OCR MEI FP1 2012 January — Question 3 6 marks

Exam BoardOCR MEI
ModuleFP1 (Further Pure Mathematics 1)
Year2012
SessionJanuary
Marks6
PaperDownload PDF ↗
Mark schemeDownload PDF ↗
TopicRoots of polynomials
TypeFactor theorem and finding roots
DifficultyModerate -0.3 This is a straightforward application of the factor theorem requiring substitution to find p, then polynomial division and solving a quadratic. While it's a Further Maths question, it follows a standard algorithmic procedure with no conceptual challenges, making it slightly easier than average overall.
Spec1.02f Solve quadratic equations: including in a function of unknown1.02j Manipulate polynomials: expanding, factorising, division, factor theorem
3 Given that \(z = 6\) is a root of the cubic equation \(z ^ { 3 } - 10 z ^ { 2 } + 37 z + p = 0\), find the value of \(p\) and the other roots.
Question 3:
AnswerMarks Guidance
AnswerMarks Guidance
\(6^3 - 10 \times 6^2 + 37 \times 6 + p = 0\)M1 Substituting in 6, or other valid method
\(\Rightarrow p = -78\)A1 cao
\(z^3 - 10z^2 + 37z - 78 = (z-6)(z^2 - 4z + 13)\)M1 Valid attempt to factorise
A1Correct quadratic factor
\(z = \dfrac{4 \pm \sqrt{16-52}}{2} = 2 \pm 3\text{j}\)M1 Valid method for solution of their 3 term quadratic
So other roots are \(2+3\text{j}\) and \(2-3\text{j}\)A1 One mark for both, cao
[6] 
## Question 3:
| Answer | Marks | Guidance |
|--------|-------|----------|
| $6^3 - 10 \times 6^2 + 37 \times 6 + p = 0$ | M1 | Substituting in 6, or other valid method |
| $\Rightarrow p = -78$ | A1 | cao |
| $z^3 - 10z^2 + 37z - 78 = (z-6)(z^2 - 4z + 13)$ | M1 | Valid attempt to factorise |
| | A1 | Correct quadratic factor |
| $z = \dfrac{4 \pm \sqrt{16-52}}{2} = 2 \pm 3\text{j}$ | M1 | Valid method for solution of their 3 term quadratic |
| So other roots are $2+3\text{j}$ and $2-3\text{j}$ | A1 | One mark for both, cao |
| **[6]** | | |

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3 Given that $z = 6$ is a root of the cubic equation $z ^ { 3 } - 10 z ^ { 2 } + 37 z + p = 0$, find the value of $p$ and the other roots.

\hfill \mbox{\textit{OCR MEI FP1 2012 Q3 [6]}}
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