| Exam Board | OCR |
|---|---|
| Module | Further Pure Core 1 (Further Pure Core 1) |
| Year | 2019 |
| Session | June |
| Marks | 4 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Roots of polynomials |
| Type | Quadratic with transformed roots |
| Difficulty | Standard +0.8 This is a Further Maths question requiring manipulation of root relationships beyond standard sum/product formulas. Part (a) is routine recall, but part (b) requires algebraic insight to express the new sum and product of transformed roots in terms of α+β and αβ, involving multiple algebraic steps with fractions. More challenging than typical A-level questions but standard for FM. |
| Spec | 4.05a Roots and coefficients: symmetric functions4.05b Transform equations: substitution for new roots |
| Answer | Marks | Guidance |
|---|---|---|
| 1 | (a) | DR |
| α+β=2, αβ=5 | B1 | |
| [1] | 1.1 | |
| (b) | DR |
| Answer | Marks |
|---|---|
| ⇒5x2 −12x+36=0 oe | M1 |
| Answer | Marks |
|---|---|
| [3] | 1.1a |
| Answer | Marks |
|---|---|
| 2.2a | Attempt both sum and |
| Answer | Marks |
|---|---|
| interpreted as quadratic. | DR so finding roots |
Question 1:
1 | (a) | DR
α+β=2, αβ=5 | B1
[1] | 1.1
(b) | DR
1 1 α+β 2 12
α+ + β+ =α+β+ =2+ =
β α αβ 5 5
1 1 1 1 36
α+ × β+ =αβ+2+ =7+ =
β α αβ 5 5
⇒5x2 −12x+36=0 oe | M1
A1
A1
[3] | 1.1a
1.1
2.2a | Attempt both sum and
product of new roots in
terms of original roots
For one of 12/5 or 36/5
For both, correctly
interpreted as quadratic. | DR so finding roots
M01 In this question you must show detailed reasoning.\\
The quadratic equation $x ^ { 2 } - 2 x + 5 = 0$ has roots $\alpha$ and $\beta$.
\begin{enumerate}[label=(\alph*)]
\item Write down the values of $\alpha + \beta$ and $\alpha \beta$.
\item Hence find a quadratic equation with roots $\alpha + \frac { 1 } { \beta }$ and $\beta + \frac { 1 } { \alpha }$.
\end{enumerate}
\hfill \mbox{\textit{OCR Further Pure Core 1 2019 Q1 [4]}}