| Exam Board | Pre-U |
|---|---|
| Module | Pre-U 9794/1 (Pre-U Mathematics Paper 1) |
| Year | 2010 |
| Session | June |
| Marks | 10 |
| Topic | Complex Numbers Arithmetic |
| Type | Solving equations involving complex fractions |
| Difficulty | Standard +0.3 This is a straightforward complex numbers question requiring standard techniques: (a) dividing complex numbers by multiplying by conjugate, (b)(i) using quadratic formula for complex roots, (b)(ii) basic modulus and argument calculations. All are routine A-level procedures with no novel insight required, making it slightly easier than average. |
| Spec | 4.02e Arithmetic of complex numbers: add, subtract, multiply, divide4.02f Convert between forms: cartesian and modulus-argument4.02i Quadratic equations: with complex roots4.02k Argand diagrams: geometric interpretation |
| Answer | Marks | Guidance |
|---|---|---|
| (a) Multiply numerator and denominator by \((2-i)\) or attempt to compare real and imaginary parts | M1 | |
| Calculate the denominator as 5 or obtain correct eqns | B1 | |
| Multiply numerator and collect real and imaginary parts or solve simultaneous eqns | M1 | |
| Obtain \(\frac{1}{5}(8+n) + i\frac{1}{5}(2n-4)\) or equiv | A1 | [4] |
| (b) (i) Attempt use of quadratic formula or comparing coefficients | M1 | |
| \(z_1 = -4+3i\) and \(z_2 = -4-3i\) | A1 | |
| Both shown correctly on diagram with -4 and ±3 clear | Dep B1 | [3] |
| (ii) Attempt Pythagoras' Theorem \( | z_1 + i | = |
| Attempt at least one argument | M1 | |
| Obtain \(\arg(z_1 + i) = \frac{3\pi}{4}\) \(\arg(z_2 + i) = -\frac{5\pi}{4}\) (Accept \(\frac{5\pi}{4}\)) and \(\sqrt{18}\) | A1 | [3] |
(a) Multiply numerator and denominator by $(2-i)$ or attempt to compare real and imaginary parts | M1 |
Calculate the denominator as 5 or obtain correct eqns | B1 |
Multiply numerator and collect real and imaginary parts or solve simultaneous eqns | M1 |
Obtain $\frac{1}{5}(8+n) + i\frac{1}{5}(2n-4)$ or equiv | A1 | [4]
(b) (i) Attempt use of quadratic formula or comparing coefficients | M1 |
$z_1 = -4+3i$ and $z_2 = -4-3i$ | A1 |
Both shown correctly on diagram with -4 and ±3 clear | Dep B1 | [3]
(ii) Attempt Pythagoras' Theorem $|z_1 + i| = |-3+3i| = \sqrt{18} = 3\sqrt{2} = |z_2 + i|$ | M1 |
Attempt at least one argument | M1 |
Obtain $\arg(z_1 + i) = \frac{3\pi}{4}$ $\arg(z_2 + i) = -\frac{5\pi}{4}$ (Accept $\frac{5\pi}{4}$) and $\sqrt{18}$ | A1 | [3]\begin{enumerate}[label=(\alph*)]
\item Solve the equation
$(2 + i)z = (4 + in)$.
Give your answer in the form $a + ib$, expressing $a$ and $b$ in terms of the real constant $n$. [4]
\item The roots of the equation $z^2 + 8z + 25 = 0$ are denoted by $z_1$ and $z_2$.
\begin{enumerate}[label=(\roman*)]
\item Find $z_1$ and $z_2$ and show these roots on an Argand diagram. [3]
\item Find the modulus and argument in radians of each of $(z_1 + 1)$ and $(z_2 + 1)$. [3]
\end{enumerate}
\end{enumerate}
\hfill \mbox{\textit{Pre-U Pre-U 9794/1 2010 Q10 [10]}}