| Exam Board | Pre-U |
|---|---|
| Module | Pre-U 9794/1 (Pre-U Mathematics Paper 1) |
| Session | Specimen |
| Marks | 10 |
| Topic | Complex Numbers Arithmetic |
| Type | Modulus-argument form conversions |
| Difficulty | Moderate -0.3 This is a straightforward multi-part question testing standard complex number techniques: solving a linear equation for z, finding modulus and argument, squaring a complex number, and verifying a trigonometric identity. All parts are routine applications of basic complex number theory with no novel insight required, making it slightly 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 |
|---|---|---|
| (ii) \( | z | = 1\) [B1] |
(i) Rearrange and attempt to solve for $z$ [M1]
Attempt to rationalise $\frac{2+i}{2-i} \times \frac{2+i}{2+i}$ [M1]
Obtain $\frac{3}{5} + \frac{4}{5}i$ [A1] **3 marks**
(ii) $|z| = 1$ [B1]
$\arg(z) = \tan^{-1}\left(\frac{4}{3}\right)$ [B1] **2 marks**
(iii) $z^2 = -\frac{7}{25} + \frac{24}{25}i$ [B1] **1 mark**
(iv) $\tan(\arg(z^2)) = -\frac{24}{7}$ [B1]
Attempt to use $\tan 2\theta = \frac{2\tan\theta}{1 - \tan^2\theta}$ [M1]
for $\tan\theta = \frac{4}{3}$ [A1]
$\frac{2 \times \frac{4}{3}}{1 - \frac{16}{9}} = -\frac{24}{7}$ [A1 (AG)] **4 marks**5 The complex number $z$ satisfies the equation $2 z - \mathrm { i } = \mathrm { i } z + 2$.\\
(i) Express $z$ in the form $a + \mathrm { i } b$ where $a$ and $b$ are rational numbers.\\
(ii) Find the exact value of $| z |$ and of $\arg ( z )$.\\
(iii) Express $z ^ { 2 }$ in the form $c + \mathrm { i } d$ where $c$ and $d$ are rational numbers.\\
(iv) Verify that $\tan ( 2 \arg ( z ) ) = \tan \left( \arg \left( z ^ { 2 } \right) \right)$ using an appropriate trigonometrical identity.
\hfill \mbox{\textit{Pre-U Pre-U 9794/1 Q5 [10]}}