| Exam Board | OCR |
|---|---|
| Module | FP1 (Further Pure Mathematics 1) |
| Year | 2006 |
| Session | June |
| Marks | 5 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Complex Numbers Arithmetic |
| Type | Quadratic from one complex root |
| Difficulty | Easy -1.2 This is a straightforward application of standard complex number theory: non-real roots occur in conjugate pairs for real coefficient polynomials, and coefficients can be found via sum/product of roots. Requires only direct recall and basic arithmetic with complex numbers, making it easier than average even for Further Maths. |
| Spec | 4.02g Conjugate pairs: real coefficient polynomials |
| Answer | Marks | Guidance |
|---|---|---|
| Part (i) \(2 + 3i\) | B1 | Conjugate seen |
| Part (ii) | M1 | Attempt to sum roots or consider \(x\) terms in expansion or substitute \(2 - 3i\) into equation and equate imaginary parts |
| A1, M1, A1 | Correct answer | |
| Part (iii) \(p = -4\) and \(q = 13\) | M1 | Attempt at product of roots or consider last term in expansion or consider real parts |
| A1, 4 | Correct answer |
**Part (i)** $2 + 3i$ | B1 | Conjugate seen
**Part (ii)** | M1 | Attempt to sum roots or consider $x$ terms in expansion or substitute $2 - 3i$ into equation and equate imaginary parts
| A1, M1, A1 | Correct answer
**Part (iii)** $p = -4$ and $q = 13$ | M1 | Attempt at product of roots or consider last term in expansion or consider real parts
| A1, 4 | Correct answer3 One root of the quadratic equation $x ^ { 2 } + p x + q = 0$, where $p$ and $q$ are real, is the complex number 2-3i.\\
(i) Write down the other root.\\
(ii) Find the values of $p$ and $q$.
\hfill \mbox{\textit{OCR FP1 2006 Q3 [5]}}