| Exam Board | Pre-U |
|---|---|
| Module | Pre-U 9794/1 (Pre-U Mathematics Paper 1) |
| Year | 2017 |
| Session | June |
| Marks | 9 |
| Topic | Complex Numbers Arithmetic |
| Type | Given a real root, find complex roots of cubic |
| Difficulty | Standard +0.3 This is a cubic equation where one real root (z=2) is easily found by inspection or trial, then factorization yields a quadratic for the complex roots. Finding moduli and arguments of the resulting roots is routine, as is sketching on an Argand diagram. Slightly easier than average due to the straightforward factorization and standard techniques throughout. |
| Spec | 4.02i Quadratic equations: with complex roots |
State $z = 2$ as a root either from the factor theorem or in a list of all 3 roots. No working required. **B1**
Attempt long division to obtain a quadratic factor. **M1**
Obtain $z^2 + 2z + 10$ **A1**
Use quadratic formula to solve their quadratic **M1**
Obtain $-1 + 3\text{i}$ and $-1 - 3\text{i}$ **A1**
State $-1 + 3\text{i}$ has modulus $\sqrt{10}$ and argument $1.89$ or $108°$ **B1**
State $-1 - 3\text{i}$ has modulus $\sqrt{10}$ and argument $-1.89$ or $4.39$ or $-108°$ or $252°$ **B1**
State $2$ has modulus $2$ and argument $0$ **B1**
Three correct points shown on an Argand diagram. **B1**
**Total: 9 marks**9 Solve the equation $z ^ { 3 } + 6 z - 20 = 0$. Find the modulus and argument of each root and illustrate the roots on an Argand diagram.
\hfill \mbox{\textit{Pre-U Pre-U 9794/1 2017 Q9 [9]}}