| Exam Board | CAIE |
|---|---|
| Module | P3 (Pure Mathematics 3) |
| Year | 2012 |
| Session | June |
| Marks | 11 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Complex Numbers Argand & Loci |
| Type | Region shading with multiple inequalities |
| Difficulty | Standard +0.3 Part (a) involves solving simultaneous equations with complex numbers using substitution and the quadratic formula—a standard technique. Part (b)(i) requires sketching standard loci (circle, half-line, vertical line) and finding their intersection—routine for P3. Part (b)(ii) needs identifying the maximum real part geometrically, which requires some visualization but follows directly from the sketch. Overall, this is a straightforward multi-part question testing standard complex number techniques without requiring 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 |
|---|---|---|
| (a) EITHER: Eliminate \(u\) or \(w\) and obtain an equation in \(w\) or in \(u\) | M1 | |
| Obtain a quadratic in \(u\) or \(w\), e.g. \(u^2 - 4iu - 5 = 0\) or \(w^2 + 4iw - 5 = 0\) | A1 | |
| Solve a 3-term quadratic for \(u\) or for \(w\) | M1 | |
| OR1: Having squared the first equation, eliminate \(u\) or \(w\) and obtain an equation in \(w\) or \(u\) | M1 | |
| Obtain a 2-term quadratic in \(u\) or \(w\), e.g. \(u^2 = -3 + 4i\) | A1 | |
| Solve a 2-term quadratic for \(u\) or for \(w\) | M1 | |
| OR2: Using \(u = a + ib, w = c + id\), equate real and imaginary parts and obtain 4 equations in \(a, b, c\) and \(d\) | M1 | |
| Obtain 4 correct equations | A1 | |
| Solve for \(a\) and \(b\), or for \(c\) and \(d\) | M1 | |
| Obtain answer \(u = 1 + 2i, w = 1 - 2i\) | A1 | |
| Obtain answer \(u = -1 + 2i, w = -1 - 2i\) and no other | A1 | [5] |
| (b)(i) Show point representing \(2 - 2i\) in relatively correct position | B1 | |
| Show a circle with centre \(2 - 2i\) and radius 2 | B1 | |
| Show line for \(\arg z = -\frac{1}{4}\pi\) | B1 | |
| Show line for \(\text{Re } z = 1\) | B1 | |
| Shade the relevant region | B1 | [5] |
| (b)(ii) State answer \(2 + \sqrt{2}\), or equivalent (accept 3.41) | B1 | [1] |
**(a)** EITHER: Eliminate $u$ or $w$ and obtain an equation in $w$ or in $u$ | M1 |
Obtain a quadratic in $u$ or $w$, e.g. $u^2 - 4iu - 5 = 0$ or $w^2 + 4iw - 5 = 0$ | A1 |
Solve a 3-term quadratic for $u$ or for $w$ | M1 |
OR1: Having squared the first equation, eliminate $u$ or $w$ and obtain an equation in $w$ or $u$ | M1 |
Obtain a 2-term quadratic in $u$ or $w$, e.g. $u^2 = -3 + 4i$ | A1 |
Solve a 2-term quadratic for $u$ or for $w$ | M1 |
OR2: Using $u = a + ib, w = c + id$, equate real and imaginary parts and obtain 4 equations in $a, b, c$ and $d$ | M1 |
Obtain 4 correct equations | A1 |
Solve for $a$ and $b$, or for $c$ and $d$ | M1 |
Obtain answer $u = 1 + 2i, w = 1 - 2i$ | A1 |
Obtain answer $u = -1 + 2i, w = -1 - 2i$ and no other | A1 | [5]
**(b)(i)** Show point representing $2 - 2i$ in relatively correct position | B1 |
Show a circle with centre $2 - 2i$ and radius 2 | B1 |
Show line for $\arg z = -\frac{1}{4}\pi$ | B1 |
Show line for $\text{Re } z = 1$ | B1 |
Shade the relevant region | B1 | [5]
**(b)(ii)** State answer $2 + \sqrt{2}$, or equivalent (accept 3.41) | B1 | [1]10
\begin{enumerate}[label=(\alph*)]
\item The complex numbers $u$ and $w$ satisfy the equations
$$u - w = 4 \mathrm { i } \quad \text { and } \quad u w = 5$$
Solve the equations for $u$ and $w$, giving all answers in the form $x + \mathrm { i } y$, where $x$ and $y$ are real.
\item \begin{enumerate}[label=(\roman*)]
\item On a sketch of an Argand diagram, shade the region whose points represent complex numbers satisfying the inequalities $| z - 2 + 2 \mathrm { i } | \leqslant 2 , \arg z \leqslant - \frac { 1 } { 4 } \pi$ and $\operatorname { Re } z \geqslant 1$, where $\operatorname { Re } z$ denotes the real part of $z$.
\item Calculate the greatest possible value of $\operatorname { Re } z$ for points lying in the shaded region.
\end{enumerate}\end{enumerate}
\hfill \mbox{\textit{CAIE P3 2012 Q10 [11]}}