| Exam Board | OCR |
|---|---|
| Module | FP1 (Further Pure Mathematics 1) |
| Year | 2015 |
| Session | June |
| Marks | 10 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Complex Numbers Arithmetic |
| Type | Square roots with follow-up application |
| Difficulty | Standard +0.3 This is a standard Further Maths FP1 question testing the routine algebraic method for finding square roots of complex numbers (equating real and imaginary parts of (a+bi)²) followed by a straightforward application of the quadratic formula. While it requires more steps than basic A-level questions and involves complex numbers (a Further Maths topic), it follows a well-practiced algorithmic procedure with no novel insight required, making it slightly easier than average overall. |
| Spec | 4.02h Square roots: of complex numbers |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| M1 | Attempt to equate real and imaginary parts of \((x+iy)^2\) and \(5+12i\) | |
| \(x^2 - y^2 = 5,\ 2xyi = 12i\) | A1 | Obtain both results or equivalent |
| M1 | Obtain and solve a quadratic in \(x^2\) or \(y^2\) or solve by inspection | |
| \(3+2i\) and \(-3-2i\) or \(\pm(3+2i)\) | A1 A1 | Obtain correct answers as complex numbers |
| [5] | S.C. \(\pm(3 \pm 2i)\) scores A1 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \(\frac{(4 \pm 2\sqrt{5+12i})}{2}\) | M1 | Solve using quadratic formula or complete square |
| A1 | Obtain correct answers, or simpler version | |
| M1 | Use result(s) from (i) | |
| \(5+2i\) and \(-1-2i\) or \(2\pm(3+2i)\) | A1 A1 | Obtain correct answers; If more than 2 roots A0 A0 |
| [5] |
## Question 7(i):
| Answer | Marks | Guidance |
|--------|-------|----------|
| | M1 | Attempt to equate real and imaginary parts of $(x+iy)^2$ and $5+12i$ |
| $x^2 - y^2 = 5,\ 2xyi = 12i$ | A1 | Obtain both results or equivalent |
| | M1 | Obtain and solve a quadratic in $x^2$ or $y^2$ or solve by inspection |
| $3+2i$ and $-3-2i$ or $\pm(3+2i)$ | A1 A1 | Obtain correct answers as complex numbers |
| **[5]** | **S.C. $\pm(3 \pm 2i)$ scores A1** | |
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## Question 7(ii):
| Answer | Marks | Guidance |
|--------|-------|----------|
| $\frac{(4 \pm 2\sqrt{5+12i})}{2}$ | M1 | Solve using quadratic formula or complete square |
| | A1 | Obtain correct answers, or simpler version |
| | M1 | Use result(s) from (i) |
| $5+2i$ and $-1-2i$ or $2\pm(3+2i)$ | A1 A1 | Obtain correct answers; If more than 2 roots A0 A0 |
| **[5]** | | |
---7 (i) Use an algebraic method to find the square roots of the complex number $5 + 12 \mathrm { i }$. You must show sufficient working to justify your answers.\\
(ii) Hence solve the quadratic equation $x ^ { 2 } - 4 x - 1 - 12 \mathrm { i } = 0$.
\hfill \mbox{\textit{OCR FP1 2015 Q7 [10]}}