| Exam Board | Pre-U |
|---|---|
| Module | Pre-U 9794/1 (Pre-U Mathematics Paper 1) |
| Session | Specimen |
| Marks | 5 |
| Topic | Complex numbers 2 |
| Type | Verify roots satisfy polynomial equations |
| Difficulty | Moderate -0.3 Part (i) is straightforward substitution and arithmetic with complex numbers. Part (ii) requires knowing that complex roots come in conjugate pairs, then performing polynomial division and solving a quadratic—all standard techniques for A-level. This is slightly easier than average as it's a guided, multi-part question with no novel insight required. |
| Spec | 4.02g Conjugate pairs: real coefficient polynomials4.02i Quadratic equations: with complex roots |
**(i)** Substitute $(1 + \text{i})$ completely into the equation M1
Obtain $(1+\text{i})^3 = 1 + 3\text{i} - 3 - \text{i}$ unsimplified B1
Obtain $(1+\text{i})^2 = 1 + 2\text{i} - 1$ unsimplified B1
Correctly simplify and show the result is 0 A1 **[4]**
**(ii)** State or imply that $[z-(1-\text{i})]$ is another factor B1
Attempt to multiply $[z-(1-\text{i})][z-(1+\text{i})]$ M1
Obtain the result $z^2 - 2z + 2$ A1
Attempt to find a real factor by division *OR* comparing coefficients M1
Obtain $(3z-2)$ and state $\frac{2}{3}$, $1+\text{i}$, $1-\text{i}$ A1 **[5]**9 (i) Show that $z = ( 1 + \mathrm { i } )$ is a root of the cubic equation $3 z ^ { 3 } - 8 z ^ { 2 } + 10 z - 4 = 0$.\\
(ii) Show that the equation $3 z ^ { 3 } - 8 z ^ { 2 } + 10 z - 4 = 0$ has a quadratic factor with real coefficients and hence solve this equation completely.
\hfill \mbox{\textit{Pre-U Pre-U 9794/1 Q9 [5]}}