| Exam Board | Pre-U |
|---|---|
| Module | Pre-U 9794/2 (Pre-U Mathematics Paper 2) |
| Year | 2014 |
| Session | June |
| Marks | 2 |
| Topic | Complex Numbers Arithmetic |
| Type | Factored form to roots |
| Difficulty | Moderate -0.8 This is a straightforward quadratic-in-disguise factorization (substituting w=z²) followed by finding four simple roots (±1, ±2i) and plotting them on an Argand diagram. Requires only routine algebraic manipulation and basic complex number knowledge, making it easier than average A-level material. |
| Spec | 4.02j Cubic/quartic equations: conjugate pairs and factor theorem4.02k Argand diagrams: geometric interpretation |
(i) $(z^2 + 4)(z^2 - 1)$
B2 [2] Stating $a = 4$, $b = -1$ (or reverse) gets B2. Allow B1 for $(z^2 - 4)(z^2 + 1)$. Allow B1 (BOD) for $(z+4)(z-1)$
(ii) [Argand diagram with points at $2i$, $-2i$, $1$, $-1$]
$\sqrt{B1}$ At least 2 correct points, following their $a$ & $b$
$\sqrt{B1}$ [2] All 4 correct points, following their $a$ & $b$ as long as one positive and one negative. NIS so B1B0 if locus drawn through points. Allow just 2 on axis as long as $2i$ seen in solution, or axis is labelled as Im7 (i) Express $z ^ { 4 } + 3 z ^ { 2 } - 4$ in the form $\left( z ^ { 2 } + a \right) \left( z ^ { 2 } + b \right)$ where $a$ and $b$ are real constants to be found.\\
(ii) Hence draw an Argand diagram showing the points that represent the roots of the equation $z ^ { 4 } + 3 z ^ { 2 } - 4 = 0$.
\hfill \mbox{\textit{Pre-U Pre-U 9794/2 2014 Q7 [2]}}