if \(b\) and \(d\) used instead of \(p\) and \(r\) |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| \(q = \alpha\beta + \alpha\gamma + \beta\gamma\) | B1 | |
| \(\alpha^2 + \beta^2 + \gamma^2 = (\alpha+\beta+\gamma)^2 - 2(\alpha\beta+\alpha\gamma+\beta\gamma)\) | M1 | Attempt to find \(q\) using \(\alpha^2+\beta^2+\gamma^2\) and \(\alpha+\beta+\gamma\), but not \(\alpha\beta\gamma\) |
| \(= (\alpha+\beta+\gamma)^2 - 2q\) | ||
| \(\Rightarrow 13 = 3^2 - 2q\) | ||
| \(\Rightarrow q = -2\) | A1 [3] | c.a.o. |
# Question 5(i):
| Answer/Working | Mark | Guidance |
|---|---|---|
| $p = -3,\ r = 7$ | B2 **[2]** | One mark for each; s.c. B1 if $b$ and $d$ used instead of $p$ and $r$ |
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# Question 5(ii):
| Answer/Working | Mark | Guidance |
|---|---|---|
| $q = \alpha\beta + \alpha\gamma + \beta\gamma$ | B1 | |
| $\alpha^2 + \beta^2 + \gamma^2 = (\alpha+\beta+\gamma)^2 - 2(\alpha\beta+\alpha\gamma+\beta\gamma)$ | M1 | Attempt to find $q$ using $\alpha^2+\beta^2+\gamma^2$ and $\alpha+\beta+\gamma$, but not $\alpha\beta\gamma$ |
| $= (\alpha+\beta+\gamma)^2 - 2q$ | | |
| $\Rightarrow 13 = 3^2 - 2q$ | | |
| $\Rightarrow q = -2$ | A1 **[3]** | c.a.o. |
---5 The equation $x ^ { 3 } + p x ^ { 2 } + q x + r = 0$ has roots $\alpha , \beta$ and $\gamma$, where
$$\begin{aligned}
\alpha + \beta + \gamma & = 3 \\
\alpha \beta \gamma & = - 7 \\
\alpha ^ { 2 } + \beta ^ { 2 } + \gamma ^ { 2 } & = 13
\end{aligned}$$
(i) Write down the values of $p$ and $r$.\\
(ii) Find the value of $q$.
\hfill \mbox{\textit{OCR MEI FP1 2008 Q5 [5]}}