| Exam Board | Pre-U |
|---|---|
| Module | Pre-U 9795/1 (Pre-U Further Mathematics Paper 1) |
| Year | 2017 |
| Session | June |
| Marks | 4 |
| Topic | Complex Numbers Arithmetic |
| Type | Pure square root finding |
| Difficulty | Moderate -0.8 This is a straightforward complex number square root problem requiring students to expand (a+ib)², equate real and imaginary parts to get two simultaneous equations, then solve. It's a standard textbook exercise with a clear method and no conceptual difficulty beyond basic complex arithmetic, making it easier than average for A-level. |
| Spec | 4.02e Arithmetic of complex numbers: add, subtract, multiply, divide |
**Question 1**
$(a + \text{i}b)^2 = (a^2 - b^2) + \text{i}.2ab$ **B1**
$(a^2 - b^2) = 21$ and $ab = -10$ **M1** Comparing real and imaginary parts
e.g. eliminating one variable and solving for the other **M1** Allow implied by e.g. $a = 5, b = 2$ (or v.v.)
$a = \pm 5,\ b = \mp 2$ **A1** Ignore any complex answers
**Total: 4 marks**1 Without using a calculator, determine the possible values of $a$ and $b$ for which $( a + \mathrm { i } b ) ^ { 2 } = 21 - 20 \mathrm { i }$.
\hfill \mbox{\textit{Pre-U Pre-U 9795/1 2017 Q1 [4]}}