| Exam Board | OCR |
|---|---|
| Module | FP1 (Further Pure Mathematics 1) |
| Year | 2010 |
| Session | January |
| Marks | 7 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Roots of polynomials |
| Type | Complex roots with real coefficients |
| Difficulty | Standard +0.3 This is a standard Further Maths question testing the fundamental theorem that complex roots come in conjugate pairs for polynomials with real coefficients. Students must identify that 5+i is also a root, use Vieta's formulas or expand (x-(5-i))(x-(5+i)) to find the quadratic factor, then determine the real root and coefficients. While it requires multiple steps, the approach is algorithmic and commonly practiced in FP1. |
| Spec | 4.02g Conjugate pairs: real coefficient polynomials4.05a Roots and coefficients: symmetric functions |
| Answer | Marks | Guidance |
|---|---|---|
| \(x = -2\) | B1 | State or use \(5 + i\) as a root |
| M1 | Use \(\sum \alpha\beta = 6\) | |
| A1, 3 | Obtain correct answer |
| Answer | Marks | Guidance |
|---|---|---|
| \(q = 52\) | M1 | Use \(p = -\sum \alpha\) |
| A1ft | Obtain correct answer, from their root | |
| M1 | Use \(q = -\alpha\beta\gamma\) | |
| A1ft, 4 | Obtain correct answer, from their root |
| Answer | Marks |
|---|---|
| M1 | Attempt to find quadratic factor |
| M1 | Attempt to expand quadratic and linear terms |
| A1A1 | Obtain correct answers |
| Answer | Marks |
|---|---|
| M1 | Substitute \((5-i)\) into equation |
| M1 | Equate real and imaginary parts |
| A1 | Obtain correct answer for \(p\) |
| A1ft | Obtain correct answer for \(q\), fit their \(p\) |
**Part (i)**
$x = -2$ | B1 | State or use $5 + i$ as a root
| M1 | Use $\sum \alpha\beta = 6$
| A1, 3 | Obtain correct answer
---
**Part (ii)**
Either:
$p = -8$
$q = 52$ | M1 | Use $p = -\sum \alpha$
| A1ft | Obtain correct answer, from their root
| M1 | Use $q = -\alpha\beta\gamma$
| A1ft, 4 | Obtain correct answer, from their root
Or:
| M1 | Attempt to find quadratic factor
| M1 | Attempt to expand quadratic and linear terms
| A1A1 | Obtain correct answers
Or:
| M1 | Substitute $(5-i)$ into equation
| M1 | Equate real and imaginary parts
| A1 | Obtain correct answer for $p$
| A1ft | Obtain correct answer for $q$, fit their $p$
---6 One root of the cubic equation $x ^ { 3 } + p x ^ { 2 } + 6 x + q = 0$, where $p$ and $q$ are real, is the complex number 5-i.\\
(i) Find the real root of the cubic equation.\\
(ii) Find the values of $p$ and $q$.
\hfill \mbox{\textit{OCR FP1 2010 Q6 [7]}}