| Exam Board | CAIE |
|---|---|
| Module | P3 (Pure Mathematics 3) |
| Year | 2007 |
| Session | November |
| Marks | 10 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Complex Numbers Argand & Loci |
| Type | Square roots of complex numbers |
| Difficulty | Standard +0.3 This is a straightforward multi-part question testing standard complex number techniques: simplifying by multiplying by conjugate, finding modulus/argument using formulas, and finding square roots by equating real/imaginary parts. All are routine A-level procedures with no novel insight required, making it slightly easier than average. |
| Spec | 1.02y Partial fractions: decompose rational functions1.04c Extend binomial expansion: rational n, |x|<1 |
| Answer | Marks | Guidance |
|---|---|---|
| (a) (i) EITHER: Carry out multiplication of numerator and denominator by \(1 + 2i\), or equivalent | M1 | |
| Obtain answer \(2 + i\), or any equivalent of the form \((a + ib)/c\) | A1 | |
| OR1: Obtain two equations in \(x\) and \(y\), and solve for \(x\) or for \(y\) | M1 | |
| Obtain answer \(2 + i\), or equivalent | A1 | |
| OR2: Using the correct processes express \(z\) in polar form | M1 | |
| Obtain answer \(2 + i\), or equivalent | A1 | [2] |
| (ii) State that the modulus of \(z\) is \(\sqrt{5}\) or \(2.24\) | B1 | |
| State that the argument of \(z\) is \(0.464\) or \(26.6°\) | B1 | [2] |
| (b) EITHER: Square \(x + iy\) and equate real and imaginary parts to 5 and \(-12\) respectively | M1 | |
| Obtain \(x^2 - y^2 = 5\) and \(2xy = -12\) | A1 | |
| Eliminate one variable and obtain an equation in the other | M1 | |
| Obtain \(x^4 - 5x^2 - 36 = 0\) or \(y^4 + 5y^2 - 36 = 0\), or 3-term equivalent | A1 | |
| Obtain answer \(3 - 2i\) | A1 | |
| Obtain second answer \(-3 + 2i\) and no others | A1 | |
| OR: Convert \(5 - 12i\) to polar form \((R, \theta)\) | M1 | |
| Use the fact that a square root has the polar form \((\sqrt{R}, \frac{1}{2}\theta)\) | M1 | |
| Obtain one root in polar form, e.g. \((\sqrt{13}, -0.588)\) or \((\sqrt{13},-33.7°)\) | A1 + A1 | |
| Obtain answer \(3 - 2i\) | A1 | |
| Obtain answer \(-3 + 2i\) and no others | A1 | [6] |
**(a)** **(i)** **EITHER:** Carry out multiplication of numerator and denominator by $1 + 2i$, or equivalent | M1 |
Obtain answer $2 + i$, or any equivalent of the form $(a + ib)/c$ | A1 |
**OR1:** Obtain two equations in $x$ and $y$, and solve for $x$ or for $y$ | M1 |
Obtain answer $2 + i$, or equivalent | A1 |
**OR2:** Using the correct processes express $z$ in polar form | M1 |
Obtain answer $2 + i$, or equivalent | A1 | [2]
**(ii)** State that the modulus of $z$ is $\sqrt{5}$ or $2.24$ | B1 |
State that the argument of $z$ is $0.464$ or $26.6°$ | B1 | [2]
**(b)** **EITHER:** Square $x + iy$ and equate real and imaginary parts to 5 and $-12$ respectively | M1 |
Obtain $x^2 - y^2 = 5$ and $2xy = -12$ | A1 |
Eliminate one variable and obtain an equation in the other | M1 |
Obtain $x^4 - 5x^2 - 36 = 0$ or $y^4 + 5y^2 - 36 = 0$, or 3-term equivalent | A1 |
Obtain answer $3 - 2i$ | A1 |
Obtain second answer $-3 + 2i$ and no others | A1 |
**OR:** Convert $5 - 12i$ to polar form $(R, \theta)$ | M1 |
Use the fact that a square root has the polar form $(\sqrt{R}, \frac{1}{2}\theta)$ | M1 |
Obtain one root in polar form, e.g. $(\sqrt{13}, -0.588)$ or $(\sqrt{13},-33.7°)$ | A1 + A1 |
Obtain answer $3 - 2i$ | A1 |
Obtain answer $-3 + 2i$ and no others | A1 | [6]8
\begin{enumerate}[label=(\alph*)]
\item The complex number $z$ is given by $z = \frac { 4 - 3 \mathrm { i } } { 1 - 2 \mathrm { i } }$.
\begin{enumerate}[label=(\roman*)]
\item Express $z$ in the form $x + \mathrm { i } y$, where $x$ and $y$ are real.
\item Find the modulus and argument of $z$.
\end{enumerate}\item Find the two square roots of the complex number 5-12i, giving your answers in the form $x + \mathrm { i } y$, where $x$ and $y$ are real.
\end{enumerate}
\hfill \mbox{\textit{CAIE P3 2007 Q8 [10]}}