| Exam Board | Edexcel |
|---|---|
| Module | F1 (Further Pure Mathematics 1) |
| Year | 2021 |
| Session | January |
| Marks | 8 |
| Paper | Download PDF ↗ |
| Topic | Roots of polynomials |
| Type | Quadratic with transformed roots |
| Difficulty | Standard +0.8 This is a Further Maths question requiring manipulation of symmetric functions of roots. Part (a) needs the identity α³+β³=(α+β)³-3αβ(α+β) using Vieta's formulas. Part (b) requires finding sum and product of transformed roots α²/β and β²/α, involving algebraic manipulation with α+β and αβ. While systematic, it demands fluency beyond standard A-level and multiple non-trivial algebraic steps. |
| Spec | 4.05a Roots and coefficients: symmetric functions4.05b Transform equations: substitution for new roots |
| VIXV SIHIANI III IM IONOO | VIAV SIHI NI JYHAM ION OO | VI4V SIHI NI JLIYM ION OO |
\begin{enumerate}
\item The equation $2 x ^ { 2 } + 5 x + 7 = 0$ has roots $\alpha$ and $\beta$
\end{enumerate}
Without solving the equation\\
(a) determine the exact value of $\alpha ^ { 3 } + \beta ^ { 3 }$\\
(b) form a quadratic equation, with integer coefficients, which has roots
$$\frac { \alpha ^ { 2 } } { \beta } \text { and } \frac { \beta ^ { 2 } } { \alpha }$$
\includegraphics[max width=\textwidth, alt={}, center]{f8660b02-384e-460f-a0e4-282ef5fef475-09_2255_50_314_34}\\
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VIXV SIHIANI III IM IONOO & VIAV SIHI NI JYHAM ION OO & VI4V SIHI NI JLIYM ION OO \\
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\hfill \mbox{\textit{Edexcel F1 2021 Q4 [8]}}