| Exam Board | Pre-U |
|---|---|
| Module | Pre-U 9795/1 (Pre-U Further Mathematics Paper 1) |
| Year | 2018 |
| Session | June |
| Marks | 8 |
| Topic | Roots of polynomials |
| Type | Equation with nonlinearly transformed roots |
| Difficulty | Challenging +1.8 This is a sophisticated roots transformation problem requiring manipulation of symmetric functions and substitution techniques. Part (i) needs algebraic insight to relate the new roots to the original ones using Vieta's formulas (sum of roots = 3). Part (ii) requires either constructing a new equation via substitution y = 3/x or computing symmetric functions of the transformed roots—both approaches demand careful algebraic manipulation across multiple steps. This goes well beyond routine application of Vieta's formulas and requires genuine problem-solving insight, though it follows a recognizable pattern for Further Maths students. |
| Spec | 4.05a Roots and coefficients: symmetric functions4.05b Transform equations: substitution for new roots |
The cubic equation $4x^3 - 12x^2 + 9x - 16 = 0$ has roots $r_1$, $r_2$ and $r_3$. A second cubic equation, with integer coefficients, has roots $R_1 = \frac{r_2 + r_3}{r_1}$, $R_2 = \frac{r_3 + r_1}{r_2}$ and $R_3 = \frac{r_1 + r_2}{r_3}$.
\begin{enumerate}[label=(\roman*)]
\item Show that $1 + R_1 = \frac{3}{r_1}$ and write down the corresponding results for the other roots. [2]
\item Using a substitution based on this result, or otherwise, find this second cubic equation. [6]
\end{enumerate}
\hfill \mbox{\textit{Pre-U Pre-U 9795/1 2018 Q6 [8]}}