| Exam Board | OCR |
|---|---|
| Module | FP1 (Further Pure Mathematics 1) |
| Year | 2006 |
| Session | June |
| Marks | 11 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Roots of polynomials |
| Type | Equation with linearly transformed roots |
| Difficulty | Standard +0.3 This is a standard Further Maths question on transformed roots using Vieta's formulas. Part (i) requires direct recall of the relationships between roots and coefficients. Parts (ii) and (iii) involve routine algebraic manipulation of sums and products of transformed roots (α+1, β+1, γ+1) using the values from part (i). While it requires careful bookkeeping across multiple steps, the technique is a textbook exercise with no novel insight needed. Slightly above average difficulty due to being Further Maths content and requiring systematic multi-step work. |
| Spec | 4.05a Roots and coefficients: symmetric functions4.05b Transform equations: substitution for new roots |
| Answer | Marks | Guidance |
|---|---|---|
| Part (i) \(\alpha + \beta + \gamma = 2\) and \(\alpha\beta\gamma = -4\) | B1, B1 | Write down correct values |
| B1, 3 | \(\alpha\beta + \beta\gamma + \gamma\alpha = 3\) | |
| Part (ii) Sum new roots: \(\alpha + 1 + \beta + 1 + \gamma + 1 = 5\) | M1 | Sum new roots |
| A1ft, A1ft, 3 | Obtain numeric value using their (i); \(p = -5\) (\(p\) is negative of their answer) | |
| Part (iii) | M1* | Expand three brackets |
| A1 | \(\alpha\beta\gamma + \alpha\beta + \beta\gamma + \gamma\alpha + \alpha + \beta + \gamma + 1\) | |
| DM1 | Use their (i) results | |
| A1ft, A1ft, 5 | Obtain 2; \(q = -2\) (\(q\) is negative of their answer) | |
| Alternative for (ii) & (iii) | M2, A1, M1, A2, A1, A1 | Substitute \(x = u - 1\) in given equation; Obtain correct unsimplified equation for \(u\); Expand; Obtain \(u^3 - 5u^2 + 10u - 2 = 0\); State correct values of \(p\) and \(q\) |
**Part (i)** $\alpha + \beta + \gamma = 2$ and $\alpha\beta\gamma = -4$ | B1, B1 | Write down correct values
| B1, 3 | $\alpha\beta + \beta\gamma + \gamma\alpha = 3$
**Part (ii)** Sum new roots: $\alpha + 1 + \beta + 1 + \gamma + 1 = 5$ | M1 | Sum new roots
| A1ft, A1ft, 3 | Obtain numeric value using their (i); $p = -5$ ($p$ is negative of their answer)
**Part (iii)** | M1* | Expand three brackets
| A1 | $\alpha\beta\gamma + \alpha\beta + \beta\gamma + \gamma\alpha + \alpha + \beta + \gamma + 1$
| DM1 | Use their (i) results
| A1ft, A1ft, 5 | Obtain 2; $q = -2$ ($q$ is negative of their answer)
**Alternative for (ii) & (iii)** | M2, A1, M1, A2, A1, A1 | Substitute $x = u - 1$ in given equation; Obtain correct unsimplified equation for $u$; Expand; Obtain $u^3 - 5u^2 + 10u - 2 = 0$; State correct values of $p$ and $q$10 The cubic equation $x ^ { 3 } - 2 x ^ { 2 } + 3 x + 4 = 0$ has roots $\alpha , \beta$ and $\gamma$.\\
(i) Write down the values of $\alpha + \beta + \gamma , \alpha \beta + \beta \gamma + \gamma \alpha$ and $\alpha \beta \gamma$.
The cubic equation $x ^ { 3 } + p x ^ { 2 } + 10 x + q = 0$, where $p$ and $q$ are constants, has roots $\alpha + 1 , \beta + 1$ and $\gamma + 1$.\\
(ii) Find the value of $p$.\\
(iii) Find the value of $q$.
\hfill \mbox{\textit{OCR FP1 2006 Q10 [11]}}