| Exam Board | Edexcel |
|---|---|
| Module | CP1 (Core Pure 1) |
| Year | 2019 |
| Session | June |
| Marks | 9 |
| 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 straightforward application of the complex conjugate root theorem for polynomials with real coefficients. Students must recognize that -1-2i and 3+i are also roots, plot all four roots, then expand (z-(-1+2i))(z-(-1-2i))(z-(3-i))(z-(3+i)) to find coefficients. While it requires multiple steps and careful algebra, it's a standard textbook exercise with no novel insight required—slightly easier than average due to its routine nature. |
| Spec | 4.02g Conjugate pairs: real coefficient polynomials4.02h Square roots: of complex numbers4.02j Cubic/quartic equations: conjugate pairs and factor theorem |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| \(z = -1-2i\) or \(z = 3+i\) | M1 | 1.2 - Identifies at least one correct complex conjugate as another root |
| \(z = -1-2i\) and \(z = 3+i\) | A1 | 1.1b - Both complex conjugate roots identified correctly |
| One complex conjugate pair correctly plotted on Argand diagram, e.g. \((-1,2)\) and \((-1,-2)\) | B1 | 1.1b - One complex conjugate pair correctly plotted |
| Both complex conjugate pairs correctly plotted: \((3,1)\), \((3,-1)\), \((-1,2)\), \((-1,-2)\). The \(3\pm i\) must be closer to real axis than \(-1\pm 2i\) | B1 | 1.1b - If no indication of coordinates, scale or complex numbers on Argand diagram this is B0 B0 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| \((z-(-1+2i))(z-(-1-2i)) = \ldots\) or \((z-(3+i))(z-(3-i)) = \ldots\) | M1 | 3.1a - Correct strategy for forming at least one quadratic factor |
| \(z^2+2z+5\) or \(z^2-6z+10\) | A1 | 1.1b - At least one correct simplified quadratic factor |
| \(z^2+2z+5\) and \(z^2-6z+10\) | A1 | 1.1b - Both simplified quadratic factors correct |
| \(f(z) = (z^2+2z+5)(z^2-6z+10)\), expands to form a quartic | M1 | 3.1a - Complete strategy to find values for \(a, b, c, d\) |
| \(f(z) = z^4-4z^3+3z^2-10z+50\) or states \(a=-4, b=3, c=-10, d=50\) | A1 | 1.1b |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| Sum roots \(= (-1+2i)+(-1-2i)+(3+i)+(3-i) = \ldots\); pair sum \(= \alpha\beta+\alpha\gamma+\alpha\delta+\beta\gamma+\beta\delta+\gamma\delta\); triple sum; product \(= \alpha\beta\gamma\delta = (-1+2i)(-1-2i)(3-i)(3+i) = \ldots\) | M1 | 3.1a - Correct strategy for finding at least three of sum, pair sum, triple sum and product |
| sum \(= 4\), pair sum \(= 3\), triple sum \(= 10\), product \(= 50\) | A1, A1 | 1.1b |
| \(a = -(\text{sum roots}) = -4\); \(b = +(\text{pair sum}) = 3\); \(c = -(\text{triple sum}) = -10\); \(d = +(\text{product}) = 50\) | M1, A1 | 3.1a, 1.1b |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| \(f(-1+2i) = (-1+2i)^4 + a(-1+2i)^3 + b(-1+2i)^2 + c(-1+2i) + d = 0\) and \(f(3+i) = (3+i)^4 + a(3+i)^3 + b(3+i)^2 + c(3+i) + d = 0\) | M1 | 3.1a - Substitutes two roots into \(f(z)=0\) and equates coefficients to form 4 equations |
| \(-7+11a-3b-c+d=0\) and \(24-2a-4b+2c=0\) | A1 | 1.1b - At least two correct equations |
| \(28+18a+8b+3c+d=0\) and \(96+26a+6b+c=0\) | A1 | 1.1b - All four correct equations |
