OCR MEI FP1 2012 June — Question 3 6 marks

Exam BoardOCR MEI
ModuleFP1 (Further Pure Mathematics 1)
Year2012
SessionJune
Marks6
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Mark schemeDownload PDF ↗
TopicRoots of polynomials
TypeRoots with special relationships
DifficultyStandard +0.8 This is a Further Maths question requiring systematic use of Vieta's formulas with algebraically complex roots. Students must set up and solve a system of equations involving the sum and product of roots (α + α/6 + α-7, etc.), which involves careful algebraic manipulation with fractions and multiple unknowns. While the technique is standard for FP1, the execution requires more algebraic sophistication than typical A-level questions.
Spec4.05a Roots and coefficients: symmetric functions4.05b Transform equations: substitution for new roots
3 The cubic equation \(3 x ^ { 3 } + 8 x ^ { 2 } + p x + q = 0\) has roots \(\alpha , \frac { \alpha } { 6 }\) and \(\alpha - 7\). Find the values of \(\alpha , p\) and \(q\).
AnswerMarks Guidance
AnswerMarks Guidance
OR, for final four marks \((x-2)(x+5)(3x-1) = 3x^3 + 8x^2 - 33x + 10 \Rightarrow p = -33\) and \(q = 10\)M1, M1, A1, A1 Express as product of factors; Expanding; \(p = -33\) cao; \(q = 10\) cao 
| Answer | Marks | Guidance |
|--------|-------|----------|
| OR, for final four marks $(x-2)(x+5)(3x-1) = 3x^3 + 8x^2 - 33x + 10 \Rightarrow p = -33$ and $q = 10$ | M1, M1, A1, A1 | Express as product of factors; Expanding; $p = -33$ cao; $q = 10$ cao |
3 The cubic equation $3 x ^ { 3 } + 8 x ^ { 2 } + p x + q = 0$ has roots $\alpha , \frac { \alpha } { 6 }$ and $\alpha - 7$. Find the values of $\alpha , p$ and $q$.

\hfill \mbox{\textit{OCR MEI FP1 2012 Q3 [6]}}
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Q1 5 Q2 7 Q3 6 Q4 4 Q5 7 Q6 7 Q7 14 Q8 10 Q9 12