| Exam Board | Edexcel |
|---|---|
| Module | CP AS (Core Pure AS) |
| Session | Specimen |
| Marks | 7 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Roots of polynomials |
| Type | Roots with special relationships |
| Difficulty | Standard +0.8 This is a Core Pure AS question requiring students to work with roots in a non-standard form (involving α and 5/α), apply Vieta's formulas strategically, and solve a resulting equation. While it uses standard polynomial theory, the special relationship between roots requires insight to set up equations efficiently, making it moderately harder than typical A-level questions but not exceptionally challenging. |
| Spec | 4.05a Roots and coefficients: symmetric functions |
| Answer | Marks | Guidance |
|---|---|---|
| M1 | Multiplies the three given roots together and sets the result equal to \(15\) or \(-15\) | 1.1b |
| A1 | Obtains a correct equation in \(a\) | 1.1b |
| M1 | Forms a quadratic equation in \(a\) and attempts to solve this equation by either completing the square or using the quadratic formula to give \(a = \ldots\) | 3.1a |
| A1 | \(a = 2 \pm i\) | 1.1b |
| A1 | Deduces the roots are \(2+i\), \(2-i\) and \(3\) | 2.2a |
| Answer | Marks | Guidance |
|---|---|---|
| M1 | Applies the process of finding sum of their three roots found in part (a) to give \(p = \ldots\) | 3.1a |
| A1 | \(p = -7\) by correct solution only (cso) | 1.1b |
| Answer | Marks | Guidance |
|---|---|---|
| M1 | Applies the process expanding \((z-3)(z^2 - \text{(their sum)}z + \text{their product})\) in order to find \(p = \ldots\) | 3.1a |
| A1 | \(p = -7\) by correct solution only (cso) | 1.1b |
# Question 1
## 1(a)
M1 | Multiplies the three given roots together and sets the result equal to $15$ or $-15$ | 1.1b
A1 | Obtains a correct equation in $a$ | 1.1b
M1 | Forms a quadratic equation in $a$ and attempts to solve this equation by either completing the square or using the quadratic formula to give $a = \ldots$ | 3.1a
A1 | $a = 2 \pm i$ | 1.1b
A1 | Deduces the roots are $2+i$, $2-i$ and $3$ | 2.2a
## 1(b)
M1 | Applies the process of finding sum of their three roots found in part (a) to give $p = \ldots$ | 3.1a
A1 | $p = -7$ by correct solution only (cso) | 1.1b
## 1(b) Alternative
M1 | Applies the process expanding $(z-3)(z^2 - \text{(their sum)}z + \text{their product})$ in order to find $p = \ldots$ | 3.1a
A1 | $p = -7$ by correct solution only (cso) | 1.1b
**Total: 7 marks**1.
$$f ( z ) = z ^ { 3 } + p z ^ { 2 } + q z - 15$$
where $p$ and $q$ are real constants.\\
Given that the equation $\mathrm { f } ( \mathrm { z } ) = 0$ has roots
$$\alpha , \frac { 5 } { \alpha } \text { and } \left( \alpha + \frac { 5 } { \alpha } - 1 \right)$$
\begin{enumerate}[label=(\alph*)]
\item solve completely the equation $\mathrm { f } ( \mathrm { z } ) = 0$
\item Hence find the value of $p$.
\end{enumerate}
\hfill \mbox{\textit{Edexcel CP AS Q1 [7]}}