OCR MEI FP1 2009 June — Question 4 6 marks

Exam BoardOCR MEI
ModuleFP1 (Further Pure Mathematics 1)
Year2009
SessionJune
Marks6
PaperDownload PDF ↗
Mark schemeDownload PDF ↗
TopicRoots of polynomials
TypeRoots with special relationships
DifficultyStandard +0.3 This is a straightforward application of Vieta's formulas for a cubic equation where the roots are given in terms of a single parameter. Students need to use sum and product of roots to set up two equations, solve for w, then find p and q. While it requires multiple steps, the approach is standard and mechanical with no conceptual difficulty beyond applying well-known formulas.
Spec4.05a Roots and coefficients: symmetric functions
4 The roots of the cubic equation \(2 x ^ { 3 } + x ^ { 2 } + p x + q = 0\) are \(2 w , - 6 w\) and \(3 w\). Find the values of the roots and the values of \(p\) and \(q\).
Question 4:
AnswerMarks Guidance
Answer/WorkingMarks Guidance
\(2w - 6w + 3w = \frac{-1}{2}\)M1 Use of sum of roots – can be implied
\(\Rightarrow w = \frac{1}{2}\)A1
\(\Rightarrow\) roots are \(1, -3, \frac{3}{2}\)A1, M1 Correct roots seen; Attempt to use relationships between roots; s.c. M1 for other valid method
\(\frac{-q}{2} = \alpha\beta\gamma = \frac{-9}{2} \Rightarrow q = 9\)
\(\frac{p}{2} = \alpha\beta + \alpha\gamma + \beta\gamma = -6 \Rightarrow p = -12\)A2(ft) [6] One mark each for \(p = -12\) and \(q = 9\) 
# Question 4:

| Answer/Working | Marks | Guidance |
|---|---|---|
| $2w - 6w + 3w = \frac{-1}{2}$ | M1 | Use of sum of roots – can be implied |
| $\Rightarrow w = \frac{1}{2}$ | A1 | |
| $\Rightarrow$ roots are $1, -3, \frac{3}{2}$ | A1, M1 | Correct roots seen; Attempt to use relationships between roots; s.c. M1 for other valid method |
| $\frac{-q}{2} = \alpha\beta\gamma = \frac{-9}{2} \Rightarrow q = 9$ | | |
| $\frac{p}{2} = \alpha\beta + \alpha\gamma + \beta\gamma = -6 \Rightarrow p = -12$ | A2(ft) **[6]** | One mark each for $p = -12$ and $q = 9$ |

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4 The roots of the cubic equation $2 x ^ { 3 } + x ^ { 2 } + p x + q = 0$ are $2 w , - 6 w$ and $3 w$. Find the values of the roots and the values of $p$ and $q$.

\hfill \mbox{\textit{OCR MEI FP1 2009 Q4 [6]}}
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