OCR C1 2009 January — Question 9 7 marks

Exam BoardOCR
ModuleC1 (Core Mathematics 1)
Year2009
SessionJanuary
Marks7
PaperDownload PDF ↗
Mark schemeDownload PDF ↗
TopicStationary points and optimisation
TypeDetermine constant from stationary point condition
DifficultyModerate -0.3 This is a straightforward application of differentiation to find stationary points. Students must differentiate, substitute x=4 into dy/dx=0 to find p, then use the second derivative test. While it requires multiple steps, each is routine and the question clearly signposts what to do, making it slightly easier than average.
Spec1.07i Differentiate x^n: for rational n and sums1.07n Stationary points: find maxima, minima using derivatives1.07o Increasing/decreasing: functions using sign of dy/dx
9 The curve \(y = x ^ { 3 } + p x ^ { 2 } + 2\) has a stationary point when \(x = 4\). Find the value of the constant \(p\) and determine whether the stationary point is a maximum or minimum point.
Question 9:
AnswerMarks Guidance
AnswerMark Guidance
\(\frac{dy}{dx} = 3x^2+2px\)M1, A1 Attempt to differentiate; correct expression cao
When \(x=4\), \(\frac{dy}{dx}=0\)M1 Setting their \(\frac{dy}{dx}=0\)
\(\therefore 3\times4^2+8p=0\)M1 Substitution of \(x=4\) into their \(\frac{dy}{dx}=0\) to evaluate \(p\)
\(8p=-48\), \(p=-6\)A1
\(\frac{d^2y}{dx^2}=6x-12\)M1 Looks at sign of \(\frac{d^2y}{dx^2}\), derived correctly from their \(\frac{dy}{dx}\), or other correct method
When \(x=4\), \(6x-12>0\), Minimum pointA1 (7) Minimum point CWO 
## Question 9:

| Answer | Mark | Guidance |
|--------|------|----------|
| $\frac{dy}{dx} = 3x^2+2px$ | M1, A1 | Attempt to differentiate; correct expression cao |
| When $x=4$, $\frac{dy}{dx}=0$ | M1 | Setting their $\frac{dy}{dx}=0$ |
| $\therefore 3\times4^2+8p=0$ | M1 | Substitution of $x=4$ into their $\frac{dy}{dx}=0$ to evaluate $p$ |
| $8p=-48$, $p=-6$ | A1 | |
| $\frac{d^2y}{dx^2}=6x-12$ | M1 | Looks at sign of $\frac{d^2y}{dx^2}$, derived correctly from their $\frac{dy}{dx}$, or other correct method |
| When $x=4$, $6x-12>0$, Minimum point | A1 (7) | Minimum point CWO |

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9 The curve $y = x ^ { 3 } + p x ^ { 2 } + 2$ has a stationary point when $x = 4$. Find the value of the constant $p$ and determine whether the stationary point is a maximum or minimum point.

\hfill \mbox{\textit{OCR C1 2009 Q9 [7]}}
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