| Exam Board | OCR |
|---|---|
| Module | FP1 (Further Pure Mathematics 1) |
| Year | 2012 |
| Session | June |
| Marks | 6 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Roots of polynomials |
| Type | Substitution to find new equation |
| Difficulty | Standard +0.3 This is a standard Further Maths transformation question requiring substitution into a polynomial and application of Vieta's formulas. While it involves multiple steps and algebraic manipulation, the techniques are routine for FP1 students—substitute, simplify to get coefficients, then use product of roots. Slightly above average difficulty due to being Further Maths content, but still a textbook-style exercise. |
| Spec | 4.05b Transform equations: substitution for new roots |
| Answer | Marks | Guidance |
|---|---|---|
| M1 | Attempt to clear fractions | |
| M1 | Attempt to expand and simplify to a quadratic | |
| \(5u^2 + 11u + 8 = 0\) | A1 | Obtain correct answer, must be an equation |
| Answer | Marks | Guidance |
|---|---|---|
| \(u = \frac{1}{x} - 1\) | B1 | State or imply by using roots of new quadratic |
| M1 | Use their \(c/a\) | |
| \(\frac{8}{5}\) | A1 FT | Obtain correct answer |
| Answer | Marks | Guidance |
|---|---|---|
| \(\frac{1}{\alpha\beta} - \frac{\alpha+\beta}{\alpha\beta} + 1\) | B1 | Express in terms of \(\alpha + \beta\) and \(\alpha\beta\) |
| M1 | Use values \(-\frac{1}{2}\) and \(\frac{5}{2}\) correctly (must be values from original equation) | |
| \(\frac{8}{5}\) | A1 | Obtain correct answer |
## Question 6:
### Part (i)
| M1 | Attempt to clear fractions
| M1 | Attempt to expand and simplify to a quadratic
$5u^2 + 11u + 8 = 0$ | A1 | Obtain correct answer, must be an equation
**[3]**
### Part (ii)
**EITHER**
$u = \frac{1}{x} - 1$ | B1 | State or imply by using roots of new quadratic
| M1 | Use their $c/a$
$\frac{8}{5}$ | A1 FT | Obtain correct answer
**[3]**
**OR**
$\frac{1}{\alpha\beta} - \frac{\alpha+\beta}{\alpha\beta} + 1$ | B1 | Express in terms of $\alpha + \beta$ and $\alpha\beta$
| M1 | Use values $-\frac{1}{2}$ and $\frac{5}{2}$ correctly (must be values from original equation)
$\frac{8}{5}$ | A1 | Obtain correct answer
---6 The quadratic equation $2 x ^ { 2 } + x + 5 = 0$ has roots $\alpha$ and $\beta$.\\
(i) Use the substitution $x = \frac { 1 } { u + 1 }$ to obtain a quadratic equation in $u$ with integer coefficients.\\
(ii) Hence, or otherwise, find the value of $\left( \frac { 1 } { \alpha } - 1 \right) \left( \frac { 1 } { \beta } - 1 \right)$.
\hfill \mbox{\textit{OCR FP1 2012 Q6 [6]}}