| Exam Board | OCR |
|---|---|
| Module | FP1 (Further Pure Mathematics 1) |
| Year | 2013 |
| Session | January |
| Marks | 8 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Complex Numbers Argand & Loci |
| Type | Roots of polynomial equations |
| Difficulty | Standard +0.8 This is a standard FP1 question on symmetric functions of roots requiring algebraic manipulation and Vieta's formulas. Part (i) is routine expansion (though algebraically intensive), and part (ii) applies the result with standard techniques. While it requires multiple steps and careful algebra, it follows a well-established pattern for this topic with no novel insight needed. |
| Spec | 4.05a Roots and coefficients: symmetric functions |
| Answer | Marks |
|---|---|
| M1, A1, A1 [3] | Attempt at complete expansion; Obtain correct unsimplified answer; Obtain given answer correctly |
| Answer | Marks | Guidance |
|---|---|---|
| Answer: Either \(\sum\alpha = -p, \sum\alpha\beta = -4, \alpha\beta\gamma = -3\) or \(\frac{16-6p}{9}\) | B1; M1, A1, M1, A1 [5] | State (anywhere) correct values for \(\sum\alpha, \sum\alpha\beta, \sum\alpha\beta\gamma\); Express given expression as a single fraction; Obtain correct expression using (i); Use their values for sum of roots etc. in their expression; Obtain correct answer |
| Or | B1, M1, A1, M1, A1 [5] | Use substitution \(1/\sqrt{u}\); Rearrange appropriately and square out; Obtain correct co-efficients of \(u'\) and \(u^2\); Use \((+/-) b/a\) from their cubic; Obtain correct answer |
### (i)
| M1, A1, A1 [3] | Attempt at complete expansion; Obtain correct unsimplified answer; Obtain given answer correctly
### (ii)
Answer: Either $\sum\alpha = -p, \sum\alpha\beta = -4, \alpha\beta\gamma = -3$ or $\frac{16-6p}{9}$ | B1; M1, A1, M1, A1 [5] | State (anywhere) correct values for $\sum\alpha, \sum\alpha\beta, \sum\alpha\beta\gamma$; Express given expression as a single fraction; Obtain correct expression using (i); Use their values for sum of roots etc. in their expression; Obtain correct answer
Or | B1, M1, A1, M1, A1 [5] | Use substitution $1/\sqrt{u}$; Rearrange appropriately and square out; Obtain correct co-efficients of $u'$ and $u^2$; Use $(+/-) b/a$ from their cubic; Obtain correct answer\begin{enumerate}[label=(\roman*)]
\item Show that $(\alpha\beta + \beta\gamma + \gamma\alpha)^2 = \alpha^2\beta^2 + \beta^2\gamma^2 + \gamma^2\alpha^2 + 2\alpha\beta\gamma(\alpha + \beta + \gamma)$. [3]
\item It is given that $\alpha$, $\beta$ and $\gamma$ are the roots of the cubic equation $x^3 + px^2 - 4x + 3 = 0$,
where $p$ is a constant. Find the value of $\frac{1}{\alpha^2} + \frac{1}{\beta^2} + \frac{1}{\gamma^2}$ in terms of $p$. [5]
\end{enumerate}
\hfill \mbox{\textit{OCR FP1 2013 Q9 [8]}}