| Exam Board | OCR |
|---|---|
| Module | FP1 (Further Pure Mathematics 1) |
| Year | 2016 |
| Session | June |
| Marks | 7 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Complex Numbers Arithmetic |
| Type | Modulus-argument form conversions |
| Difficulty | Standard +0.3 This is a straightforward Further Maths FP1 question requiring conversion from modulus-argument form to Cartesian form using standard formulas, followed by algebraic manipulation with complex conjugates. While it involves multiple steps and exact values, the techniques are routine for FP1 students with no novel problem-solving required. Slightly above average difficulty due to the algebraic manipulation in part (ii), but still a standard textbook exercise. |
| Spec | 4.02b Express complex numbers: cartesian and modulus-argument forms4.02d Exponential form: re^(i*theta) |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \(x = 2\sqrt{3}\cos(-\frac{\pi}{3})\), \(y = 2\sqrt{3}\sin(-\frac{\pi}{3})\), \(x^2+y^2=12\), \(y=-x\sqrt{3}\) | M1 | Correct trig expression for \(x\) or \(y\), allow positive angle or 2 equations for \(x\) and \(y\), not involving trig |
| \(\sqrt{3} - 3\text{i}\) | A1 | Obtain correct answer as a complex number; extra answers not rejected gets A0 |
| [2] |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \(\sqrt{3} + 3\text{i}\) | B1ft | Correct conjugate seen, ft from their \(z\) in (i) |
| \(-1-(4\sqrt{3})\text{i}\) | M1 | Expand denominator |
| A1 | Correct value seen | |
| \(-\frac{1}{49}+\frac{4\sqrt{3}}{49}\text{i}\) | M1 | Attempt to rationalise |
| A1 | Obtain correct answer a.e.e.f. | |
| [5] | N.B. if 2 answers given in (i) award marks for better solution in (ii) |
## Question 2:
### Part (i):
| Answer | Marks | Guidance |
|--------|-------|----------|
| $x = 2\sqrt{3}\cos(-\frac{\pi}{3})$, $y = 2\sqrt{3}\sin(-\frac{\pi}{3})$, $x^2+y^2=12$, $y=-x\sqrt{3}$ | M1 | Correct trig expression for $x$ or $y$, allow positive angle or 2 equations for $x$ and $y$, not involving trig |
| $\sqrt{3} - 3\text{i}$ | A1 | Obtain correct answer as a complex number; extra answers not rejected gets A0 |
| **[2]** | | |
### Part (ii):
| Answer | Marks | Guidance |
|--------|-------|----------|
| $\sqrt{3} + 3\text{i}$ | B1ft | Correct conjugate seen, ft from their $z$ in (i) |
| $-1-(4\sqrt{3})\text{i}$ | M1 | Expand denominator |
| | A1 | Correct value seen |
| $-\frac{1}{49}+\frac{4\sqrt{3}}{49}\text{i}$ | M1 | Attempt to rationalise |
| | A1 | Obtain correct answer a.e.e.f. |
| **[5]** | | N.B. if 2 answers given in (i) award marks for better solution in (ii) |
---2 The complex number $z$ has modulus $2 \sqrt { 3 }$ and argument $- \frac { 1 } { 3 } \pi$. Giving your answers in the form $x + \mathrm { i } y$, where $x$ and $y$ are exact real numbers, and showing clearly how you obtain them, find\\
(i) $z$,\\
(ii) $\frac { 1 } { \left( z ^ { * } - 5 \mathrm { i } \right) ^ { 2 } }$.
\hfill \mbox{\textit{OCR FP1 2016 Q2 [7]}}