| Exam Board | SPS |
|---|---|
| Module | SPS FM Pure (SPS FM Pure) |
| Year | 2023 |
| Session | November |
| Marks | 8 |
| Topic | Roots of polynomials |
| Type | Equation with linearly transformed roots |
| Difficulty | Standard +0.8 This is a Further Maths question on transforming polynomial equations using roots. Part (a) requires applying Vieta's formulas to find a symmetric function of roots (straightforward but requires care). Part (b) requires a substitution to find a new polynomial equation, which is a standard FM technique but involves algebraic manipulation across multiple steps. The 'show detailed reasoning' requirement and the need for systematic algebraic work place this moderately above average difficulty. |
| Spec | 4.05a Roots and coefficients: symmetric functions4.05b Transform equations: substitution for new roots |
In this question you must show detailed reasoning.
The roots of the equation $2x^3 - 5x + 7 = 0$ are $\alpha$, $\beta$ and $\gamma$.
(a) Find $\frac{1}{\alpha} + \frac{1}{\beta} + \frac{1}{\gamma}$. [4]
(b) Find an equation with integer coefficients whose roots are $2\alpha - 1$, $2\beta - 1$ and $2\gamma - 1$. [4]
\hfill \mbox{\textit{SPS SPS FM Pure 2023 Q3 [8]}}