| Exam Board | Pre-U |
|---|---|
| Module | Pre-U 9794/2 (Pre-U Mathematics Paper 2) |
| Year | 2013 |
| Session | June |
| Marks | 10 |
| Topic | Complex Numbers Arithmetic |
| Type | Given a real root, find complex roots of cubic |
| Difficulty | Moderate -0.3 This is a straightforward multi-part question requiring verification by substitution, factorization of a cubic given one root, then solving a quadratic (likely using the formula to get complex roots), and plotting on an Argand diagram. All steps are standard techniques with no novel insight required, making it slightly easier than average. |
| Spec | 4.02i Quadratic equations: with complex roots4.02k Argand diagrams: geometric interpretation |
(i) Substitute $z = -1$ and convincingly obtain 0 — B1 **[1]**
(ii) 3 term quadratic — M1; $z^3 + 5z^2 + 9z + 5 = (z+1)(z^2 + 4z + 5) = 0$ — A1; Solve $z^2 + 4z + 5 = 0$ — M1; Obtain $-2 + i$ and $-2 - i$ — A1 **[4]**
(iii) Argand diagram showing their three roots — B1 ft **[5]**
**Total: [10]**4 (i) Verify that $z = - 1$ is a root of the equation $z ^ { 3 } + 5 z ^ { 2 } + 9 z + 5 = 0$.\\
(ii) Find the two complex roots of the equation $z ^ { 3 } + 5 z ^ { 2 } + 9 z + 5 = 0$.\\
(iii) Show all three roots on an Argand diagram.
\hfill \mbox{\textit{Pre-U Pre-U 9794/2 2013 Q4 [10]}}