| Exam Board | CAIE |
|---|---|
| Module | P3 (Pure Mathematics 3) |
| Year | 2021 |
| Session | June |
| Marks | 7 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Complex Numbers Arithmetic |
| Type | Parametric polynomials with root conditions |
| Difficulty | Standard +0.8 This question requires solving a quadratic with complex coefficients using the quadratic formula, then applying geometric constraints (points on imaginary axis, equilateral triangle) to derive relationships between parameters. Part (a) is routine, but parts (b) and (c) require translating geometric conditions into algebraic constraints on complex roots, involving modulus and argument considerations. This is moderately challenging but within standard A-level Further Maths scope. |
| Spec | 4.02i Quadratic equations: with complex roots4.02k Argand diagrams: geometric interpretation |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| Use quadratic formula and \(i^2 = -1\) | M1 | |
| Obtain answers \(pi + \sqrt{q - p^2}\) and \(pi - \sqrt{q - p^2}\) | A1 | Accept \(\dfrac{2pi \pm \sqrt{-4p^2 + 4q}}{2}\) and ISW |
| Total | 2 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| State or imply that the discriminant must be negative | M1 | |
| State condition \(q < p^2\) | A1 | |
| Total | 2 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| Carry out a correct method for finding a relation, e.g. use the fact that the argument of one of the roots is \((\pm)60°\) | M1 | |
| State a correct relation in any form, e.g. \(\dfrac{p}{\sqrt{q-p^2}} = (\pm)\sqrt{3}\) | A1 | |
| Simplify to \(q = \dfrac{4}{3}p^2\) | A1 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| Carry out a correct method, e.g. use the fact that the sides have equal length | M1 | |
| State a correct relation in any form, e.g. \(4(q - p^2) = p^2 + q - p^2\) | A1 | |
| Simplify to \(q = \dfrac{4}{3}p^2\) | A1 | |
| Total | 3 |
## Question 5(a):
| Answer | Marks | Guidance |
|--------|-------|----------|
| Use quadratic formula and $i^2 = -1$ | M1 | |
| Obtain answers $pi + \sqrt{q - p^2}$ and $pi - \sqrt{q - p^2}$ | A1 | Accept $\dfrac{2pi \pm \sqrt{-4p^2 + 4q}}{2}$ and ISW |
| **Total** | **2** | |
## Question 5(b):
| Answer | Marks | Guidance |
|--------|-------|----------|
| State or imply that the discriminant must be negative | M1 | |
| State condition $q < p^2$ | A1 | |
| **Total** | **2** | |
## Question 5(c):
| Answer | Marks | Guidance |
|--------|-------|----------|
| Carry out a correct method for finding a relation, e.g. use the fact that the argument of one of the roots is $(\pm)60°$ | M1 | |
| State a correct relation in any form, e.g. $\dfrac{p}{\sqrt{q-p^2}} = (\pm)\sqrt{3}$ | A1 | |
| Simplify to $q = \dfrac{4}{3}p^2$ | A1 | |
**Alternative method:**
| Answer | Marks | Guidance |
|--------|-------|----------|
| Carry out a correct method, e.g. use the fact that the sides have equal length | M1 | |
| State a correct relation in any form, e.g. $4(q - p^2) = p^2 + q - p^2$ | A1 | |
| Simplify to $q = \dfrac{4}{3}p^2$ | A1 | |
| **Total** | **3** | |
---5
\begin{enumerate}[label=(\alph*)]
\item Solve the equation $z ^ { 2 } - 2 p \mathrm { i } z - q = 0$, where $p$ and $q$ are real constants.\\
In an Argand diagram with origin $O$, the roots of this equation are represented by the distinct points $A$ and $B$.
\item Given that $A$ and $B$ lie on the imaginary axis, find a relation between $p$ and $q$.
\item Given instead that triangle $O A B$ is equilateral, express $q$ in terms of $p$.
\end{enumerate}
\hfill \mbox{\textit{CAIE P3 2021 Q5 [7]}}