| Exam Board | Edexcel |
|---|---|
| Module | F1 (Further Pure Mathematics 1) |
| Year | 2018 |
| 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 systematic application of Vieta's formulas and algebraic manipulation of symmetric functions. Parts (a) and (b) are standard technique, but part (c) requires finding sum and product of complex transformed roots (α + α/β² and β + β/α²), involving multiple algebraic steps with fractions and careful manipulation. The multi-step nature and algebraic complexity place it above average difficulty, though it follows established methods without requiring novel insight. |
| Spec | 4.05a Roots and coefficients: symmetric functions4.05b Transform equations: substitution for new roots |
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\item The quadratic equation
\end{enumerate}
$$3 x ^ { 2 } + 2 x + 5 = 0$$
has roots $\alpha$ and $\beta$.
Without solving the equation,\\
(a) find the value of $\alpha ^ { 2 } + \beta ^ { 2 }$\\
(b) show that $\alpha ^ { 3 } + \beta ^ { 3 } = \frac { 82 } { 27 }$\\
(c) find a quadratic equation which has roots
$$\left( \alpha + \frac { \alpha } { \beta ^ { 2 } } \right) \text { and } \left( \beta + \frac { \beta } { \alpha ^ { 2 } } \right)$$
giving your answer in the form $p x ^ { 2 } + q x + r = 0$, where $p , q$ and $r$ are integers.
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\hfill \mbox{\textit{Edexcel F1 2018 Q4 [8]}}