| Exam Board | OCR |
|---|---|
| Module | FP1 (Further Pure Mathematics 1) |
| Year | 2007 |
| Session | January |
| Marks | 7 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Complex Numbers Arithmetic |
| Type | Factored form to roots |
| Difficulty | Moderate -0.8 This is a straightforward Further Pure 1 question requiring routine verification of factorization, solving a quadratic using the formula (with complex roots), and plotting three points on an Argand diagram. While it involves complex numbers, all techniques are standard and mechanical with no problem-solving insight required. It's easier than average even for FP1 standards. |
| Spec | 4.02i Quadratic equations: with complex roots4.02k Argand diagrams: geometric interpretation |
| Answer | Marks | Guidance |
|---|---|---|
| (i) Show given answer correctly | B1 | (1 mark) |
| (ii) Attempt to solve quadratic equation or substitute \(x + iy\) and equate real and imaginary parts | M1 | |
| \(-1 \pm i\sqrt{3}\) | A1 A1 | Obtain answers as complex numbers, simplified |
| (iii) Correct root on \(x\) axis, co-ords. shown | B1 | |
| Other roots in 2nd and 3rd quadrants | B1 | |
| Correct lengths and angles or co-ordinates or complex numbers shown | B1 | (3 marks, 7 total) |
(i) Show given answer correctly | B1 | (1 mark)
(ii) Attempt to solve quadratic equation or substitute $x + iy$ and equate real and imaginary parts | M1 |
$-1 \pm i\sqrt{3}$ | A1 A1 | Obtain answers as complex numbers, simplified
(iii) Correct root on $x$ axis, co-ords. shown | B1 |
Other roots in 2nd and 3rd quadrants | B1 |
Correct lengths and angles or co-ordinates or complex numbers shown | B1 | (3 marks, 7 total)5 (i) Verify that $z ^ { 3 } - 8 = ( z - 2 ) \left( z ^ { 2 } + 2 z + 4 \right)$.\\
(ii) Solve the quadratic equation $z ^ { 2 } + 2 z + 4 = 0$, giving your answers exactly in the form $x + \mathrm { i } y$. Show clearly how you obtain your answers.\\
(iii) Show on an Argand diagram the roots of the cubic equation $z ^ { 3 } - 8 = 0$.
\hfill \mbox{\textit{OCR FP1 2007 Q5 [7]}}