| Solves simultaneous equations to find at least one constant | M1 | 3.1a |
| \(a=-4, b=3, c=-10, d=50\) | A1 | 1.1b |
# Question 1:
## Part (a):
| Answer/Working | Mark | Guidance |
|---|---|---|
| $z = -1-2i$ or $z = 3+i$ | M1 | 1.2 - Identifies at least one correct complex conjugate as another root |
| $z = -1-2i$ and $z = 3+i$ | A1 | 1.1b - Both complex conjugate roots identified correctly |
| One complex conjugate pair correctly plotted on Argand diagram, e.g. $(-1,2)$ and $(-1,-2)$ | B1 | 1.1b - One complex conjugate pair correctly plotted |
| Both complex conjugate pairs correctly plotted: $(3,1)$, $(3,-1)$, $(-1,2)$, $(-1,-2)$. The $3\pm i$ must be closer to real axis than $-1\pm 2i$ | B1 | 1.1b - If no indication of coordinates, scale or complex numbers on Argand diagram this is B0 B0 |
## Part (b) - Way 1:
| Answer/Working | Mark | Guidance |
|---|---|---|
| $(z-(-1+2i))(z-(-1-2i)) = \ldots$ or $(z-(3+i))(z-(3-i)) = \ldots$ | M1 | 3.1a - Correct strategy for forming at least one quadratic factor |
| $z^2+2z+5$ or $z^2-6z+10$ | A1 | 1.1b - At least one correct simplified quadratic factor |
| $z^2+2z+5$ and $z^2-6z+10$ | A1 | 1.1b - Both simplified quadratic factors correct |
| $f(z) = (z^2+2z+5)(z^2-6z+10)$, expands to form a quartic | M1 | 3.1a - Complete strategy to find values for $a, b, c, d$ |
| $f(z) = z^4-4z^3+3z^2-10z+50$ or states $a=-4, b=3, c=-10, d=50$ | A1 | 1.1b |
## Part (b) - Way 2:
| Answer/Working | Mark | Guidance |
|---|---|---|
| Sum roots $= (-1+2i)+(-1-2i)+(3+i)+(3-i) = \ldots$; pair sum $= \alpha\beta+\alpha\gamma+\alpha\delta+\beta\gamma+\beta\delta+\gamma\delta$; triple sum; product $= \alpha\beta\gamma\delta = (-1+2i)(-1-2i)(3-i)(3+i) = \ldots$ | M1 | 3.1a - Correct strategy for finding at least three of sum, pair sum, triple sum and product |
| sum $= 4$, pair sum $= 3$, triple sum $= 10$, product $= 50$ | A1, A1 | 1.1b |
| $a = -(\text{sum roots}) = -4$; $b = +(\text{pair sum}) = 3$; $c = -(\text{triple sum}) = -10$; $d = +(\text{product}) = 50$ | M1, A1 | 3.1a, 1.1b |
## Part (b) - Way 3:
| Answer/Working | Mark | Guidance |
|---|---|---|
| $f(-1+2i) = (-1+2i)^4 + a(-1+2i)^3 + b(-1+2i)^2 + c(-1+2i) + d = 0$ and $f(3+i) = (3+i)^4 + a(3+i)^3 + b(3+i)^2 + c(3+i) + d = 0$ | M1 | 3.1a - Substitutes two roots into $f(z)=0$ and equates coefficients to form 4 equations |
| $-7+11a-3b-c+d=0$ and $24-2a-4b+2c=0$ | A1 | 1.1b - At least two correct equations |
| $28+18a+8b+3c+d=0$ and $96+26a+6b+c=0$ | A1 | 1.1b - All four correct equations |
| Solves simultaneous equations to find at least one constant | M1 | 3.1a |
| $a=-4, b=3, c=-10, d=50$ | A1 | 1.1b |
*Note: Correct answer only will score 5/5*
---1.
$$f ( z ) = z ^ { 4 } + a z ^ { 3 } + b z ^ { 2 } + c z + d$$
where $a , b , c$ and $d$ are real constants.\\
Given that $- 1 + 2 \mathrm { i }$ and $3 - \mathrm { i }$ are two roots of the equation $\mathrm { f } ( \mathrm { z } ) = 0$
\begin{enumerate}[label=(\alph*)]
\item show all the roots of $f ( z ) = 0$ on a single Argand diagram,
\item find the values of $a , b , c$ and $d$.
\end{enumerate}
\hfill \mbox{\textit{Edexcel CP1 2019 Q1 [9]}}