| Exam Board | OCR MEI |
|---|---|
| Module | FP1 (Further Pure Mathematics 1) |
| Year | 2006 |
| Session | January |
| Marks | 6 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Roots of polynomials |
| Type | Equation with linearly transformed roots |
| Difficulty | Standard +0.3 This is a straightforward application of standard Further Maths techniques: using Vieta's formulas to read off symmetric functions (routine recall), then forming a new equation with scaled roots using the substitution y=2x. While it's a Further Maths topic, the question requires only direct application of well-practiced methods with no problem-solving insight or complex manipulation. |
| Spec | 4.05a Roots and coefficients: symmetric functions4.05b Transform equations: substitution for new roots |
| Answer | Marks | Guidance |
|---|---|---|
| \(\alpha + \beta + \gamma = -3\) | B1 | |
| \(\alpha\beta + \beta\gamma + \gamma\alpha = -7\) | B1 | |
| \(\alpha\beta\gamma = -1\) | B1 | (allow all three on one line for B2) |
| Answer | Marks | Guidance |
|---|---|---|
| Sum of new roots \(= 2(\alpha+\beta+\gamma) = -6\) | M1 | |
| Sum of products in pairs \(= 4(\alpha\beta+\beta\gamma+\gamma\alpha) = -28\) | M1 | |
| Product of new roots \(= 8\alpha\beta\gamma = -8\) | A1 | |
| New equation: \(x^3 + 6x^2 - 28x + 8 = 0\) | A1 | cao |
# Question 5:
**(i)**
$\alpha + \beta + \gamma = -3$ | B1 |
$\alpha\beta + \beta\gamma + \gamma\alpha = -7$ | B1 |
$\alpha\beta\gamma = -1$ | B1 | (allow all three on one line for B2)
**(ii)**
Sum of new roots $= 2(\alpha+\beta+\gamma) = -6$ | M1 |
Sum of products in pairs $= 4(\alpha\beta+\beta\gamma+\gamma\alpha) = -28$ | M1 |
Product of new roots $= 8\alpha\beta\gamma = -8$ | A1 |
New equation: $x^3 + 6x^2 - 28x + 8 = 0$ | A1 | cao
---5 The cubic equation $x ^ { 3 } + 3 x ^ { 2 } - 7 x + 1 = 0$ has roots $\alpha , \beta$ and $\gamma$.\\
(i) Write down the values of $\alpha + \beta + \gamma , \alpha \beta + \beta \gamma + \gamma \alpha$ and $\alpha \beta \gamma$.\\
(ii) Find the cubic equation with roots $2 \alpha , 2 \beta$ and $2 \gamma$, simplifying your answer as far as possible.
\hfill \mbox{\textit{OCR MEI FP1 2006 Q5 [6]}}