| Exam Board | OCR |
|---|---|
| Module | FP1 (Further Pure Mathematics 1) |
| Year | 2010 |
| Session | January |
| Marks | 5 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Roots of polynomials |
| Type | Substitution to find new equation |
| Difficulty | Standard +0.3 This is a straightforward Further Maths question requiring routine algebraic substitution and application of Vieta's formulas. The substitution x = u - 1 is explicitly given, making part (i) mechanical expansion. Part (ii) requires recognizing that if α, β, γ are roots of the original equation, then α+1, β+1, γ+1 are roots of the new equation, so their product equals -constant/leading coefficient. While this is Further Maths content, it's a standard textbook exercise with no novel insight required. |
| Spec | 4.05a Roots and coefficients: symmetric functions4.05b Transform equations: substitution for new roots |
2 The cubic equation $2 x ^ { 3 } + 3 x - 3 = 0$ has roots $\alpha , \beta$ and $\gamma$.\\
(i) Use the substitution $x = u - 1$ to find a cubic equation in $u$ with integer coefficients.\\
(ii) Hence find the value of $( \alpha + 1 ) ( \beta + 1 ) ( \gamma + 1 )$.
\hfill \mbox{\textit{OCR FP1 2010 Q2 [5]}}