| Exam Board | OCR MEI |
|---|---|
| Module | FP1 (Further Pure Mathematics 1) |
| Year | 2011 |
| Session | June |
| Marks | 11 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Complex Numbers Arithmetic |
| Type | Given two complex roots, find all roots |
| Difficulty | Standard +0.3 This is a standard Further Pure 1 question testing the fundamental principle that complex roots of polynomials with real coefficients occur in conjugate pairs. Part (i) requires simple reasoning (4 roots needed, not 3), part (ii) involves routine expansion of factors, and part (iii) is straightforward plotting and finding a circle through four points. While it's Further Maths content, it's a textbook application with no novel insight required, making it slightly easier than average overall. |
| Spec | 4.02g Conjugate pairs: real coefficient polynomials4.02j Cubic/quartic equations: conjugate pairs and factor theorem4.02k Argand diagrams: geometric interpretation |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| Because a cubic can only have a maximum of two complex roots, which must form a conjugate pair. | E1 | |
| [1] |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| \(2+j,\ -1-2j\) | B1, B1 | |
| \(P(z) = (z-(2-j))(z-(2+j))(z-(-1+2j))(z-(-1-2j))\) | M1 | Use of factor theorem |
| \(= ((z-2)^2+1)((z+1)^2+4)\) | M1 | Attempt to multiply out factors |
| \(= (z^2-4z+5)(z^2+2z+5)\) | ||
| \(= z^4 - 2z^3 + 2z^2 - 10z + 25\) | A4 | \(-1\) for each incorrect coefficient |
| OR using root relationships: | ||
| \(\alpha+\beta+\gamma+\delta = 2 \Rightarrow a = -2\) | M2 | M1 for attempt to use all 4 root relationships; M2 for all correct |
| \(\alpha\beta+\alpha\gamma+\alpha\delta+\beta\gamma+\beta\delta+\gamma\delta = 2 \Rightarrow b = 2\) | B1 | \(a = -2\) |
| \(\alpha\beta\gamma+\alpha\beta\delta+\alpha\gamma\delta+\beta\gamma\delta = 10 \Rightarrow c = -10\) | A3 | \(b, c, d\) correct \(-1\) for each incorrect |
| \(\alpha\beta\gamma\delta = 25 \Rightarrow d = 25\) | \(-1\) for \(P(z)\) not explicit, following A4 or B1A3 | |
| \(\Rightarrow P(z) = z^4 - 2z^3 + 2z^2 - 10z + 25\) | ||
| [8] |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| Argand diagram with all four roots correctly plotted and labelled | B1 | All correct with annotation on axes or labels |
| \( | z | = \sqrt{5}\) |
| [2] |
# Question 8(i):
| Answer | Mark | Guidance |
|--------|------|----------|
| Because a cubic can only have a maximum of two complex roots, which must form a conjugate pair. | E1 | |
| | **[1]** | |
---
# Question 8(ii):
| Answer | Mark | Guidance |
|--------|------|----------|
| $2+j,\ -1-2j$ | B1, B1 | |
| $P(z) = (z-(2-j))(z-(2+j))(z-(-1+2j))(z-(-1-2j))$ | M1 | Use of factor theorem |
| $= ((z-2)^2+1)((z+1)^2+4)$ | M1 | Attempt to multiply out factors |
| $= (z^2-4z+5)(z^2+2z+5)$ | | |
| $= z^4 - 2z^3 + 2z^2 - 10z + 25$ | A4 | $-1$ for each incorrect coefficient |
| **OR** using root relationships: | | |
| $\alpha+\beta+\gamma+\delta = 2 \Rightarrow a = -2$ | M2 | M1 for attempt to use all 4 root relationships; M2 for all correct |
| $\alpha\beta+\alpha\gamma+\alpha\delta+\beta\gamma+\beta\delta+\gamma\delta = 2 \Rightarrow b = 2$ | B1 | $a = -2$ |
| $\alpha\beta\gamma+\alpha\beta\delta+\alpha\gamma\delta+\beta\gamma\delta = 10 \Rightarrow c = -10$ | A3 | $b, c, d$ correct $-1$ for each incorrect |
| $\alpha\beta\gamma\delta = 25 \Rightarrow d = 25$ | | $-1$ for $P(z)$ not explicit, following A4 or B1A3 |
| $\Rightarrow P(z) = z^4 - 2z^3 + 2z^2 - 10z + 25$ | | |
| | **[8]** | |
---
# Question 8(iii):
| Answer | Mark | Guidance |
|--------|------|----------|
| Argand diagram with all four roots correctly plotted and labelled | B1 | All correct with annotation on axes or labels |
| $|z| = \sqrt{5}$ | B1 | |
| | **[2]** | |
---8 A polynomial $\mathrm { P } ( z )$ has real coefficients. Two of the roots of $\mathrm { P } ( z ) = 0$ are $2 - \mathrm { j }$ and $- 1 + 2 \mathrm { j }$.\\
(i) Explain why $\mathrm { P } ( z )$ cannot be a cubic.
You are given that $\mathrm { P } ( z )$ is a quartic.\\
(ii) Write down the other roots of $\mathrm { P } ( z ) = 0$ and hence find $\mathrm { P } ( z )$ in the form $z ^ { 4 } + a z ^ { 3 } + b z ^ { 2 } + c z + d$.\\
(iii) Show the roots of $\mathrm { P } ( z ) = 0$ on an Argand diagram and give, in terms of $z$, the equation of the circle they lie on.
\hfill \mbox{\textit{OCR MEI FP1 2011 Q8 [11]}}