| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| Show \(u\) in a relatively correct position | B1 | |
| Show \(u^*\) in a relatively correct position | B1 | |
| Show \(u^* - u\) in a relatively correct position | B1 | |
| State or imply that \(OABC\) is a parallelogram | B1 | |
| Total: 4 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| *EITHER*: Substitute for \(u\) and multiply numerator and denominator by \(3 + i\), or equivalent | M1 | |
| Simplify the numerator to \(8 + 6i\) or the denominator to \(10\) | A1 | |
| Obtain final answer \(\frac{4}{5} + \frac{3}{5}i\), or equivalent | A1 | |
| *OR*: Substitute for \(u\), obtain two equations in \(x\) and \(y\) and solve for \(x\) or for \(y\) | (M1) | |
| Obtain \(x = \frac{4}{5}\) or \(y = \frac{3}{5}\), or equivalent | (A1) | |
| Obtain final answer \(\frac{4}{5} + \frac{3}{5}i\), or equivalent | (A1) | |
| Total: 3 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| State or imply \(\arg(u^*/u) = \tan^{-1}(\frac{3}{4})\) | B1 | |
| Substitute exact arguments in \(\arg(u^*/u) = \arg u^* - \arg u\) | M1 | |
| Fully justify the given statement using exact values | A1 | |
| Total: 3 marks |
## Question 9(i):
| Answer | Mark | Guidance |
|--------|------|----------|
| Show $u$ in a relatively correct position | B1 | |
| Show $u^*$ in a relatively correct position | B1 | |
| Show $u^* - u$ in a relatively correct position | B1 | |
| State or imply that $OABC$ is a parallelogram | B1 | |
| **Total: 4** | | |
## Question 9(ii):
| Answer | Mark | Guidance |
|--------|------|----------|
| *EITHER*: Substitute for $u$ and multiply numerator and denominator by $3 + i$, or equivalent | M1 | |
| Simplify the numerator to $8 + 6i$ or the denominator to $10$ | A1 | |
| Obtain final answer $\frac{4}{5} + \frac{3}{5}i$, or equivalent | A1 | |
| *OR*: Substitute for $u$, obtain two equations in $x$ and $y$ and solve for $x$ or for $y$ | (M1) | |
| Obtain $x = \frac{4}{5}$ or $y = \frac{3}{5}$, or equivalent | (A1) | |
| Obtain final answer $\frac{4}{5} + \frac{3}{5}i$, or equivalent | (A1) | |
| **Total: 3** | | |
## Question 9(iii):
| Answer | Mark | Guidance |
|--------|------|----------|
| State or imply $\arg(u^*/u) = \tan^{-1}(\frac{3}{4})$ | B1 | |
| Substitute exact arguments in $\arg(u^*/u) = \arg u^* - \arg u$ | M1 | |
| Fully justify the given statement using exact values | A1 | |
| **Total: 3 marks** | | |
---9 The complex number $3 - \mathrm { i }$ is denoted by $u$. Its complex conjugate is denoted by $u ^ { * }$.\\
(i) On an Argand diagram with origin $O$, show the points $A , B$ and $C$ representing the complex numbers $u , u ^ { * }$ and $u ^ { * } - u$ respectively. What type of quadrilateral is $O A B C$ ?\\
(ii) Showing your working and without using a calculator, express $\frac { u ^ { * } } { u }$ in the form $x + \mathrm { i } y$, where $x$ and $y$ are real.\\
(iii) By considering the argument of $\frac { u ^ { * } } { u }$, prove that
$$\tan ^ { - 1 } \left( \frac { 3 } { 4 } \right) = 2 \tan ^ { - 1 } \left( \frac { 1 } { 3 } \right) .$$
\includegraphics[max width=\textwidth, alt={}, center]{d4a7604c-9e2c-47ef-a496-8697bc88fdd4-18_360_758_260_689}
The diagram shows the curve $y = \frac { x ^ { 2 } } { 1 + x ^ { 3 } }$ for $x \geqslant 0$, and its maximum point $M$. The shaded region $R$ is enclosed by the curve, the $x$-axis and the lines $x = 1$ and $x = p$.\\
(i) Find the exact value of the $x$-coordinate of $M$.\\
(ii) Calculate the value of $p$ for which the area of $R$ is equal to 1 . Give your answer correct to 3 significant figures.\\
\hfill \mbox{\textit{CAIE P3 Q9 [10]}}