| Exam Board | OCR |
|---|---|
| Module | FP1 (Further Pure Mathematics 1) |
| Year | 2008 |
| Session | June |
| Marks | 7 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Roots of polynomials |
| Type | Complex roots with real coefficients |
| Difficulty | Moderate -0.5 This is a straightforward application of the complex conjugate root theorem and Vieta's formulas. Students need to recognize that (3-i) must be the third root, then use sum and product of roots to find coefficients—routine for FP1 with no conceptual challenges beyond standard recall. |
| Spec | 4.02g Conjugate pairs: real coefficient polynomials4.05a Roots and coefficients: symmetric functions |
| Answer | Marks | Guidance |
|---|---|---|
| (i) \(3 - i\) | B1 | Conjugate stated |
| 1 mark | ||
| (ii) EITHER | M1 | Use sum of roots |
| A1 | Obtain correct answer | |
| M1 | Use sum of pairs of roots | |
| A1 | Obtain correct answer | |
| M1 | Use product of roots | |
| A1 | Obtain correct answers | |
| \(a = -8, b = 22, c = -20\) | ||
| OR | M1 | Attempt to find a quadratic factor |
| A1 | Obtain correct factor | |
| M1 | Expand linear and quadratic factors | |
| A1A1A1 | Obtain correct answers | |
| \(a = -8, b = 22, c = -20\) | ||
| OR | M1 | Substitute 1 imaginary & the real root into eqn |
| M1 | Equate real and imaginary parts | |
| M1 | Attempt to solve 3 eqns | |
| A1A1A1 | Obtain correct answers | |
| \(a = -8, b = 22, c = -20\) | 6 marks |
**(i)** $3 - i$ | B1 | Conjugate stated
| **1 mark**
**(ii)** EITHER | M1 | Use sum of roots
| A1 | Obtain correct answer
| M1 | Use sum of pairs of roots
| A1 | Obtain correct answer
| M1 | Use product of roots
| A1 | Obtain correct answers
$a = -8, b = 22, c = -20$ | |
OR | M1 | Attempt to find a quadratic factor
| A1 | Obtain correct factor
| M1 | Expand linear and quadratic factors
| A1A1A1 | Obtain correct answers
$a = -8, b = 22, c = -20$ | |
OR | M1 | Substitute 1 imaginary & the real root into eqn
| M1 | Equate real and imaginary parts
| M1 | Attempt to solve 3 eqns
| A1A1A1 | Obtain correct answers
$a = -8, b = 22, c = -20$ | **6 marks**6 The cubic equation $x ^ { 3 } + a x ^ { 2 } + b x + c = 0$, where $a , b$ and $c$ are real, has roots ( $3 + \mathrm { i }$ ) and 2 .\\
(i) Write down the other root of the equation.\\
(ii) Find the values of $a , b$ and $c$.
\hfill \mbox{\textit{OCR FP1 2008 Q6 [7]}}