| Exam Board | OCR |
|---|---|
| Module | FP1 (Further Pure Mathematics 1) |
| Year | 2007 |
| Session | June |
| Marks | 11 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Complex Numbers Arithmetic |
| Type | Square roots with follow-up application |
| Difficulty | Standard +0.3 This is a standard Further Maths FP1 question testing routine techniques: finding square roots of complex numbers algebraically (setting (a+bi)²=16+30i and solving) and applying the quadratic formula. While it requires more steps than basic A-level questions and involves complex numbers (a Further Maths topic), both parts follow well-established procedures with no novel insight required, making it slightly easier than average overall. |
| Spec | 4.02h Square roots: of complex numbers4.02i Quadratic equations: with complex roots |
| Answer | Marks | Guidance |
|---|---|---|
| (i) \(x^2 - y^2 = 16\) and \(xy = 15\) | M1 A1A1 | Attempt to equate real and imaginary parts of \((x + iy)^2\) and \(16 + 30i\). Obtain each result. |
| \(\pm(5 + 3i)\) | M1 M1 A1 | Eliminate to obtain a quadratic in \(x^2\) or \(y^2\). Solve to obtain \(x = (\pm) 5\) or \(y = (\pm) 3\). Obtain correct answers as complex numbers. |
| (ii) \(z = 1 \pm \sqrt{16 + 30i}\) | M1* | Use quadratic formula or complete the square. |
| \(6 + 3i, -4 - 3i\) | A1 *M1dep A1 A1ft | Simplify to this stage. Use answers from (i). Obtain correct answers. |
(i) $x^2 - y^2 = 16$ and $xy = 15$ | M1 A1A1 | Attempt to equate real and imaginary parts of $(x + iy)^2$ and $16 + 30i$. Obtain each result. |
$\pm(5 + 3i)$ | M1 M1 A1 | Eliminate to obtain a quadratic in $x^2$ or $y^2$. Solve to obtain $x = (\pm) 5$ or $y = (\pm) 3$. Obtain correct answers as complex numbers. |
(ii) $z = 1 \pm \sqrt{16 + 30i}$ | M1* | Use quadratic formula or complete the square. |
$6 + 3i, -4 - 3i$ | A1 *M1dep A1 A1ft | Simplify to this stage. Use answers from (i). Obtain correct answers. |
**Total: 6 + 5 = 11 marks**10 (i) Use an algebraic method to find the square roots of the complex number $16 + 30 \mathrm { i }$.\\
(ii) Use your answers to part (i) to solve the equation $z ^ { 2 } - 2 z - ( 15 + 30 \mathrm { i } ) = 0$, giving your answers in the form $x + \mathrm { i } y$.
\hfill \mbox{\textit{OCR FP1 2007 Q10 [11]}}