| Exam Board | Edexcel |
|---|---|
| Module | CP1 (Core Pure 1) |
| Year | 2022 |
| Session | June |
| Marks | 6 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Complex Numbers Arithmetic |
| Type | Given one complex root of cubic or quartic, find all roots |
| Difficulty | Moderate -0.3 This is a standard Core Pure 1 question testing routine knowledge that complex roots come in conjugate pairs for polynomials with real coefficients. Part (a) requires simple recall, parts (b)(i)-(ii) involve straightforward algebraic manipulation using Vieta's formulas or expansion, and part (c) is basic plotting. While multi-part, each step follows a well-practiced procedure with no novel problem-solving required, making it slightly easier than average. |
| Spec | 4.02g Conjugate pairs: real coefficient polynomials4.02k Argand diagrams: geometric interpretation |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| \(2 + 3i\) | B1 | AO 1.1b |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| \(z^* = 2+3i\) so \(z + z^* = 4\), \(zz^* = 13\); using \(z + z^* + \alpha = 0 \Rightarrow \alpha = \ldots\) or \(\alpha zz^* = -52 \Rightarrow \alpha = -\frac{52}{13} = \ldots\) or forming \((z-(2+3i))(z-(2-3i)) = \ldots\) leading to \((z^2 - 4z + 13)(z+4) \Rightarrow z = \ldots\) | M1 | AO 3.1a |
| \(z = 2 \pm 3i, \ -4\) | A1 | AO 1.1b |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| \((z^2 - 4z + 13)(z+4)\) expand to find value of \(a\); or \(a =\) pair sum \(= -4(2+3i+2-3i)+13 = \ldots\); or \(f(-4)/f(2\pm 3i) = 0 \Rightarrow \ldots \Rightarrow a = \ldots\) | M1 | AO 1.1b |
| \(a = -3\) | A1 | AO 2.2a |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| Argand diagram showing three points: \(2+3i\), \(2-3i\), and \(-4\) plotted correctly | B1ft | AO 1.1b |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| \(2 + 3i\) | B1 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| Complete method to find third root | M1 | Forms quadratic factor and uses to find linear factor; or uses sum of roots \(= 0\) or product of roots \(= \pm 52\) with complex roots to find third; factor theorem also valid |
| All three roots clearly stated | A1 | All three must be clearly stated in (b), not just seen on diagram in part (c) |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| Complete method to find value of \(a\) | M1 | Multiplies out quadratic and linear factors to find coefficient of \(z\); or uses pair sum; or uses factor theorem with one of the roots |
| Correct value of \(a\) deduced | A1 | May be seen as the \(z\) coefficient in the cubic; need not be extracted but if it is must be correct |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| Correctly plots all three roots | B1ft | \(-4\) must be further from \(O\) than 2; two complex roots symmetric about real axis in quadrants 1 and 4; accept \((0, -4)\) labelled on real axis in correct place |
## Question 1:
### Part (a)
| Answer | Mark | Guidance |
|--------|------|----------|
| $2 + 3i$ | B1 | AO 1.1b |
**Total: (1)**
---
### Part (b)(i)
| Answer | Mark | Guidance |
|--------|------|----------|
| $z^* = 2+3i$ so $z + z^* = 4$, $zz^* = 13$; using $z + z^* + \alpha = 0 \Rightarrow \alpha = \ldots$ or $\alpha zz^* = -52 \Rightarrow \alpha = -\frac{52}{13} = \ldots$ or forming $(z-(2+3i))(z-(2-3i)) = \ldots$ leading to $(z^2 - 4z + 13)(z+4) \Rightarrow z = \ldots$ | M1 | AO 3.1a |
| $z = 2 \pm 3i, \ -4$ | A1 | AO 1.1b |
### Part (b)(ii)
| Answer | Mark | Guidance |
|--------|------|----------|
| $(z^2 - 4z + 13)(z+4)$ expand to find value of $a$; or $a =$ pair sum $= -4(2+3i+2-3i)+13 = \ldots$; or $f(-4)/f(2\pm 3i) = 0 \Rightarrow \ldots \Rightarrow a = \ldots$ | M1 | AO 1.1b |
| $a = -3$ | A1 | AO 2.2a |
**Total: (4)**
---
### Part (c)
| Answer | Mark | Guidance |
|--------|------|----------|
| Argand diagram showing three points: $2+3i$, $2-3i$, and $-4$ plotted correctly | B1ft | AO 1.1b |
**Total: (1)**
---
**Overall total: (6 marks)**
# Question 1:
## Part (a)
| Answer | Mark | Guidance |
|--------|------|----------|
| $2 + 3i$ | B1 | |
## Part (b)(i)
| Answer | Mark | Guidance |
|--------|------|----------|
| Complete method to find third root | M1 | Forms quadratic factor and uses to find linear factor; or uses sum of roots $= 0$ or product of roots $= \pm 52$ with complex roots to find third; factor theorem also valid |
| All three roots clearly stated | A1 | All three must be clearly stated in (b), not just seen on diagram in part (c) |
## Part (b)(ii)
| Answer | Mark | Guidance |
|--------|------|----------|
| Complete method to find value of $a$ | M1 | Multiplies out quadratic and linear factors to find coefficient of $z$; or uses pair sum; or uses factor theorem with one of the roots |
| Correct value of $a$ deduced | A1 | May be seen as the $z$ coefficient in the cubic; need not be extracted but if it is must be correct |
## Part (c)
| Answer | Mark | Guidance |
|--------|------|----------|
| Correctly plots all three roots | B1ft | $-4$ must be further from $O$ than 2; two complex roots symmetric about real axis in quadrants 1 and 4; accept $(0, -4)$ labelled on real axis in correct place |
---1.
$$\mathrm { f } ( \mathrm { z } ) = \mathrm { z } ^ { 3 } + a \mathrm { z } + 52 \quad \text { where } a \text { is a real constant }$$
Given that $2 - 3 \mathrm { i }$ is a root of the equation $\mathrm { f } ( \mathrm { z } ) = 0$
\begin{enumerate}[label=(\alph*)]
\item write down the other complex root.
\item Hence
\begin{enumerate}[label=(\roman*)]
\item solve completely $\mathrm { f } ( \mathrm { z } ) = 0$
\item determine the value of $a$
\end{enumerate}\item Show all the roots of the equation $\mathrm { f } ( \mathrm { z } ) = 0$ on a single Argand diagram.
\end{enumerate}
\hfill \mbox{\textit{Edexcel CP1 2022 Q1 [6]}}