| Exam Board | OCR |
|---|---|
| Module | FP1 (Further Pure Mathematics 1) |
| Year | 2008 |
| Session | January |
| Marks | 8 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Roots of polynomials |
| Type | Equation with nonlinearly transformed roots |
| Difficulty | Standard +0.8 This is a substantial Further Maths question combining polynomial roots transformation with telescoping series. Part (i) requires forming a quadratic from transformed roots using sum/product formulas (α³+β³ and α³β³), which is standard FP1 but multi-step. Parts (ii)-(iv) involve partial fractions, telescoping series, and solving for N—all requiring careful algebraic manipulation. The length, multiple techniques, and Further Maths context place this moderately above average difficulty. |
| Spec | 4.05a Roots and coefficients: symmetric functions4.06b Method of differences: telescoping series |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| \(\alpha^3 + 3\alpha^2\beta + 3\alpha\beta^2 + \beta^3\) | M1 | Correct binomial expansion seen |
| A1 | 2 marks Obtain given answer with no errors seen |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| Either \(\alpha + \beta = 5\), \(\alpha\beta = 7\) | B1 B1 | State or use correct values |
| \(\alpha^3 + \beta^3 = 20\) | M1 | Find numeric value for \(\alpha^3 + \beta^3\) |
| A1 | Obtain correct answer | |
| Use new sum and product correctly in quadratic expression | M1 | |
| \(x^2 - 20x + 343 = 0\) | A1ft | 6 marks Obtain correct equation |
| *Or* substitute \(x = u^{\frac{1}{3}}\) | M1 A1 | Obtain correct answer |
| \(u^{\frac{2}{3}} - 5u^{\frac{1}{3}} + 7 = 0\) | M2 | Complete method for removing fractional powers |
| \(u^3 - 20u + 343 = 0\) | A2 | 8 marks Obtain correct answer |
# Question 9:
## Part (i)
| Answer | Mark | Guidance |
|--------|------|----------|
| $\alpha^3 + 3\alpha^2\beta + 3\alpha\beta^2 + \beta^3$ | M1 | Correct binomial expansion seen |
| | A1 | **2 marks** Obtain given answer with no errors seen |
## Part (ii)
| Answer | Mark | Guidance |
|--------|------|----------|
| Either $\alpha + \beta = 5$, $\alpha\beta = 7$ | B1 B1 | State or use correct values |
| $\alpha^3 + \beta^3 = 20$ | M1 | Find numeric value for $\alpha^3 + \beta^3$ |
| | A1 | Obtain correct answer |
| Use new sum and product correctly in quadratic expression | M1 | |
| $x^2 - 20x + 343 = 0$ | A1ft | **6 marks** Obtain correct equation |
| *Or* substitute $x = u^{\frac{1}{3}}$ | M1 A1 | Obtain correct answer |
| $u^{\frac{2}{3}} - 5u^{\frac{1}{3}} + 7 = 0$ | M2 | Complete method for removing fractional powers |
| $u^3 - 20u + 343 = 0$ | A2 | **8 marks** Obtain correct answer |
---9 (i) Show that $\alpha ^ { 3 } + \beta ^ { 3 } = ( \alpha + \beta ) ^ { 3 } - 3 \alpha \beta ( \alpha + \beta )$.\\
(ii) The quadratic equation $x ^ { 2 } - 5 x + 7 = 0$ has roots $\alpha$ and $\beta$. Find a quadratic equation with roots $\alpha ^ { 3 }$ and $\beta ^ { 3 }$.\\
(i) Show that $\frac { 2 } { r } - \frac { 1 } { r + 1 } - \frac { 1 } { r + 2 } = \frac { 3 r + 4 } { r ( r + 1 ) ( r + 2 ) }$.\\
(ii) Hence find an expression, in terms of $n$, for
$$\sum _ { r = 1 } ^ { n } \frac { 3 r + 4 } { r ( r + 1 ) ( r + 2 ) }$$
(iii) Hence write down the value of $\sum _ { r = 1 } ^ { \infty } \frac { 3 r + 4 } { r ( r + 1 ) ( r + 2 ) }$.\\
(iv) Given that $\sum _ { r = N + 1 } ^ { \infty } \frac { 3 r + 4 } { r ( r + 1 ) ( r + 2 ) } = \frac { 7 } { 10 }$, find the value of $N$.
\hfill \mbox{\textit{OCR FP1 2008 Q9 [8]}}