| Exam Board | Edexcel |
|---|---|
| Module | CP2 (Core Pure 2) |
| Session | Specimen |
| Marks | 8 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Roots of polynomials |
| Type | Symmetric functions of roots |
| Difficulty | Moderate -0.3 This is a standard symmetric functions question requiring direct application of Vieta's formulas with routine algebraic manipulation. Part (i) uses sum of reciprocals = (αβ+αγ+βγ)/αβγ, part (ii) involves substituting y=x+2, and part (iii) uses (α+β+γ)²-2(αβ+αγ+βγ). All three are textbook exercises with no novel insight required, making it slightly easier than average for A-level. |
| Spec | 4.05a Roots and coefficients: symmetric functions |
| Answer | Marks | Guidance |
|---|---|---|
| B1 | Identifies the correct values for all 3 expressions (can score anywhere) | 3.1a |
| M1 | Uses a correct identity | 1.1b |
| A1ft | Correct value (follow through their 8, 28 and 32) | 1.1b |
| Answer | Marks | Guidance |
|---|---|---|
| M1 | Attempts to expand | 1.1b |
| A1 | Correct expansion | 1.1b |
| A1 | Correct value | 1.1b |
| Answer | Marks | Guidance |
|---|---|---|
| M1 | Substitutes \(x - 2\) for \(x\) in the given cubic | 1.1b |
| A1 | Calculates the correct constant term | 1.1b |
| A1 | Changes sign and so obtains the correct value | 1.1b |
| Answer | Marks | Guidance |
|---|---|---|
| M1 | Establishes the correct identity | 3.1a |
| A1ft | Correct value (follow through their 8, 28 and 32) | 1.1b |
# Question 1
## 1(i)
B1 | Identifies the correct values for all 3 expressions (can score anywhere) | 3.1a
M1 | Uses a correct identity | 1.1b
A1ft | Correct value (follow through their 8, 28 and 32) | 1.1b
**(3 marks)**
## 1(ii)
M1 | Attempts to expand | 1.1b
A1 | Correct expansion | 1.1b
A1 | Correct value | 1.1b
**Alternative:**
M1 | Substitutes $x - 2$ for $x$ in the given cubic | 1.1b
A1 | Calculates the correct constant term | 1.1b
A1 | Changes sign and so obtains the correct value | 1.1b
**(3 marks)**
## 1(iii)
M1 | Establishes the correct identity | 3.1a
A1ft | Correct value (follow through their 8, 28 and 32) | 1.1b
**(2 marks)**
---
**(8 marks total)**\begin{enumerate}
\item The roots of the equation
\end{enumerate}
$$x ^ { 3 } - 8 x ^ { 2 } + 28 x - 32 = 0$$
are $\alpha , \beta$ and $\gamma$\\
Without solving the equation, find the value of\\
(i) $\frac { 1 } { \alpha } + \frac { 1 } { \beta } + \frac { 1 } { \gamma }$\\
(ii) $( \alpha + 2 ) ( \beta + 2 ) ( \gamma + 2 )$\\
(iii) $\alpha ^ { 2 } + \beta ^ { 2 } + \gamma ^ { 2 }$
\hfill \mbox{\textit{Edexcel CP2 Q1 [8]}}