| Exam Board | AQA |
|---|---|
| Module | FP1 (Further Pure Mathematics 1) |
| Year | 2008 |
| Session | January |
| Marks | 12 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Roots of polynomials |
| Type | Quadratic with transformed roots |
| Difficulty | Standard +0.8 This FP1 question requires systematic application of root transformation theory (finding sum and product of α³ and β³ from those of α and β), complex number manipulation, and connecting abstract algebraic results to concrete calculations. While the individual techniques are standard for Further Maths, the multi-part structure requiring both symbolic manipulation without solving and verification through complex arithmetic makes it moderately challenging, above average difficulty but not requiring exceptional insight. |
| Spec | 1.02d Quadratic functions: graphs and discriminant conditions4.02a Complex numbers: real/imaginary parts, modulus, argument4.02e Arithmetic of complex numbers: add, subtract, multiply, divide4.02i Quadratic equations: with complex roots4.05a Roots and coefficients: symmetric functions4.05b Transform equations: substitution for new roots |
| Answer | Marks | Guidance |
|---|---|---|
| Working/Answer | Marks | Guidance |
| \(\alpha + \beta = 2,\; \alpha\beta = 4\) | B1B1 | |
| \(\alpha^3 + \beta^3 = (2)^3 - 3(4)(2) = -16\) | M1A1 | |
| \(\alpha^3\beta^3 = (4)^3 = 64\), hence result | M1A1 | convincingly shown (AG) |
| Subtotal | 6 |
| Answer | Marks | Guidance |
|---|---|---|
| Working/Answer | Marks | Guidance |
| Discriminant \(= 0\), so roots equal | B1E1 | or by factorisation |
| Subtotal | 2 |
| Answer | Marks | Guidance |
|---|---|---|
| Working/Answer | Marks | Guidance |
| \(x = \frac{2 \pm \sqrt{4-16}}{2}\) | M1 | or by completing the square |
| \(= 1 \pm \frac{1}{2}i\sqrt{12}\) | A1 | |
| Subtotal | 2 |
| Answer | Marks | Guidance |
|---|---|---|
| Working/Answer | Marks | Guidance |
| \(\alpha,\,\beta = 1 \pm i\sqrt{3}\) and \(\alpha^3 = \beta^3\), hence result | E2 | |
| Subtotal | 2 | |
| Total | 12 |
## Question 8(a)(i):
| Working/Answer | Marks | Guidance |
|---|---|---|
| $\alpha + \beta = 2,\; \alpha\beta = 4$ | B1B1 | |
| $\alpha^3 + \beta^3 = (2)^3 - 3(4)(2) = -16$ | M1A1 | |
| $\alpha^3\beta^3 = (4)^3 = 64$, hence result | M1A1 | convincingly shown (AG) |
| **Subtotal** | **6** | |
## Question 8(a)(ii):
| Working/Answer | Marks | Guidance |
|---|---|---|
| Discriminant $= 0$, so roots equal | B1E1 | or by factorisation |
| **Subtotal** | **2** | |
## Question 8(b):
| Working/Answer | Marks | Guidance |
|---|---|---|
| $x = \frac{2 \pm \sqrt{4-16}}{2}$ | M1 | or by completing the square |
| $= 1 \pm \frac{1}{2}i\sqrt{12}$ | A1 | |
| **Subtotal** | **2** | |
## Question 8(c):
| Working/Answer | Marks | Guidance |
|---|---|---|
| $\alpha,\,\beta = 1 \pm i\sqrt{3}$ and $\alpha^3 = \beta^3$, hence result | E2 | |
| **Subtotal** | **2** | |
| **Total** | **12** | |
---8
\begin{enumerate}[label=(\alph*)]
\item \begin{enumerate}[label=(\roman*)]
\item It is given that $\alpha$ and $\beta$ are the roots of the equation
$$x ^ { 2 } - 2 x + 4 = 0$$
Without solving this equation, show that $\alpha ^ { 3 }$ and $\beta ^ { 3 }$ are the roots of the equation
$$x ^ { 2 } + 16 x + 64 = 0$$
(6 marks)
\item State, giving a reason, whether the roots of the equation
$$x ^ { 2 } + 16 x + 64 = 0$$
are real and equal, real and distinct, or non-real.
\end{enumerate}\item Solve the equation
$$x ^ { 2 } - 2 x + 4 = 0$$
\item Use your answers to parts (a) and (b) to show that
$$( 1 + \mathrm { i } \sqrt { 3 } ) ^ { 3 } = ( 1 - \mathrm { i } \sqrt { 3 } ) ^ { 3 }$$
\end{enumerate}
\hfill \mbox{\textit{AQA FP1 2008 Q8 [12]}}