| Exam Board | CAIE |
|---|---|
| Module | P3 (Pure Mathematics 3) |
| Year | 2019 |
| Session | March |
| Marks | 10 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Complex Numbers Argand & Loci |
| Type | Region shading with multiple inequalities |
| Difficulty | Standard +0.3 Part (a) is a standard complex quadratic requiring the quadratic formula with complex coefficients—routine algebraic manipulation. Part (b) involves sketching standard loci: a perpendicular bisector and two half-lines from a point, then shading their intersection. Both parts are textbook exercises requiring careful execution but no novel insight, making this slightly easier than average. |
| Spec | 4.02i Quadratic equations: with complex roots4.02k Argand diagrams: geometric interpretation4.02o Loci in Argand diagram: circles, half-lines |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| Use quadratic formula to solve for \(z\) | M1 | |
| Use \(i^2=-1\) throughout | M1 | |
| Obtain correct answer in any form | A1 | |
| Multiply numerator and denominator by \(1-i\), or equivalent | M1 | |
| Obtain final answer, e.g. \(1-i\) | A1 | |
| Obtain second final answer, e.g. \(\frac{5}{2}+\frac{1}{2}i\) | A1 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| Show the point representing \(u\) in relatively correct position | B1 | |
| Show the horizontal line through \(z=i\) | B1 | |
| Show correct half-lines from \(u\), one of gradient 1 and the other vertical | B1ft | |
| Shade the correct region | B1 |
## Question 7(a):
| Answer | Mark | Guidance |
|--------|------|----------|
| Use quadratic formula to solve for $z$ | M1 | |
| Use $i^2=-1$ throughout | M1 | |
| Obtain correct answer in any form | A1 | |
| Multiply numerator and denominator by $1-i$, or equivalent | M1 | |
| Obtain final answer, e.g. $1-i$ | A1 | |
| Obtain second final answer, e.g. $\frac{5}{2}+\frac{1}{2}i$ | A1 | |
## Question 7(b):
| Answer | Mark | Guidance |
|--------|------|----------|
| Show the point representing $u$ in relatively correct position | B1 | |
| Show the horizontal line through $z=i$ | B1 | |
| Show correct half-lines from $u$, one of gradient 1 and the other vertical | B1ft | |
| Shade the correct region | B1 | |7
\begin{enumerate}[label=(\alph*)]
\item Showing all working and without using a calculator, solve the equation
$$( 1 + \mathrm { i } ) z ^ { 2 } - ( 4 + 3 \mathrm { i } ) z + 5 + \mathrm { i } = 0$$
Give your answers in the form $x + \mathrm { i } y$, where $x$ and $y$ are real.
\item The complex number $u$ is given by
$$u = - 1 - \mathrm { i }$$
On a sketch of an Argand diagram show the point representing $u$. Shade the region whose points represent complex numbers satisfying the inequalities $| z | < | z - 2 \mathrm { i } |$ and $\frac { 1 } { 4 } \pi < \arg ( z - u ) < \frac { 1 } { 2 } \pi$.
\end{enumerate}
\hfill \mbox{\textit{CAIE P3 2019 Q7 [10]}}