| Exam Board | OCR |
|---|---|
| Module | Further Pure Core 2 (Further Pure Core 2) |
| Year | 2021 |
| Session | June |
| Marks | 6 |
| Topic | Roots of polynomials |
| Type | Equation with nonlinearly transformed roots |
| Difficulty | Challenging +1.2 This is a Further Maths question on transformed roots requiring systematic application of Newton's identities and algebraic manipulation of symmetric functions. While it involves multiple steps and careful algebra, the techniques are standard for FM students who have learned this topic—no novel insight is required, just methodical execution of learned procedures. |
| Spec | 4.05a Roots and coefficients: symmetric functions4.05b Transform equations: substitution for new roots |
1 In this question you must show detailed reasoning.\\
The roots of the equation $3 x ^ { 3 } - 2 x ^ { 2 } - 5 x - 4 = 0$ are $\alpha , \beta$ and $\gamma$.
\begin{enumerate}[label=(\alph*)]
\item Find a cubic equation with integer coefficients whose roots are $\alpha ^ { 2 } , \beta ^ { 2 }$ and $\gamma ^ { 2 }$.
\item Find the exact value of $\frac { \alpha ^ { 2 } \beta ^ { 2 } + \beta ^ { 2 } \gamma ^ { 2 } + \gamma ^ { 2 } \alpha ^ { 2 } } { \alpha \beta \gamma }$.
\end{enumerate}
\hfill \mbox{\textit{OCR Further Pure Core 2 2021 Q1 [6]}}