| Exam Board | CAIE |
|---|---|
| Module | P3 (Pure Mathematics 3) |
| Year | 2006 |
| Session | November |
| Marks | 10 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Complex Numbers Argand & Loci |
| Type | Optimization of modulus on loci |
| Difficulty | Standard +0.3 This is a standard multi-part complex numbers question covering routine techniques: simplifying a complex fraction by multiplying by conjugate (i), finding modulus and argument from Cartesian form (ii), sketching a circle locus (iii), and finding minimum distance from origin to circle (iv). The optimization in part (iv) is straightforward geometry (distance from origin to circle center minus radius). All steps are textbook exercises with no novel insight required, making it slightly easier than average. |
| Spec | 1.07k Differentiate trig: sin(kx), cos(kx), tan(kx)1.07n Stationary points: find maxima, minima using derivatives1.08i Integration by parts1.09d Newton-Raphson method |
| Answer | Marks | Guidance |
|---|---|---|
| (i) EITHER: Multiply numerator and denominator by \(2 + i\), or equivalent | M1 | |
| Simplify numerator to \(5 + 5i\) or denominator to \(5\) | A1 | |
| Obtain answer \(1 + i\) | A1 | |
| OR: Obtain two equations in \(x\) and \(y\), and solve for \(x\) or for \(y\) | M1 | |
| Obtain \(x = 1\) | A1 | |
| Obtain \(y = 1\) | A1 | |
| OR: Using correct processes express \(u\) in polar form | M1 | |
| Obtain \(u = \sqrt{2}(\cos 45° + i\sin 45°)\), or equivalent | A1 | |
| Obtain answer \(1 + i\) | A1 | |
| (ii) State that the modulus is \(\sqrt{2}\) or 1.41 | B1/' | |
| State that the argument is \(45°\) or \(t\) (or 0.785) | B1/' | |
| (iii) Show the point representing \(u\) in a relatively correct position | B1/' | |
| Show a circle with centre at the point representing \(u\) | B1/' | |
| Indicate or imply the radius is \(1\) | B1 | |
| [NB: If the Argand diagram has unequal scales the locus is not circular in appearance, but an ellipse with centre \(u\) and equal axes parallel to the axes of the diagram earns B1/', and B1 if both semi-axes are indicated or implied to be equal to 1. In such a situation only award B1/' for a circle with centre \(u\) and a horizontal or vertical radius indicated or implied to be 1.] | ||
| (iv) Carry out complete strategy for calculating \(\min | z | \) for the locus |
| Obtain answer \(\sqrt{2} - 1\) (or 0.414) | A1/' | |
| [The f.t. is on the value of u.] | ||
| 3 | ||
| 2 | ||
| 3 | ||
| 2 |
(i) EITHER: Multiply numerator and denominator by $2 + i$, or equivalent | M1 |
Simplify numerator to $5 + 5i$ or denominator to $5$ | A1 |
Obtain answer $1 + i$ | A1 |
OR: Obtain two equations in $x$ and $y$, and solve for $x$ or for $y$ | M1 |
Obtain $x = 1$ | A1 |
Obtain $y = 1$ | A1 |
OR: Using correct processes express $u$ in polar form | M1 |
Obtain $u = \sqrt{2}(\cos 45° + i\sin 45°)$, or equivalent | A1 |
Obtain answer $1 + i$ | A1 |
(ii) State that the modulus is $\sqrt{2}$ or 1.41 | B1/' |
State that the argument is $45°$ or $t$ (or 0.785) | B1/' |
(iii) Show the point representing $u$ in a relatively correct position | B1/' |
Show a circle with centre at the point representing $u$ | B1/' |
Indicate or imply the radius is $1$ | B1 |
[NB: If the Argand diagram has unequal scales the locus is not circular in appearance, but an ellipse with centre $u$ and equal axes parallel to the axes of the diagram earns B1/', and B1 if both semi-axes are indicated or implied to be equal to 1. In such a situation only award B1/' for a circle with centre $u$ and a horizontal or vertical radius indicated or implied to be 1.] | | |
(iv) Carry out complete strategy for calculating $\min |z|$ for the locus | M1 |
Obtain answer $\sqrt{2} - 1$ (or 0.414) | A1/' |
[The f.t. is on the value of u.] | | |
| | 3 |
| | 2 |
| | 3 |
| | 2 |9 The complex number $u$ is given by
$$u = \frac { 3 + \mathrm { i } } { 2 - \mathrm { i } }$$
(i) Express $u$ in the form $x + \mathrm { i } y$, where $x$ and $y$ are real.\\
(ii) Find the modulus and argument of $u$.\\
(iii) Sketch an Argand diagram showing the point representing the complex number $u$. Show on the same diagram the locus of the point representing the complex number $z$ such that $| z - u | = 1$.\\
(iv) Using your diagram, calculate the least value of $| z |$ for points on this locus.
\hfill \mbox{\textit{CAIE P3 2006 Q9 [10]}}