| Exam Board | Edexcel |
|---|---|
| Module | CP2 (Core Pure 2) |
| Year | 2023 |
| Session | June |
| Marks | 11 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Complex Numbers Argand & Loci |
| Type | Roots of polynomial equations |
| Difficulty | Challenging +1.2 This is a multi-part question requiring understanding that complex roots of real polynomials come in conjugate pairs, so three distinct roots on a line must lie on the real axis (part a), or if one is non-real, the line passes through conjugate pairs. Part (b) is straightforward application of conjugate root theorem. Parts (c) and (d) involve logical deduction about shared roots and basic polynomial manipulation. While lengthy, each step follows standard A-level techniques without requiring novel insight or difficult problem-solving. |
| Spec | 4.02a Complex numbers: real/imaginary parts, modulus, argument4.02g Conjugate pairs: real coefficient polynomials |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| The real axis / horizontal line through \((0,0)\) / line \(y=0\) | B1 | One correct line described |
| Other possibility: all three roots have same real part, lying on vertical line/perpendicular to real axis, \(x=k\) where \(k\) is real | B1 | Two correct lines described. Special case: "any line is possible" scores B1B1 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| Other roots are \(\frac{3}{2}\) and \(\frac{3}{2}-\frac{3}{2}i\) | B1 | Interprets conclusion from (a) by identifying the correct two roots |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| Common root must be \(\frac{3}{2}\) | B1 | Deduces the real root is the one in common |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| Sets product of roots \(= -12\) using \(\frac{3}{2}\times -4\times\alpha = -12\), or forms \(g(z)=\left(z-\frac{3}{2}\right)(z+4)(z-\alpha)\) | M1 | Sets product of roots \(=-12\). Alternatively forms equation for \(g(z)\) using the roots |
| Solves to find third root: \(\frac{3}{2}\times -4\times\alpha=-12 \Rightarrow \alpha=2\) | M1 | Solves their equation. Condone use of 12. Alternatively multiply constant and set \(=12\) |
| \(\alpha = 2\) | A1 | Correct third root |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| \(8\left\{z-\frac{3}{2}\right\}\left(z-\frac{3}{2}-\frac{3}{2}i\right)\left(z-\frac{3}{2}+\frac{3}{2}i\right) = 8\left\{z-\frac{3}{2}\right\}\left(z^2-3z+\frac{9}{2}\right)\), giving \(f(z)=8z^3-36z^2+72z-54\) | M1 | Uses roots of \(f(z)\) to form cubic, expands to at least linear term times quadratic. Must have factor of 8 |
| Sets \(f(z)=g(z)\): \(8z^3-36z^2+72z-54 = \left(z-\frac{3}{2}\right)(z+4)(z-2)\), leading to \(7z^2-26z+44=0\) or \(7z^3-\frac{73}{2}z^2+83z-66=0\) | M1 | Sets expressions equal, cancels/factors common term to achieve quadratic in \(z\) |
| Solutions: \(z=\frac{3}{2},\ \dfrac{13\pm i\sqrt{139}}{7}\) | A1 | All three correct solutions. Note: decimals \(\frac{13}{7}\pm 1.68...i\) is A0 |
## Question 8:
### Part (a):
| Answer/Working | Mark | Guidance |
|---|---|---|
| The real axis / horizontal line through $(0,0)$ / line $y=0$ | B1 | One correct line described |
| Other possibility: all three roots have same real part, lying on vertical line/perpendicular to real axis, $x=k$ where $k$ is real | B1 | Two correct lines described. Special case: "any line is possible" scores B1B1 |
### Part (b):
| Answer/Working | Mark | Guidance |
|---|---|---|
| Other roots are $\frac{3}{2}$ and $\frac{3}{2}-\frac{3}{2}i$ | B1 | Interprets conclusion from (a) by identifying the correct two roots |
### Part (c)(i):
| Answer/Working | Mark | Guidance |
|---|---|---|
| Common root must be $\frac{3}{2}$ | B1 | Deduces the real root is the one in common |
### Part (c)(ii):
| Answer/Working | Mark | Guidance |
|---|---|---|
| Sets product of roots $= -12$ using $\frac{3}{2}\times -4\times\alpha = -12$, or forms $g(z)=\left(z-\frac{3}{2}\right)(z+4)(z-\alpha)$ | M1 | Sets product of roots $=-12$. Alternatively forms equation for $g(z)$ using the roots |
| Solves to find third root: $\frac{3}{2}\times -4\times\alpha=-12 \Rightarrow \alpha=2$ | M1 | Solves their equation. Condone use of 12. Alternatively multiply constant and set $=12$ |
| $\alpha = 2$ | A1 | Correct third root |
### Part (d):
| Answer/Working | Mark | Guidance |
|---|---|---|
| $8\left\{z-\frac{3}{2}\right\}\left(z-\frac{3}{2}-\frac{3}{2}i\right)\left(z-\frac{3}{2}+\frac{3}{2}i\right) = 8\left\{z-\frac{3}{2}\right\}\left(z^2-3z+\frac{9}{2}\right)$, giving $f(z)=8z^3-36z^2+72z-54$ | M1 | Uses roots of $f(z)$ to form cubic, expands to at least linear term times quadratic. Must have factor of 8 |
| Sets $f(z)=g(z)$: $8z^3-36z^2+72z-54 = \left(z-\frac{3}{2}\right)(z+4)(z-2)$, leading to $7z^2-26z+44=0$ or $7z^3-\frac{73}{2}z^2+83z-66=0$ | M1 | Sets expressions equal, cancels/factors common term to achieve quadratic in $z$ |
| Solutions: $z=\frac{3}{2},\ \dfrac{13\pm i\sqrt{139}}{7}$ | A1 | All three correct solutions. Note: decimals $\frac{13}{7}\pm 1.68...i$ is A0 |\begin{enumerate}
\item Given that a cubic equation has three distinct roots that all lie on the same straight line in the complex plane,\\
(a) describe the possible lines the roots can lie on.
\end{enumerate}
$$f ( z ) = 8 z ^ { 3 } + b z ^ { 2 } + c z + d$$
where $b , c$ and $d$ are real constants.\\
The roots of $f ( z )$ are distinct and lie on a straight line in the complex plane.\\
Given that one of the roots is $\frac { 3 } { 2 } + \frac { 3 } { 2 } \mathrm { i }$\\
(b) state the other two roots of $\mathrm { f } ( \mathrm { z } )$
$$g ( z ) = z ^ { 3 } + P z ^ { 2 } + Q z + 12$$
where $P$ and $Q$ are real constants, has 3 distinct roots.\\
The roots of $g ( z )$ lie on a different straight line in the complex plane than the roots of $\mathrm { f } ( \mathrm { z } )$
Given that
\begin{itemize}
\item $f ( z )$ and $g ( z )$ have one root in common
\item one of the roots of $\mathrm { g } ( \mathrm { z } )$ is - 4\\
(c) (i) write down the value of the common root,\\
(ii) determine the value of the other root of $\mathrm { g } ( \mathrm { z } )$\\
(d) Hence solve the equation $\mathrm { f } ( \mathrm { z } ) = \mathrm { g } ( \mathrm { z } )$
\end{itemize}
\hfill \mbox{\textit{Edexcel CP2 2023 Q8 [11]}}