| Exam Board | Edexcel |
|---|---|
| Module | F1 (Further Pure Mathematics 1) |
| Year | 2024 |
| Session | June |
| Marks | 9 |
| Paper | Download PDF ↗ |
| Topic | Roots of polynomials |
| Type | Quadratic with transformed roots |
| Difficulty | Challenging +1.2 This is a standard Further Maths transformed roots question requiring systematic application of Vieta's formulas and algebraic manipulation. Part (a) is routine (finding sum/product of reciprocal roots), while part (b) requires careful but methodical work with the given transformation. The multi-step nature and algebraic complexity place it above average, but it follows a well-established template without requiring novel insight. |
| Spec | 4.05a Roots and coefficients: symmetric functions4.05b Transform equations: substitution for new roots |
\begin{enumerate}
\item The equation $5 x ^ { 2 } - 4 x + 2 = 0$ has roots $\frac { 1 } { p }$ and $\frac { 1 } { q }$\\
(a) Without solving the equation,\\
(i) show that $p q = \frac { 5 } { 2 }$\\
(ii) determine the value of $p + q$\\
(b) Hence, without finding the values of $p$ and $q$, determine a quadratic equation with roots
\end{enumerate}
$$\frac { p } { p ^ { 2 } + 1 } \text { and } \frac { q } { q ^ { 2 } + 1 }$$
giving your answer in the form $a x ^ { 2 } + b x + c = 0$ where $a , b$ and $c$ are integers.
\hfill \mbox{\textit{Edexcel F1 2024 Q5 [9]}}