| Exam Board | CAIE |
|---|---|
| Module | P3 (Pure Mathematics 3) |
| Year | 2014 |
| Session | November |
| Marks | 8 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Complex Numbers Arithmetic |
| Type | Solving equations involving complex fractions |
| Difficulty | Standard +0.3 Part (i) is straightforward substitution and complex fraction simplification by multiplying by the conjugate—a standard technique. Part (ii) requires setting w=z to form a quadratic equation in z, which is slightly more involved but still a routine exercise in algebraic manipulation with complex numbers. Both parts test standard A-level techniques without requiring novel insight. |
| Spec | 4.02e Arithmetic of complex numbers: add, subtract, multiply, divide |
| Answer | Marks | Guidance |
|---|---|---|
| (i) Substitute \(z = 1 + i\) and obtain \(w = \frac{1 + 2i}{1 + i}\) | B1 | |
| EITHER: Multiply numerator and denominator by the conjugate of the denominator, or equivalent | M1 | |
| Simplify numerator to \(3 + i\) or denominator to 2 | A1 | |
| Obtain final answer \(\frac{3}{2} + \frac{1}{2}i\), or equivalent | A1 | |
| OR: Obtain two equations in \(x\) and \(y\), and solve for \(x\) or for \(y\) | M1 | |
| Obtain \(x = \frac{3}{2}\) or \(y = \frac{1}{2}\), or equivalent | A1 | |
| Obtain final answer \(\frac{3}{2} + \frac{1}{2}i\), or equivalent | A1 | [4] |
| (ii) EITHER: Substitute \(w = z\) and obtain a 3-term quadratic equation in \(z\), e.g. \(iz^2 + z - i = 0\) | B1 | |
| Solve a 3-term quadratic for \(z\) or substitute \(z = x + iy\) and use a correct method to solve for \(x\) and \(y\) | M1 | |
| OR: Substitute \(w = x + iy\) and obtain two correct equations in \(x\) and \(y\) by equating real and imaginary parts | B1 | |
| Solve for \(x\) and \(y\) | M1 | |
| Obtain a correct solution in any form, e.g. \(z = \frac{-1 \pm \sqrt{3}i}{2i}\) | A1 | |
| Obtain final answer \(-\frac{\sqrt{3}}{2} + \frac{1}{2}i\) | A1 | [4] |
**(i)** Substitute $z = 1 + i$ and obtain $w = \frac{1 + 2i}{1 + i}$ | B1 |
**EITHER:** Multiply numerator and denominator by the conjugate of the denominator, or equivalent | M1 |
Simplify numerator to $3 + i$ or denominator to 2 | A1 |
Obtain final answer $\frac{3}{2} + \frac{1}{2}i$, or equivalent | A1 |
**OR:** Obtain two equations in $x$ and $y$, and solve for $x$ or for $y$ | M1 |
Obtain $x = \frac{3}{2}$ or $y = \frac{1}{2}$, or equivalent | A1 |
Obtain final answer $\frac{3}{2} + \frac{1}{2}i$, or equivalent | A1 | [4]
**(ii)** **EITHER:** Substitute $w = z$ and obtain a 3-term quadratic equation in $z$, e.g. $iz^2 + z - i = 0$ | B1 |
Solve a 3-term quadratic for $z$ or substitute $z = x + iy$ and use a correct method to solve for $x$ and $y$ | M1 |
**OR:** Substitute $w = x + iy$ and obtain two correct equations in $x$ and $y$ by equating real and imaginary parts | B1 |
Solve for $x$ and $y$ | M1 |
Obtain a correct solution in any form, e.g. $z = \frac{-1 \pm \sqrt{3}i}{2i}$ | A1 |
Obtain final answer $-\frac{\sqrt{3}}{2} + \frac{1}{2}i$ | A1 | [4]5 Throughout this question the use of a calculator is not permitted.
The complex numbers $w$ and $z$ satisfy the relation
$$w = \frac { z + \mathrm { i } } { \mathrm { i } z + 2 }$$
(i) Given that $z = 1 + \mathrm { i }$, find $w$, giving your answer in the form $x + \mathrm { i } y$, where $x$ and $y$ are real.\\
(ii) Given instead that $w = z$ and the real part of $z$ is negative, find $z$, giving your answer in the form $x + \mathrm { i } y$, where $x$ and $y$ are real.
\hfill \mbox{\textit{CAIE P3 2014 Q5 [8]}}