| Exam Board | Pre-U |
|---|---|
| Module | Pre-U 9794/1 (Pre-U Mathematics Paper 1) |
| Year | 2011 |
| Session | June |
| Marks | 9 |
| Topic | Complex Numbers Argand & Loci |
| Type | Complex arithmetic operations |
| Difficulty | Moderate -0.3 Part (a) is straightforward conversion from modulus-argument form to Cartesian form using standard formulas. Part (b) requires expanding the product of two complex numbers, equating real and imaginary parts to form simultaneous equations, then solving a quadratic—all standard techniques for A-level complex numbers with no novel insight required. The 9-mark total and multi-step nature elevate it slightly above trivial, but it remains easier than average. |
| Spec | 4.02a Complex numbers: real/imaginary parts, modulus, argument4.02b Express complex numbers: cartesian and modulus-argument forms4.02e Arithmetic of complex numbers: add, subtract, multiply, divide |
| Answer | Marks | Guidance |
|---|---|---|
| (a) \(\text{Re } z = -1\) \(\text{Im } z = -\sqrt{3}\) | B1, B1 | State \(-1\). State with or without \(i\), \(-\sqrt{3}\) |
| (b) \(uv = (1 + ia)(b - i)\) | B1 | State \(uv\) |
| \(a + b = 7\) | M1 | Attempt to equate real and imaginary parts Allow aef |
| \(ab - 1 = 9\) | A1 | |
| \(a^2 - 7a + 10 = 0\) | M1 | Solve simultaneous eqns to obtain a quadratic |
| \((a-2)(a-5)\) | A1 | Attempt soln of quadratic |
| \(a = 2\) \(b = 5\) | A1 | [9] |
(a) $\text{Re } z = -1$ $\text{Im } z = -\sqrt{3}$ | B1, B1 | State $-1$. State with or without $i$, $-\sqrt{3}$
(b) $uv = (1 + ia)(b - i)$ | B1 | State $uv$
$a + b = 7$ | M1 | Attempt to equate real and imaginary parts Allow aef
$ab - 1 = 9$ | A1 |
$a^2 - 7a + 10 = 0$ | M1 | Solve simultaneous eqns to obtain a quadratic
$(a-2)(a-5)$ | A1 | Attempt soln of quadratic
$a = 2$ $b = 5$ | A1 | [9] | Obtain $a = 2$ and $b = 5$\begin{enumerate}[label=(\alph*)]
\item The complex number $z$ is such that $|z| = 2$ and $\arg z = -\frac{3}{4}\pi$. Find the exact value of the real part of $z$ and of the imaginary part of $z$. [2]
\item The complex numbers $u$ and $v$ are such that
$$u = 1 + ia \quad \text{and} \quad v = b - i,$$
where $a$ and $b$ are real and $a < b$. Given that $uv = 7 + 9i$, find the values of $a$ and $b$. [7]
\end{enumerate}
\hfill \mbox{\textit{Pre-U Pre-U 9794/1 2011 Q10 [9]}}