| Exam Board | CAIE |
|---|---|
| Module | P3 (Pure Mathematics 3) |
| Year | 2016 |
| Session | November |
| Marks | 10 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Complex Numbers Arithmetic |
| Type | Quadratic with complex coefficients |
| Difficulty | Standard +0.3 Part (a) requires applying the quadratic formula with complex coefficients and simplifying to x+iy form—straightforward but involves careful arithmetic with complex numbers. Part (b) is standard Argand diagram work identifying a disc intersection with an angular sector. Both parts are routine A-level further maths exercises requiring technique but no novel insight. |
| 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/Working | Mark | Guidance |
| Use quadratic formula to solve for \(w\) | M1 | |
| Use \(i^2=-1\) | M1 | |
| Obtain one of the answers \(w=\dfrac{1}{2i+1}\) and \(w=-\dfrac{5}{2i+1}\) | A1 | |
| Multiply numerator and denominator of an answer by \(-2i+1\), or equivalent | M1 | |
| Obtain final answers \(\frac{1}{5}-\frac{2}{5}i\) and \(-1+2i\) | A1 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| Multiply the equation by \(1-2i\) | M1 | |
| Use \(i^2=-1\) | M1 | |
| Obtain \(5w^2+4w(1-2i)-(1-2i)^2=0\), or equivalent | A1 | |
| Use quadratic formula or factorise to solve for \(w\) | M1 | |
| Obtain final answers \(\frac{1}{5}-\frac{2}{5}i\) and \(-1+2i\) | A1 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| Substitute \(w=x+iy\) and form equations for real and imaginary parts | M1 | |
| Use \(i^2=-1\) | M1 | |
| Obtain \((x^2-y^2)-4xy+4x-1=0\) and \(2(x^2-y^2)+2xy+4y+2=0\) o.e. | A1 | |
| Form equation in \(x\) only or \(y\) only and solve | M1 | |
| Obtain final answers \(\frac{1}{5}-\frac{2}{5}i\) and \(-1+2i\) | A1 | Total: [5] |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| Show a circle with centre \(1+i\) | B1 | |
| Show a circle with radius \(2\) | B1 | |
| Show half-line \(\arg z = \frac{1}{4}\pi\) | B1 | |
| Show half-line \(\arg z = -\frac{1}{4}\pi\) | B1 | |
| Shade the correct region | B1 | Total: [5] |
# Question 9:
## Part (a):
### EITHER:
| Answer/Working | Mark | Guidance |
|---|---|---|
| Use quadratic formula to solve for $w$ | M1 | |
| Use $i^2=-1$ | M1 | |
| Obtain one of the answers $w=\dfrac{1}{2i+1}$ and $w=-\dfrac{5}{2i+1}$ | A1 | |
| Multiply numerator and denominator of an answer by $-2i+1$, or equivalent | M1 | |
| Obtain final answers $\frac{1}{5}-\frac{2}{5}i$ and $-1+2i$ | A1 | |
### OR1:
| Answer/Working | Mark | Guidance |
|---|---|---|
| Multiply the equation by $1-2i$ | M1 | |
| Use $i^2=-1$ | M1 | |
| Obtain $5w^2+4w(1-2i)-(1-2i)^2=0$, or equivalent | A1 | |
| Use quadratic formula or factorise to solve for $w$ | M1 | |
| Obtain final answers $\frac{1}{5}-\frac{2}{5}i$ and $-1+2i$ | A1 | |
### OR2:
| Answer/Working | Mark | Guidance |
|---|---|---|
| Substitute $w=x+iy$ and form equations for real and imaginary parts | M1 | |
| Use $i^2=-1$ | M1 | |
| Obtain $(x^2-y^2)-4xy+4x-1=0$ and $2(x^2-y^2)+2xy+4y+2=0$ o.e. | A1 | |
| Form equation in $x$ only or $y$ only and solve | M1 | |
| Obtain final answers $\frac{1}{5}-\frac{2}{5}i$ and $-1+2i$ | A1 | Total: [5] |
## Part (b):
| Answer/Working | Mark | Guidance |
|---|---|---|
| Show a circle with centre $1+i$ | B1 | |
| Show a circle with radius $2$ | B1 | |
| Show half-line $\arg z = \frac{1}{4}\pi$ | B1 | |
| Show half-line $\arg z = -\frac{1}{4}\pi$ | B1 | |
| Shade the correct region | B1 | Total: [5] |
---\begin{enumerate}[label=(\alph*)]
\item Solve the equation $( 1 + 2 \mathrm { i } ) w ^ { 2 } + 4 w - ( 1 - 2 \mathrm { i } ) = 0$, giving your answers in the form $x + \mathrm { i } y$, where $x$ and $y$ are real.
\item On a sketch of an Argand diagram, shade the region whose points represent complex numbers satisfying the inequalities $| z - 1 - \mathrm { i } | \leqslant 2$ and $- \frac { 1 } { 4 } \pi \leqslant \arg z \leqslant \frac { 1 } { 4 } \pi$.
\end{enumerate}
\hfill \mbox{\textit{CAIE P3 2016 Q9 [10]}}