| Exam Board | CAIE |
|---|---|
| Module | P3 (Pure Mathematics 3) |
| Year | 2009 |
| Session | November |
| Marks | 10 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Complex Numbers Argand & Loci |
| Type | Region shading with multiple inequalities |
| Difficulty | Standard +0.3 This is a multi-part question covering standard complex number techniques: finding coefficients using conjugate roots, calculating modulus/argument, and shading a region defined by two inequalities. The locus |z| < |z-2| is a standard perpendicular bisector (half-plane), and arg(z-u) defines a sector. While requiring multiple steps, each component uses routine A-level methods without requiring novel insight or particularly challenging geometric reasoning. |
| Spec | 4.02a Complex numbers: real/imaginary parts, modulus, argument4.02c Complex notation: z, z*, Re(z), Im(z), |z|, arg(z)4.02i Quadratic equations: with complex roots4.02j Cubic/quartic equations: conjugate pairs and factor theorem4.02k Argand diagrams: geometric interpretation4.02o Loci in Argand diagram: circles, half-lines |
| Answer | Marks |
|---|---|
| Substitute \(x = -2 + i\) in the equation and attempt expansion of \((-2 + i)^3\) | M1 |
| Use \(i^2 = -1\) correctly at least once and solve for \(k\) | M1 |
| Obtain \(k = 20\) | A1 |
| [3] |
| Answer | Marks |
|---|---|
| State that the other complex root is \(-2 - i\) | B1 |
| [1] |
| Answer | Marks |
|---|---|
| Obtain modulus \(\sqrt{5}\) | B1 |
| Obtain argument 153.4° or 2.68 radians | B1 |
| [2] |
| Answer | Marks |
|---|---|
| Show point representing \(u\) in relatively correct position in an Argand diagram | B1 |
| Show vertical line through \(z = 1\) | B1 |
| Show the correct half-lines from \(u\) of gradient zero and 1 | B1 |
| Shade the relevant region | B1 |
| Answer | Marks |
|---|---|
| State that the other complex root is \(-2 - i\) | B1 |
| State quadratic factor \(x^2 + 4x + 5\) | B1 |
| Divide cubic by 3-term quadratic, equate remainder to zero and solve for \(k\), or, using 3-term quadratic, factorise cubic and obtain \(k\) | M1 |
| Obtain \(k = 20\) | A1 |
| [4] |
**(i)**
| Substitute $x = -2 + i$ in the equation and attempt expansion of $(-2 + i)^3$ | M1 |
| Use $i^2 = -1$ correctly at least once and solve for $k$ | M1 |
| Obtain $k = 20$ | A1 |
| [3] |
**(ii)**
| State that the other complex root is $-2 - i$ | B1 |
| [1] |
**(iii)**
| Obtain modulus $\sqrt{5}$ | B1 |
| Obtain argument 153.4° or 2.68 radians | B1 |
| [2] |
**(iv)**
| Show point representing $u$ in relatively correct position in an Argand diagram | B1 |
| Show vertical line through $z = 1$ | B1 |
| Show the correct half-lines from $u$ of gradient zero and 1 | B1 |
| Shade the relevant region | B1 |
[SR: For parts (i) and (ii) allow the following alternative method:
State that the other complex root is $-2 - i$ | B1 |
State quadratic factor $x^2 + 4x + 5$ | B1 |
Divide cubic by 3-term quadratic, equate remainder to zero and solve for $k$, or, using 3-term quadratic, factorise cubic and obtain $k$ | M1 |
Obtain $k = 20$ | A1 |
[4] |7 The complex number $- 2 + \mathrm { i }$ is denoted by $u$.\\
(i) Given that $u$ is a root of the equation $x ^ { 3 } - 11 x - k = 0$, where $k$ is real, find the value of $k$.\\
(ii) Write down the other complex root of this equation.\\
(iii) Find the modulus and argument of $u$.\\
(iv) Sketch an Argand diagram showing the point representing $u$. Shade the region whose points represent the complex numbers $z$ satisfying both the inequalities
$$| z | < | z - 2 | \quad \text { and } \quad 0 < \arg ( z - u ) < \frac { 1 } { 4 } \pi$$
\hfill \mbox{\textit{CAIE P3 2009 Q7 [10]}}