| Exam Board | CAIE |
|---|---|
| Module | P3 (Pure Mathematics 3) |
| Year | 2013 |
| 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, involving careful arithmetic with complex numbers but following a standard procedure. Part (b) involves converting exponential form to Cartesian form, finding conjugates, plotting points, and calculating triangle area using coordinate geometry—all routine A-level techniques with no novel insight required. This is slightly easier than average due to the straightforward application of standard methods. |
| Spec | 4.02d Exponential form: re^(i*theta)4.02i Quadratic equations: with complex roots4.02k Argand diagrams: geometric interpretation |
| Answer | Marks | Guidance |
|---|---|---|
| (a) Solve using formula, including simplification under square root sign | M1* | |
| Obtain \(\frac{-2+4i}{2(2-i)}\) or similarly simplified equivalents | A1 | |
| Multiply by \(\frac{2+i}{2+i}\) or equivalent in at least one case | M1(d*M) | |
| Obtain final answer \(-\frac{4}{5} + \frac{3}{5}i\) | A1 | |
| Obtain final answer \(-i\) | A1 | [5] |
| (b) Show \(w\) in first quadrant with modulus and argument relatively correct | B1 | |
| Show \(w'\) in second quadrant with modulus and argument relatively correct | B1 | |
| Show \(w''\) in fourth quadrant with modulus and argument relatively correct | B1 | |
| Use correct method for area of triangle | M1 | |
| Obtain \(10\) by calculation | A1 | [5] |
**(a)** Solve using formula, including simplification under square root sign | M1* |
Obtain $\frac{-2+4i}{2(2-i)}$ or similarly simplified equivalents | A1 |
Multiply by $\frac{2+i}{2+i}$ or equivalent in at least one case | M1(d*M) |
Obtain final answer $-\frac{4}{5} + \frac{3}{5}i$ | A1 |
Obtain final answer $-i$ | A1 | [5]
**(b)** Show $w$ in first quadrant with modulus and argument relatively correct | B1 |
Show $w'$ in second quadrant with modulus and argument relatively correct | B1 |
Show $w''$ in fourth quadrant with modulus and argument relatively correct | B1 |
Use correct method for area of triangle | M1 |
Obtain $10$ by calculation | A1 | [5]9
\begin{enumerate}[label=(\alph*)]
\item Without using a calculator, use the formula for the solution of a quadratic equation to solve
$$( 2 - \mathrm { i } ) z ^ { 2 } + 2 z + 2 + \mathrm { i } = 0$$
Give your answers in the form $a + b \mathrm { i }$.
\item The complex number $w$ is defined by $w = 2 \mathrm { e } ^ { \frac { 1 } { 4 } \pi \mathrm { i } }$. In an Argand diagram, the points $A , B$ and $C$ represent the complex numbers $w , w ^ { 3 }$ and $w ^ { * }$ respectively (where $w ^ { * }$ denotes the complex conjugate of $w$ ). Draw the Argand diagram showing the points $A , B$ and $C$, and calculate the area of triangle $A B C$.
\end{enumerate}
\hfill \mbox{\textit{CAIE P3 2013 Q9 [10]}}