| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| Expand and simplify as far as \(iw^2 = -8i\) or equivalent | B1 | |
| Obtain first answer \(i\sqrt{8}\), or equivalent | B1 | |
| Obtain second answer \(-i\sqrt{8}\), or equivalent and no others | B1 | [3] |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| Draw circle with centre in first quadrant | M1 | |
| Draw correct circle with interior shaded or indicated | A1 | [2] |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| Identify ends of diameter corresponding to line through origin and centre | M1 | |
| Obtain \(p = 3.66\) and \(q = 7.66\) | A1 | |
| Show tangents from origin to circle | M1 | |
| Evaluate \(\sin^{-1}\left(\frac{1}{4}\sqrt{2}\right)\) | M1 | |
| Obtain \(\alpha = \frac{1}{4}\pi - \sin^{-1}\left(\frac{1}{4}\sqrt{2}\right)\) or equivalent and hence \(0.424\) | A1 | |
| Obtain \(\beta = \frac{1}{4}\pi + \sin^{-1}\left(\frac{1}{4}\sqrt{2}\right)\) or equivalent and hence \(1.15\) | A1 | [6] |
# Question 10:
## Part (a)
| Answer/Working | Mark | Guidance |
|---|---|---|
| Expand and simplify as far as $iw^2 = -8i$ or equivalent | B1 | |
| Obtain first answer $i\sqrt{8}$, or equivalent | B1 | |
| Obtain second answer $-i\sqrt{8}$, or equivalent and no others | B1 | [3] |
## Part (b)(i)
| Answer/Working | Mark | Guidance |
|---|---|---|
| Draw circle with centre in first quadrant | M1 | |
| Draw correct circle with interior shaded or indicated | A1 | [2] |
## Part (b)(ii)
| Answer/Working | Mark | Guidance |
|---|---|---|
| Identify ends of diameter corresponding to line through origin and centre | M1 | |
| Obtain $p = 3.66$ and $q = 7.66$ | A1 | |
| Show tangents from origin to circle | M1 | |
| Evaluate $\sin^{-1}\left(\frac{1}{4}\sqrt{2}\right)$ | M1 | |
| Obtain $\alpha = \frac{1}{4}\pi - \sin^{-1}\left(\frac{1}{4}\sqrt{2}\right)$ or equivalent and hence $0.424$ | A1 | |
| Obtain $\beta = \frac{1}{4}\pi + \sin^{-1}\left(\frac{1}{4}\sqrt{2}\right)$ or equivalent and hence $1.15$ | A1 | [6] |10
\begin{enumerate}[label=(\alph*)]
\item Without using a calculator, solve the equation $\mathrm { i } w ^ { 2 } = ( 2 - 2 \mathrm { i } ) ^ { 2 }$.
\item \begin{enumerate}[label=(\roman*)]
\item Sketch an Argand diagram showing the region $R$ consisting of points representing the complex numbers $z$ where
$$| z - 4 - 4 i | \leqslant 2$$
\item For the complex numbers represented by points in the region $R$, it is given that
$$p \leqslant | z | \leqslant q \quad \text { and } \quad \alpha \leqslant \arg z \leqslant \beta$$
Find the values of $p , q , \alpha$ and $\beta$, giving your answers correct to 3 significant figures.
\end{enumerate}\end{enumerate}
\hfill \mbox{\textit{CAIE P3 2012 Q10 [11]}}