| Exam Board | OCR |
|---|---|
| Module | FP1 (Further Pure Mathematics 1) |
| Year | 2007 |
| Session | January |
| Marks | 8 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Roots of polynomials |
| Type | Quadratic with transformed roots |
| Difficulty | Standard +0.3 This is a standard FP1 question on transformed roots using sum/product relationships. Parts (i) and (ii) are routine applications of Vieta's formulas and algebraic manipulation. Part (iii) requires finding sum and product of the new roots using established results, which is a textbook technique. While it involves multiple steps, each is straightforward with no novel insight required—slightly easier than average for Further Maths content. |
| Spec | 4.05a Roots and coefficients: symmetric functions4.05b Transform equations: substitution for new roots |
| Answer | Marks | Guidance |
|---|---|---|
| (i) \(\alpha + \beta = -5\), \(\alpha\beta = 10\) | B1 B1 | State correct values |
| (ii) \(\alpha^2 + \beta^2 = 5\) | M1 | Use \((\alpha + \beta)^2 - 2\alpha\beta\) |
| A1 | Obtain given answer correctly, using value of \(-5\) | |
| (iii) Product of roots \(= 1\) | B1 | |
| M1 | Attempt to find sum of roots | |
| A1 | Obtain \(\frac{5}{10}\) or equivalent | |
| \(x^2 - \frac{1}{2}x + 1 = 0\) | B1ft | Write down required quadratic equation, or any multiple. |
(i) $\alpha + \beta = -5$, $\alpha\beta = 10$ | B1 B1 | State correct values
(ii) $\alpha^2 + \beta^2 = 5$ | M1 | Use $(\alpha + \beta)^2 - 2\alpha\beta$
| A1 | Obtain given answer correctly, using value of $-5$
(iii) Product of roots $= 1$ | B1 |
| M1 | Attempt to find sum of roots
| A1 | Obtain $\frac{5}{10}$ or equivalent
$x^2 - \frac{1}{2}x + 1 = 0$ | B1ft | Write down required quadratic equation, or any multiple. | (4 marks, 8 total)7 The quadratic equation $x ^ { 2 } + 5 x + 10 = 0$ has roots $\alpha$ and $\beta$.\\
(i) Write down the values of $\alpha + \beta$ and $\alpha \beta$.\\
(ii) Show that $\alpha ^ { 2 } + \beta ^ { 2 } = 5$.\\
(iii) Hence find a quadratic equation which has roots $\frac { \alpha } { \beta }$ and $\frac { \beta } { \alpha }$.
\hfill \mbox{\textit{OCR FP1 2007 Q7 [8]}}