| Exam Board | CAIE |
|---|---|
| Module | P3 (Pure Mathematics 3) |
| Year | 2015 |
| Session | June |
| Marks | 9 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Complex Numbers Argand & Loci |
| Type | Circle equations in complex form |
| Difficulty | Standard +0.3 This is a multi-part question with standard complex number techniques: (i) simplifying by expanding denominator (routine), (ii) finding range of p from argument inequalities (standard geometric interpretation), (iii) finding circle equation through three points including origin (textbook exercise). All parts use well-practiced methods with no novel insight required, making it slightly easier than average. |
| Spec | 4.02e Arithmetic of complex numbers: add, subtract, multiply, divide4.02i Quadratic equations: with complex roots4.02k Argand diagrams: geometric interpretation |
| Answer | Marks | Guidance |
|---|---|---|
| (i) | ||
| Either | ||
| Expand \((2-i)^2\) to obtain \(3 - 4i\) or unsimplified equivalent | B1 | |
| Multiply by \(\frac{3+4i}{3+4i}\) and simplify to \(x + iy\) form or equivalent | M1 | |
| Confirm given answer \(2 + 4i\) | A1 | |
| Or | ||
| Expand \((2-i)^2\) to obtain \(3 - 4i\) or unsimplified equivalent | B1 | |
| Obtain two equations in \(x\) and \(y\) and solve for \(x\) or \(y\) | M1 | |
| Confirm given answer \(2 + 4i\) | A1 | [3] |
| (ii) | ||
| Identify \(4 + 4\) or \(-4 + 4i\) as point at either end or state \(p = 2\) or state \(p = -6\) | B1 | |
| Use appropriate method to find both critical values of \(p\) | M1 | |
| State \(-6 \leq p \leq 2\) | A1 | [3] |
| (iii) | ||
| Identify equation as of form \( | z-d | = a\) or equivalent |
| Form correct equation for \(a\) not involving modulus, e.g. \((a-2)^2 + 4^2 = a^2\) | A1 | |
| State \( | z-\bar{5} | = 5\) |
**(i)** |
**Either** |
Expand $(2-i)^2$ to obtain $3 - 4i$ or unsimplified equivalent | B1 |
Multiply by $\frac{3+4i}{3+4i}$ and simplify to $x + iy$ form or equivalent | M1 |
Confirm given answer $2 + 4i$ | A1 |
**Or** |
Expand $(2-i)^2$ to obtain $3 - 4i$ or unsimplified equivalent | B1 |
Obtain two equations in $x$ and $y$ and solve for $x$ or $y$ | M1 |
Confirm given answer $2 + 4i$ | A1 | [3]
**(ii)** |
Identify $4 + 4$ or $-4 + 4i$ as point at either end or state $p = 2$ or state $p = -6$ | B1 |
Use appropriate method to find both critical values of $p$ | M1 |
State $-6 \leq p \leq 2$ | A1 | [3]
**(iii)** |
Identify equation as of form $|z-d| = a$ or equivalent | M1 |
Form correct equation for $a$ not involving modulus, e.g. $(a-2)^2 + 4^2 = a^2$ | A1 |
State $|z-\bar{5}| = 5$ | A1 | [3]8 The complex number $w$ is defined by $w = \frac { 22 + 4 \mathrm { i } } { ( 2 - \mathrm { i } ) ^ { 2 } }$.\\
(i) Without using a calculator, show that $w = 2 + 4 \mathrm { i }$.\\
(ii) It is given that $p$ is a real number such that $\frac { 1 } { 4 } \pi \leqslant \arg ( w + p ) \leqslant \frac { 3 } { 4 } \pi$. Find the set of possible values of $p$.\\
(iii) The complex conjugate of $w$ is denoted by $w ^ { * }$. The complex numbers $w$ and $w ^ { * }$ are represented in an Argand diagram by the points $S$ and $T$ respectively. Find, in the form $| z - a | = k$, the equation of the circle passing through $S , T$ and the origin.
\hfill \mbox{\textit{CAIE P3 2015 Q8 [9]}}