| Exam Board | Edexcel |
|---|---|
| Module | CP AS (Core Pure AS) |
| Year | 2023 |
| Session | June |
| Marks | 7 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Roots of polynomials |
| Type | Complex roots with real coefficients |
| Difficulty | Standard +0.3 This is a standard Core Pure question testing the conjugate root theorem and basic polynomial manipulation. Part (a) requires using the fact that complex roots come in conjugate pairs for real coefficients, then finding the third root and using Vieta's formulas or expansion. Parts (b) and (c) are routine applications. While it requires multiple steps, all techniques are standard textbook exercises with no novel insight needed, making it slightly easier than average. |
| Spec | 4.02g Conjugate pairs: real coefficient polynomials4.02k Argand diagrams: geometric interpretation |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| \(z^* = -3 - 4i\); uses \((z-(-3+4i))(z-(-3-4i)) = z^2 + pz + q\); writes \(f(z) = (z^2+pz+q)(z+r)\) | M1 | 3.1a |
| \((z^2+6z+25)(z+7)\) | A1 | 1.1b |
| Multiplies out \((z^2+6z+25)(z+7) = \alpha z^2 + \beta z \ldots\) | M1 | 1.1b |
| \(z^3+13z^2+67z+175\) or \(a=13, b=67\) | A1 | 1.1b |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| Uses complex conjugate and product of roots \(= -175\) to find third root | M1 | 3.1a |
| Third root \(= -7\) | A1 | 1.1b |
| Uses sum of roots \(= -a\) to find \(a\) or uses pair sum \(= b\) to find \(b\); or expands \((z-(-3+4i))(z-(-3-4i))(z-\text{their third root})\) | M1 | 1.1b |
| \(a=13, b=67\) | A1 | 1.1b |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| \((-3+4i)^3 + a(-3+4i)^2 + b(-3+4i)+175=0 \Rightarrow 117+44i+a(-7-24i)+b(-3+4i)+175=0\); equates real and imaginary parts to form two linear simultaneous equations | M1 | 3.1a |
| \(117-7a-3b+175=0 \Rightarrow -7a-3b=-292\); \(44-24a+4b=0 \Rightarrow -24a+4b=-44\) | A1 | 1.1b |
| Solves simultaneously to find values for \(a\) or \(b\) | M1 | 1.1b |
| \(a=13, b=67\) | A1 | 1.1b |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| Correctly plotting \(-3+4i,\ -3-4i\) | B1 | 1.1b |
| Correctly plotting \(-7\) | B1 | 2.2a |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| \(-5+4i,\ -5-4i,\ -9\) | B1ft | 2.2a — ft: \(-5+4i\), \(-5-4i\) and subtracts 2 from real root shown on Argand diagram |
## Question 2(a):
| Answer/Working | Mark | Guidance |
|---|---|---|
| $z^* = -3 - 4i$; uses $(z-(-3+4i))(z-(-3-4i)) = z^2 + pz + q$; writes $f(z) = (z^2+pz+q)(z+r)$ | M1 | 3.1a |
| $(z^2+6z+25)(z+7)$ | A1 | 1.1b |
| Multiplies out $(z^2+6z+25)(z+7) = \alpha z^2 + \beta z \ldots$ | M1 | 1.1b |
| $z^3+13z^2+67z+175$ or $a=13, b=67$ | A1 | 1.1b |
**Alternative 1:**
| Answer/Working | Mark | Guidance |
|---|---|---|
| Uses complex conjugate and product of roots $= -175$ to find third root | M1 | 3.1a |
| Third root $= -7$ | A1 | 1.1b |
| Uses sum of roots $= -a$ to find $a$ **or** uses pair sum $= b$ to find $b$; **or** expands $(z-(-3+4i))(z-(-3-4i))(z-\text{their third root})$ | M1 | 1.1b |
| $a=13, b=67$ | A1 | 1.1b |
**Alternative 2:**
| Answer/Working | Mark | Guidance |
|---|---|---|
| $(-3+4i)^3 + a(-3+4i)^2 + b(-3+4i)+175=0 \Rightarrow 117+44i+a(-7-24i)+b(-3+4i)+175=0$; equates real and imaginary parts to form two linear simultaneous equations | M1 | 3.1a |
| $117-7a-3b+175=0 \Rightarrow -7a-3b=-292$; $44-24a+4b=0 \Rightarrow -24a+4b=-44$ | A1 | 1.1b |
| Solves simultaneously to find values for $a$ or $b$ | M1 | 1.1b |
| $a=13, b=67$ | A1 | 1.1b |
---
## Question 2(b):
| Answer/Working | Mark | Guidance |
|---|---|---|
| Correctly plotting $-3+4i,\ -3-4i$ | B1 | 1.1b |
| Correctly plotting $-7$ | B1 | 2.2a |
---
## Question 2(c):
| Answer/Working | Mark | Guidance |
|---|---|---|
| $-5+4i,\ -5-4i,\ -9$ | B1ft | 2.2a — ft: $-5+4i$, $-5-4i$ and subtracts 2 from real root shown on Argand diagram |
---\begin{enumerate}
\item $\mathrm { f } ( \mathrm { z } ) = \mathrm { z } ^ { 3 } + a \mathrm { z } ^ { 2 } + b \mathrm { z } + 175 \quad$ where $a$ and $b$ are real constants
\end{enumerate}
Given that $- 3 + 4 \mathrm { i }$ is a root of the equation $\mathrm { f } ( \mathrm { z } ) = 0$\\
(a) determine the value of $a$ and the value of $b$.\\
(b) Show all the roots of the equation $\mathrm { f } ( \mathrm { z } ) = 0$ on a single Argand diagram.\\
(c) Write down the roots of the equation $\mathrm { f } ( \mathrm { z } + 2 ) = 0$
\hfill \mbox{\textit{Edexcel CP AS 2023 Q2 [7]}}