| Exam Board | CAIE |
|---|---|
| Module | P3 (Pure Mathematics 3) |
| Year | 2012 |
| Session | June |
| Marks | 8 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Complex Numbers Arithmetic |
| Type | Argument relationships and tangent identities |
| Difficulty | Standard +0.3 This is a multi-part question involving standard complex number operations (expressing in Cartesian form, plotting on Argand diagram) and using argument properties to prove a tangent identity. While it requires understanding of arg(z₁z₂) = arg(z₁) + arg(z₂) and connecting this to arctangent, the steps are relatively straightforward once the connection is recognized. The proof is guided and uses routine A-level techniques, making it slightly easier than average. |
| Spec | 1.05i Inverse trig functions: arcsin, arccos, arctan domains and graphs4.02a Complex numbers: real/imaginary parts, modulus, argument4.02c Complex notation: z, z*, Re(z), Im(z), |z|, arg(z)4.02k Argand diagrams: geometric interpretation |
| Answer | Marks |
|---|---|
| Multiply numerator and denominator by \(1 + 3i\), or equivalent | M1 |
| Simplify numerator to \(-5 + 5i\), or denominator to 10, or equivalent | A1 |
| Obtain final answer \(-\frac{1}{2} + \frac{1}{2}i\), or equivalent | A1 |
| Answer | Marks | Guidance |
|---|---|---|
| Obtain two equations in \(x\) and \(y\), and solve for \(x\) or \(y\) | M1 | |
| Obtain \(x = -\frac{1}{2}\) or \(y = \frac{1}{2}\), or equivalent | A1 | |
| Obtain final answer \(-\frac{1}{2} + \frac{1}{2}i\), or equivalent | A1 | [3] |
| Answer | Marks | Guidance |
|---|---|---|
| Show \(B\) and \(C\) in relatively correct positions in an Argand diagram | B1 | |
| Show \(u\) in a relatively correct position | B1⬇ | [2] |
| Answer | Marks | Guidance |
|---|---|---|
| Substitute exact arguments in the LHS \(\arg(1 + 2i) - \arg(1 - 3i) = \arg u\), or equivalent | M1 | |
| Obtain and use \(\arg u = \frac{3}{4}\pi\) | A1 | |
| Obtain the given result correctly | A1 | [3] |
**(i)**
**Main scheme (EITHER):**
| Multiply numerator and denominator by $1 + 3i$, or equivalent | M1 | |
| Simplify numerator to $-5 + 5i$, or denominator to 10, or equivalent | A1 | |
| Obtain final answer $-\frac{1}{2} + \frac{1}{2}i$, or equivalent | A1 | |
**OR:**
| Obtain two equations in $x$ and $y$, and solve for $x$ or $y$ | M1 | |
| Obtain $x = -\frac{1}{2}$ or $y = \frac{1}{2}$, or equivalent | A1 | |
| Obtain final answer $-\frac{1}{2} + \frac{1}{2}i$, or equivalent | A1 | [3] |
**(ii)**
| Show $B$ and $C$ in relatively correct positions in an Argand diagram | B1 | |
| Show $u$ in a relatively correct position | B1⬇ | [2] |
**(iii)**
| Substitute exact arguments in the LHS $\arg(1 + 2i) - \arg(1 - 3i) = \arg u$, or equivalent | M1 | |
| Obtain and use $\arg u = \frac{3}{4}\pi$ | A1 | |
| Obtain the given result correctly | A1 | [3] |
---(i) Express $u$ in the form $x + \mathrm { i } y$, where $x$ and $y$ are real.\\
(ii) Show on a sketch of an Argand diagram the points $A , B$ and $C$ representing the complex numbers $u , 1 + 2 \mathrm { i }$ and $1 - 3 \mathrm { i }$ respectively.\\
(iii) By considering the arguments of $1 + 2 \mathrm { i }$ and $1 - 3 \mathrm { i }$, show that
$$\tan ^ { - 1 } 2 + \tan ^ { - 1 } 3 = \frac { 3 } { 4 } \pi$$
\hfill \mbox{\textit{CAIE P3 2012 Q7 [8]}}