| Exam Board | OCR |
|---|---|
| Module | FP1 (Further Pure Mathematics 1) |
| Year | 2008 |
| Session | June |
| Marks | 11 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Complex Numbers Arithmetic |
| Type | Square roots with follow-up application |
| Difficulty | Standard +0.8 This is a Further Maths question requiring multiple techniques: algebraic method for complex square roots (equating real/imaginary parts), then recognizing the quartic as a quadratic in x², and finally applying the results from parts (i) and (ii). While systematic, it requires good algebraic manipulation and the insight to connect all three parts, placing it moderately above average difficulty. |
| Spec | 4.02e Arithmetic of complex numbers: add, subtract, multiply, divide4.02h Square roots: of complex numbers |
| Answer | Marks | Guidance |
|---|---|---|
| (i) \(x^2 - y^2 = 5\) and \(xy = 6\) | A1 | Obtain both results |
| M1 | Eliminate to obtain a quadratic in \(x^2\) or \(x^2\) | |
| M1 | Solve a 3 term quadratic & obtain \(x\) or \(y\) | |
| A1 | Obtain correct answers as complex nos. | |
| 5 marks | ||
| (ii) \(\pm(3 + 2i)\) | B1B1 | Correct real and imaginary parts |
| 2 marks | ||
| (iii) \(x^2 = 5 \pm 12i\) | M1 | Attempt to solve a quadratic equation |
| \(x = \pm(3 \pm 2i)\) | A1 | Obtain correct answers |
| A1A1 | Each pair of correct answers a.e.f. | |
| 4 marks |
**(i)** $x^2 - y^2 = 5$ and $xy = 6$ | A1 | Obtain both results
| M1 | Eliminate to obtain a quadratic in $x^2$ or $x^2$
| M1 | Solve a 3 term quadratic & obtain $x$ or $y$
| A1 | Obtain correct answers as complex nos.
| **5 marks**
**(ii)** $\pm(3 + 2i)$ | B1B1 | Correct real and imaginary parts
| **2 marks**
**(iii)** $x^2 = 5 \pm 12i$ | M1 | Attempt to solve a quadratic equation
$x = \pm(3 \pm 2i)$ | A1 | Obtain correct answers
| A1A1 | Each pair of correct answers a.e.f.
| **4 marks**9 (i) Use an algebraic method to find the square roots of the complex number $5 + 12 \mathrm { i }$.\\
(ii) Find $( 3 - 2 \mathrm { i } ) ^ { 2 }$.\\
(iii) Hence solve the quartic equation $x ^ { 4 } - 10 x ^ { 2 } + 169 = 0$.
\hfill \mbox{\textit{OCR FP1 2008 Q9 [11]}}