| Exam Board | OCR MEI |
|---|---|
| Module | FP1 (Further Pure Mathematics 1) |
| Year | 2005 |
| Session | June |
| Marks | 5 |
| 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 requiring straightforward application of sum/product formulas and algebraic manipulation. Part (i) is direct recall, part (ii) uses the identity α²+β²=(α+β)²-2αβ, and part (iii) applies the standard substitution method. While it's Further Maths content, the techniques are routine and well-practiced, making it slightly easier than average overall. |
| Spec | 4.05a Roots and coefficients: symmetric functions4.05b Transform equations: substitution for new roots |
| Answer | Marks | Guidance |
|---|---|---|
| \(\alpha + \beta = 2,\ \alpha\beta = 4\) | B1 [1] | Both. Accept method involving calculation of roots |
| Answer | Marks | Guidance |
|---|---|---|
| \(\alpha^2 + \beta^2 = (\alpha+\beta)^2 - 2\alpha\beta = 4 - 8 = -4\) | M1, A1(ft) | Accept method involving calculation of roots |
| Answer | Marks | Guidance |
|---|---|---|
| Sum of roots \(= 2\alpha + 2\beta = 2(\alpha+\beta) = 4\) | M1 | Or substitution method |
| Product of roots \(= 2\alpha \times 2\beta = 4\alpha\beta = 16\) | — | — |
| \(x^2 - 4x + 16 = 0\) | A1(ft) [5] | The \(= 0\), or equivalent, is necessary for final A1 |
## Question 4:
### Part (i)
$\alpha + \beta = 2,\ \alpha\beta = 4$ | B1 **[1]** | Both. Accept method involving calculation of roots
### Part (ii)
$\alpha^2 + \beta^2 = (\alpha+\beta)^2 - 2\alpha\beta = 4 - 8 = -4$ | M1, A1(ft) | Accept method involving calculation of roots
### Part (iii)
Sum of roots $= 2\alpha + 2\beta = 2(\alpha+\beta) = 4$ | M1 | Or substitution method
Product of roots $= 2\alpha \times 2\beta = 4\alpha\beta = 16$ | — | —
$x^2 - 4x + 16 = 0$ | A1(ft) **[5]** | The $= 0$, or equivalent, is necessary for final A1
---4 The quadratic equation $x ^ { 2 } - 2 x + 4 = 0$ has roots $\alpha$ and $\beta$.\\
(i) Write down the values of $\alpha + \beta$ and $\alpha \beta$.\\
(ii) Hence find the value of $\alpha ^ { 2 } + \beta ^ { 2 }$.\\
(iii) Find a quadratic equation which has roots $2 \alpha$ and $2 \beta$.
\hfill \mbox{\textit{OCR MEI FP1 2005 Q4 [5]}}