| Exam Board | Pre-U |
|---|---|
| Module | Pre-U 9795/1 (Pre-U Further Mathematics Paper 1) |
| Year | 2011 |
| Session | June |
| Marks | 5 |
| Topic | Roots of polynomials |
| Type | Sum of powers of roots |
| Difficulty | Standard +0.3 This is a straightforward application of Vieta's formulas and Newton's identities. Part (i) requires the standard technique of using (α+β+γ)² = α²+β²+γ² + 2(αβ+βγ+γα), which is a common textbook exercise. Part (ii) requires recognizing that a negative sum of squares implies complex roots, which is a simple deduction once part (i) is complete. The question is slightly above average difficulty due to the two-part structure and the need to interpret the result, but remains a standard Further Maths question with no novel problem-solving required. |
| Spec | 4.05a Roots and coefficients: symmetric functions |
| Answer | Marks |
|---|---|
| (i) Noting \(\alpha + \beta + \gamma = -1\) and \(\alpha\beta + \beta\gamma + \gamma\alpha = 7\) (\(\alpha\beta\gamma = 1\)) | B1, M1 |
| \(\alpha^2 + \beta^2 + \gamma^2 = (\alpha + \beta + \gamma)^2 - 2(\alpha\beta + \beta\gamma + \gamma\alpha) = -13\) GIVEN ANSWER legit | A1 |
| [3] | |
| ALTERNATIVE Substitute \(x = \sqrt{y}\) to find eqn. with roots \(\alpha^2, \beta^2, \gamma^2\) | M1 |
| \(y^3 + 13y^2 + 51y - 1 = 0\) | A1 |
| \(\alpha^2 + \beta^2 + \gamma^2 = (-13)/1 = -13\) | A1 |
| [3] | |
| (ii) Eqn. has at least one non-real (complex) root | B1 |
| one real and two complex (conjugate) roots | B1 |
| [2] |
**(i)** Noting $\alpha + \beta + \gamma = -1$ and $\alpha\beta + \beta\gamma + \gamma\alpha = 7$ ($\alpha\beta\gamma = 1$) | B1, M1 |
$\alpha^2 + \beta^2 + \gamma^2 = (\alpha + \beta + \gamma)^2 - 2(\alpha\beta + \beta\gamma + \gamma\alpha) = -13$ **GIVEN ANSWER legit** | A1 |
| [3] |
**ALTERNATIVE** Substitute $x = \sqrt{y}$ to find eqn. with roots $\alpha^2, \beta^2, \gamma^2$ | M1 |
$y^3 + 13y^2 + 51y - 1 = 0$ | A1 |
$\alpha^2 + \beta^2 + \gamma^2 = (-13)/1 = -13$ | A1 |
| [3] |
**(ii)** Eqn. has at least one non-real (complex) root | B1 |
one real and two complex (conjugate) roots | B1 |
| [2] |
---The cubic equation $x^3 + x^2 + 7x - 1 = 0$ has roots $\alpha$, $\beta$ and $\gamma$.
\begin{enumerate}[label=(\roman*)]
\item Show that $\alpha^2 + \beta^2 + \gamma^2 = -13$. [3]
\item State what can be deduced about the nature of these roots. [2]
\end{enumerate}
\hfill \mbox{\textit{Pre-U Pre-U 9795/1 2011 Q2 [5]}}