| Exam Board | CAIE |
|---|---|
| Module | P3 (Pure Mathematics 3) |
| Year | 2021 |
| Session | June |
| Marks | 10 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Complex numbers 2 |
| Type | Find conjugate roots from polynomial |
| Difficulty | Standard +0.3 This is a straightforward complex roots question requiring verification by substitution (routine calculation) and finding remaining roots using the conjugate root theorem and polynomial division. While it involves multiple steps, the techniques are standard A-level procedures with no novel insight required, making it slightly easier than average. |
| Spec | 4.02g Conjugate pairs: real coefficient polynomials4.02i Quadratic equations: with complex roots |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| Substitute \(-1+\sqrt{2}\mathrm{i}\) and attempt expansions of the \(z^2\) and \(z^4\) terms | M1 | |
| Use \(\mathrm{i}^2 = -1\) at least once | M1 | |
| Complete the verification correctly | A1 | |
| Total | 3 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| State second root \(-1-\sqrt{2}\mathrm{i}\) | A1 | |
| Carry out a method to find a quadratic factor with zeros \(-1 \pm \sqrt{2}\mathrm{i}\) | M1 | |
| Obtain \(z^2 + 2z + 3\) | A1 | |
| Commence division and reach partial quotient \(z^2 + kz\) | M1 | |
| Obtain second quadratic factor \(z^2 - 2z + 4\) | A1 | |
| Solve a 3-term quadratic and use \(\mathrm{i}^2 = -1\) | M1 | |
| Obtain roots \(1+\sqrt{3}\mathrm{i}\) and \(1-\sqrt{3}\mathrm{i}\) | A1 | |
| Total | 7 |
## Question 10(a):
| Answer | Mark | Guidance |
|--------|------|----------|
| Substitute $-1+\sqrt{2}\mathrm{i}$ and attempt expansions of the $z^2$ and $z^4$ terms | M1 | |
| Use $\mathrm{i}^2 = -1$ at least once | M1 | |
| Complete the verification correctly | A1 | |
| **Total** | **3** | |
## Question 10(b):
| Answer | Mark | Guidance |
|--------|------|----------|
| State second root $-1-\sqrt{2}\mathrm{i}$ | A1 | |
| Carry out a method to find a quadratic factor with zeros $-1 \pm \sqrt{2}\mathrm{i}$ | M1 | |
| Obtain $z^2 + 2z + 3$ | A1 | |
| Commence division and reach partial quotient $z^2 + kz$ | M1 | |
| Obtain second quadratic factor $z^2 - 2z + 4$ | A1 | |
| Solve a 3-term quadratic and use $\mathrm{i}^2 = -1$ | M1 | |
| Obtain roots $1+\sqrt{3}\mathrm{i}$ and $1-\sqrt{3}\mathrm{i}$ | A1 | |
| **Total** | **7** | |10
\begin{enumerate}[label=(\alph*)]
\item Verify that $- 1 + \sqrt { 2 } \mathrm { i }$ is a root of the equation $z ^ { 4 } + 3 z ^ { 2 } + 2 z + 12 = 0$.
\item Find the other roots of this equation.\\
If you use the following lined page to complete the answer(s) to any question(s), the question number(s) must be clearly shown.
\end{enumerate}
\hfill \mbox{\textit{CAIE P3 2021 Q10 [10]}}