| Exam Board | OCR |
|---|---|
| Module | FP1 (Further Pure Mathematics 1) |
| Year | 2012 |
| Session | January |
| Marks | 12 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Roots of polynomials |
| Type | Equation with nonlinearly transformed roots |
| Difficulty | Standard +0.3 This is a standard Further Maths question on symmetric functions and transformed roots. Part (i) is direct application of formulas relating coefficients to sums of roots. Part (ii) requires using identities to express sums/products of squared roots in terms of the original symmetric functions, which is a well-practiced technique in FP1. The working is methodical rather than requiring novel insight. |
| Spec | 4.05a Roots and coefficients: symmetric functions |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \(\alpha+\beta+\gamma = 3\) | B1 | State correct value |
| \(\alpha\beta+\beta\gamma+\gamma\alpha = 2\) | B1 | State correct value |
| \(\alpha\beta\gamma = -\frac{2}{3}\) | B1 | State correct value |
| [3] |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \(c = (\pm)\alpha^2\beta^2\gamma^2\) | M1 | |
| \(c = -\frac{4}{9}\) | A1FT | Obtain given correct answer (FT for sign error in (i)) |
| \(\sum\alpha^2 = (\sum\alpha)^2 - 2\sum\alpha\beta\) | M1 | Use correct expression |
| \(= 5\) | A1FT | Obtain correct value (FT for sign error in (i)) |
| \(a = -5\) | A1FT | Obtain answer correctly; sign change done correctly |
| \(\sum\alpha^2\beta^2 = (\sum\alpha\beta)^2 - 2\alpha\beta\gamma\sum\alpha\) | M1* | Attempt to find an identity |
| \(= 4\) (correct identity) | A1 | |
| Use appropriate values | DM1 | |
| \(b = 8\) | A1 | Obtain correct answer cao |
| [9] |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| State or use correct substitution | B1 | |
| Rearrange, fractional indices isolated | M1 | |
| Square both sides | DM1 | |
| Expand and simplify | DM1 | |
| \(9y^3 - 45y^2 + 72y - 4 = 0\) | A1 | Obtain correct equation |
| Use coefficients of their cubic | M1 | |
| \(c = -\frac{4}{9}\) | A1 | Obtain given answer correctly |
| \(a = -5\) | A1FT | Obtain correct answer |
| \(b = 8\) | A1FT | Obtain correct answer; SC mixture of methods only A1FT for \(a\) and \(b\) |
| [9] |
## Question 10(i):
| Answer | Marks | Guidance |
|--------|-------|----------|
| $\alpha+\beta+\gamma = 3$ | B1 | State correct value |
| $\alpha\beta+\beta\gamma+\gamma\alpha = 2$ | B1 | State correct value |
| $\alpha\beta\gamma = -\frac{2}{3}$ | B1 | State correct value |
| **[3]** | | |
---
## Question 10(ii):
**EITHER method:**
| Answer | Marks | Guidance |
|--------|-------|----------|
| $c = (\pm)\alpha^2\beta^2\gamma^2$ | M1 | |
| $c = -\frac{4}{9}$ | A1FT | Obtain given correct answer (FT for sign error in (i)) |
| $\sum\alpha^2 = (\sum\alpha)^2 - 2\sum\alpha\beta$ | M1 | Use correct expression |
| $= 5$ | A1FT | Obtain correct value (FT for sign error in (i)) |
| $a = -5$ | A1FT | Obtain answer correctly; sign change done correctly |
| $\sum\alpha^2\beta^2 = (\sum\alpha\beta)^2 - 2\alpha\beta\gamma\sum\alpha$ | M1* | Attempt to find an identity |
| $= 4$ (correct identity) | A1 | |
| Use appropriate values | DM1 | |
| $b = 8$ | A1 | Obtain correct answer cao |
| **[9]** | | |
**OR method:**
| Answer | Marks | Guidance |
|--------|-------|----------|
| State or use correct substitution | B1 | |
| Rearrange, fractional indices isolated | M1 | |
| Square both sides | DM1 | |
| Expand and simplify | DM1 | |
| $9y^3 - 45y^2 + 72y - 4 = 0$ | A1 | Obtain correct equation |
| Use coefficients of their cubic | M1 | |
| $c = -\frac{4}{9}$ | A1 | Obtain given answer correctly |
| $a = -5$ | A1FT | Obtain correct answer |
| $b = 8$ | A1FT | Obtain correct answer; **SC** mixture of methods only A1FT for $a$ and $b$ |
| **[9]** | | |
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Could you upload the correct pages?10 The cubic equation $3 x ^ { 3 } - 9 x ^ { 2 } + 6 x + 2 = 0$ has roots $\alpha , \beta$ and $\gamma$.\\
(i) Write down the values of $\alpha + \beta + \gamma , \alpha \beta + \beta \gamma + \gamma \alpha$ and $\alpha \beta \gamma$.
The cubic equation $x ^ { 3 } + a x ^ { 2 } + b x + c = 0$ has roots $\alpha ^ { 2 } , \beta ^ { 2 }$ and $\gamma ^ { 2 }$.\\
(ii) Show that $c = - \frac { 4 } { 9 }$ and find the values of $a$ and $b$.
\section*{THERE ARE NO QUESTIONS WRITTEN ON THIS PAGE}
\hfill \mbox{\textit{OCR FP1 2012 Q10 [12]}}