| Exam Board | Edexcel |
|---|---|
| Module | CP1 (Core Pure 1) |
| Year | 2024 |
| Session | June |
| Marks | 8 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Roots of polynomials |
| Type | Symmetric functions of roots |
| Difficulty | Moderate -0.5 This is a standard symmetric functions question requiring direct application of Vieta's formulas in part (a), then routine algebraic manipulation in part (b). While it involves multiple steps, each technique (finding common denominators, expanding products, using the identity for sum of squares) is well-practiced in Core Pure 1 with no novel insight required, making it slightly easier than average. |
| Spec | 4.05a Roots and coefficients: symmetric functions |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| \(\alpha+\beta+\gamma=\frac{3}{2},\ \alpha\beta+\alpha\gamma+\beta\gamma=6,\ \alpha\beta\gamma=-\frac{7}{2}\) | B1 | Correct values stated |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| \(\frac{2}{\alpha}+\frac{2}{\beta}+\frac{2}{\gamma}=\frac{2(\alpha\beta+\alpha\gamma+\beta\gamma)}{\alpha\beta\gamma}=2\times\frac{\text{"6"}}{\text{"-7/2"}}\) | M1 | Correct identity with attempt to substitute values |
| \(=-\frac{24}{7}\) | A1ft | Correct value, follow through from part (a) |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| \((\alpha-1)(\beta-1)(\gamma-1)=(\alpha\beta-(\alpha+\beta)+1)(\gamma-1)=\ldots\) | M1 | Attempts to expand product fully (allow sign slips, at most one incorrect/missing term) |
| \(=\alpha\beta\gamma-(\alpha\beta+\alpha\gamma+\beta\gamma)+\alpha+\beta+\gamma-1\) | A1 | Correct expansion in terms of product, pair sum and sum |
| \(=-9\) | A1 | Correct value |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| \(\alpha^2+\beta^2+\gamma^2=(\alpha+\beta+\gamma)^2-2(\alpha\beta+\alpha\gamma+\beta\gamma) = \left(\text{"}\frac{3}{2}\text{"}\right)^2-2\text{"6"}\) | M1 | Correct identity with attempt to substitute values |
| \(=\frac{9}{4}-2(6)=-\frac{39}{4}\) | A1ft | Correct value, follow through from part (a) |
# Question 2:
## Part (a):
| Answer/Working | Mark | Guidance |
|---|---|---|
| $\alpha+\beta+\gamma=\frac{3}{2},\ \alpha\beta+\alpha\gamma+\beta\gamma=6,\ \alpha\beta\gamma=-\frac{7}{2}$ | B1 | Correct values stated |
**(1 mark)**
## Part (b)(i):
| Answer/Working | Mark | Guidance |
|---|---|---|
| $\frac{2}{\alpha}+\frac{2}{\beta}+\frac{2}{\gamma}=\frac{2(\alpha\beta+\alpha\gamma+\beta\gamma)}{\alpha\beta\gamma}=2\times\frac{\text{"6"}}{\text{"-7/2"}}$ | M1 | Correct identity with attempt to substitute values |
| $=-\frac{24}{7}$ | A1ft | Correct value, follow through from part (a) |
## Part (b)(ii):
| Answer/Working | Mark | Guidance |
|---|---|---|
| $(\alpha-1)(\beta-1)(\gamma-1)=(\alpha\beta-(\alpha+\beta)+1)(\gamma-1)=\ldots$ | M1 | Attempts to expand product fully (allow sign slips, at most one incorrect/missing term) |
| $=\alpha\beta\gamma-(\alpha\beta+\alpha\gamma+\beta\gamma)+\alpha+\beta+\gamma-1$ | A1 | Correct expansion in terms of product, pair sum and sum |
| $=-9$ | A1 | Correct value |
## Part (b)(iii):
| Answer/Working | Mark | Guidance |
|---|---|---|
| $\alpha^2+\beta^2+\gamma^2=(\alpha+\beta+\gamma)^2-2(\alpha\beta+\alpha\gamma+\beta\gamma) = \left(\text{"}\frac{3}{2}\text{"}\right)^2-2\text{"6"}$ | M1 | Correct identity with attempt to substitute values |
| $=\frac{9}{4}-2(6)=-\frac{39}{4}$ | A1ft | Correct value, follow through from part (a) |
**(7 marks total for (b))**
**(8 marks)**
---\begin{enumerate}
\item The roots of the equation
\end{enumerate}
$$2 x ^ { 3 } - 3 x ^ { 2 } + 12 x + 7 = 0$$
are $\alpha , \beta$ and $\gamma$\\
Without solving the equation,\\
(a) write down the value of each of
$$\alpha + \beta + \gamma \quad \alpha \beta + \alpha \gamma + \beta \gamma \quad \alpha \beta \gamma$$
(b) Use the answers to part (a) to determine the value of\\
(i) $\frac { 2 } { \alpha } + \frac { 2 } { \beta } + \frac { 2 } { \gamma }$\\
(ii) $( \alpha - 1 ) ( \beta - 1 ) ( \gamma - 1 )$\\
(iii) $\alpha ^ { 2 } + \beta ^ { 2 } + \gamma ^ { 2 }$
\hfill \mbox{\textit{Edexcel CP1 2024 Q2 [8]}}