| Exam Board | CAIE |
|---|---|
| Module | P3 (Pure Mathematics 3) |
| Year | 2023 |
| Session | June |
| Marks | 6 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Complex numbers 2 |
| Type | Modulus and argument calculations |
| Difficulty | Standard +0.3 This question requires expanding a cubic expression in complex form and equating real parts to solve for y, then finding the argument. While it involves multiple steps (expansion, simplification, solving a cubic equation, and finding argument), these are all standard techniques for A-level Further Maths. The algebra is somewhat involved but straightforward, making it slightly easier than average for a Further Maths question. |
| Spec | 4.02a Complex numbers: real/imaginary parts, modulus, argument4.02c Complex notation: z, z*, Re(z), Im(z), |z|, arg(z)4.02e Arithmetic of complex numbers: add, subtract, multiply, divide |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| Substitute \(2 + yi\) in \(a^3 - a^2 - 2a\) and attempt expansions of \(a^2\) and \(a^3\) | M1 | \(a^2 = 4 + 4yi - y^2\), \(a^3 = 8 + 12yi - 6y^2 - y^3i\). If using \(a(a^2 - a - 2)\) must then expand fully. Must see working |
| Use \(i^2 = -1\) | M1 | Seen at least once (e.g. in squaring) |
| Obtain final answer \(-5y^2 + (6y - y^3)i\) | A1 | Or simplified equivalent e.g. \(6yi - 5y^2 - y^3i\). Do not ISW |
| Total: 3 | No evidence of working for square or cube can score SC B1 for correct answer |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| Equate *their* \(-5y^2\) to \(-20\) and solve for \(y\) | M1 | Need to obtain a value for \(y\). Available even if *their* \(y\) is not real |
| Obtain \(y = -2\) | A1 | From correct work. Allow after incorrect f(a) if real part was correct. Condone \(\pm 2\) with positive not rejected |
| Obtain final answer \(\arg a = -\frac{\pi}{4}\) | A1 | Correct only (must have rejected \(y\) positive). OE e.g. \(-\frac{\pi}{4} \pm 2n\pi\). Accept \(-0.785\), \(5.50\). Allow after incorrect f(a) if real part was correct. Accept degrees. Do not ISW |
| Total: 3 |
## Question 5(a):
| Answer | Mark | Guidance |
|--------|------|----------|
| Substitute $2 + yi$ in $a^3 - a^2 - 2a$ and attempt expansions of $a^2$ and $a^3$ | M1 | $a^2 = 4 + 4yi - y^2$, $a^3 = 8 + 12yi - 6y^2 - y^3i$. If using $a(a^2 - a - 2)$ must then expand fully. Must see working |
| Use $i^2 = -1$ | M1 | Seen at least once (e.g. in squaring) |
| Obtain **final** answer $-5y^2 + (6y - y^3)i$ | A1 | Or simplified equivalent e.g. $6yi - 5y^2 - y^3i$. Do not ISW |
| **Total: 3** | | No evidence of working for square or cube can score SC B1 for correct answer |
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## Question 5(b):
| Answer | Mark | Guidance |
|--------|------|----------|
| Equate *their* $-5y^2$ to $-20$ and solve for $y$ | M1 | Need to obtain a value for $y$. Available even if *their* $y$ is not real |
| Obtain $y = -2$ | A1 | From correct work. Allow after incorrect f(a) if real part was correct. Condone $\pm 2$ with positive not rejected |
| Obtain **final** answer $\arg a = -\frac{\pi}{4}$ | A1 | Correct only (must have rejected $y$ positive). OE e.g. $-\frac{\pi}{4} \pm 2n\pi$. Accept $-0.785$, $5.50$. Allow after incorrect f(a) if real part was correct. Accept degrees. Do not ISW |
| **Total: 3** | | |
---5 The complex number $2 + y \mathrm { i }$ is denoted by $a$, where $y$ is a real number and $y < 0$. It is given that $\mathrm { f } ( a ) = a ^ { 3 } - a ^ { 2 } - 2 a$.
\begin{enumerate}[label=(\alph*)]
\item Find a simplified expression for $\mathrm { f } ( a )$ in terms of $y$.
\item Given that $\operatorname { Re } ( \mathrm { f } ( a ) ) = - 20$, find $\arg a$.
\end{enumerate}
\hfill \mbox{\textit{CAIE P3 2023 Q5 [6]}}