| Exam Board | OCR |
|---|---|
| Module | FP1 (Further Pure Mathematics 1) |
| Year | 2016 |
| Session | June |
| Marks | 6 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Roots of polynomials |
| Type | Symmetric functions of roots |
| Difficulty | Standard +0.3 This is a straightforward application of Vieta's formulas followed by algebraic manipulation. Part (i) is direct recall (α+β = -1/k, αβ = 1), and part (ii) requires expanding the expression and substituting known values—a standard textbook exercise with no novel insight required, making it slightly easier than average. |
| Spec | 4.05a Roots and coefficients: symmetric functions |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \(\alpha+\beta = -\frac{1}{k}\), \(\alpha\beta = 1\) | B1 | State correct values |
| [1] |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| *Either:* \(\alpha\beta + \frac{1}{\alpha\beta} + \frac{(\alpha+\beta)^2 - 2\alpha\beta}{\alpha\beta}\) | M1 | Expand expression |
| M1 | Use correct process for \(\alpha^2+\beta^2\) | |
| A1 | Obtain correct expression | |
| *Or:* \((\alpha+\beta)(\beta+\alpha)\) | M2 | State \(\alpha=\frac{1}{\beta}\) and \(\beta=\frac{1}{\alpha}\) and substitute into given expression |
| A1 | Obtain correct expression | |
| \(\frac{1}{k^2}\) or \(k^{-2}\) | M1 | Use their value(s) in their expression |
| A1 | Obtain correct single term answer | |
| [5] |
## Question 3:
### Part (i):
| Answer | Marks | Guidance |
|--------|-------|----------|
| $\alpha+\beta = -\frac{1}{k}$, $\alpha\beta = 1$ | B1 | State correct values |
| **[1]** | | |
### Part (ii):
| Answer | Marks | Guidance |
|--------|-------|----------|
| *Either:* $\alpha\beta + \frac{1}{\alpha\beta} + \frac{(\alpha+\beta)^2 - 2\alpha\beta}{\alpha\beta}$ | M1 | Expand expression |
| | M1 | Use correct process for $\alpha^2+\beta^2$ |
| | A1 | Obtain correct expression |
| *Or:* $(\alpha+\beta)(\beta+\alpha)$ | M2 | State $\alpha=\frac{1}{\beta}$ and $\beta=\frac{1}{\alpha}$ and substitute into given expression |
| | A1 | Obtain correct expression |
| $\frac{1}{k^2}$ or $k^{-2}$ | M1 | Use their value(s) in their expression |
| | A1 | Obtain correct single term answer |
| **[5]** | | |
---3 The quadratic equation $k x ^ { 2 } + x + k = 0$ has roots $\alpha$ and $\beta$.\\
(i) Write down the values of $\alpha + \beta$ and $\alpha \beta$.\\
(ii) Find the value of $\left( \alpha + \frac { 1 } { \alpha } \right) \left( \beta + \frac { 1 } { \beta } \right)$ in terms of $k$.
\hfill \mbox{\textit{OCR FP1 2016 Q3 [6]}}