CAIE P1 2013 June — Question 6 7 marks

Exam BoardCAIE
ModuleP1 (Pure Mathematics 1)
Year2013
SessionJune
Marks7
PaperDownload PDF ↗
Mark schemeDownload PDF ↗
TopicProduct & Quotient Rules
TypeFind stationary points and nature
DifficultyStandard +0.3 This is a straightforward optimization problem requiring substitution to express u as a function of one variable, then applying the product rule to find du/dx, solving du/dx=0, and using the second derivative test. While it involves multiple steps, each is standard and the problem setup clearly guides the approach with no conceptual surprises.
Spec1.07n Stationary points: find maxima, minima using derivatives1.07r Chain rule: dy/dx = dy/du * du/dx and connected rates
6 The non-zero variables \(x , y\) and \(u\) are such that \(u = x ^ { 2 } y\). Given that \(y + 3 x = 9\), find the stationary value of \(u\) and determine whether this is a maximum or a minimum value.
Question 6:
AnswerMarks Guidance
Answer/WorkingMarks Guidance
\(u = x^2y \quad y + 3x = 9\)M1 Expressing \(u\) in terms of 1 variable
\(u = x^2(9-3x)\) or \(\left(\frac{9-y}{3}\right)^2 y\)
\(\frac{du}{dx} = 18x - 9x^2\) or \(\frac{du}{dy} = 27 - 12y + y^2\)DM1A1 Knowing to differentiate
\(= 0\) when \(x = 2\) or \(y = 3 \rightarrow u = 12\)DM1 A1 Setting differential to 0
\(\frac{d^2u}{dx^2} = 18 - 18x\) — veDM1 A1 [7] Any valid method 
## Question 6:

| Answer/Working | Marks | Guidance |
|---|---|---|
| $u = x^2y \quad y + 3x = 9$ | M1 | Expressing $u$ in terms of 1 variable |
| $u = x^2(9-3x)$ or $\left(\frac{9-y}{3}\right)^2 y$ | | |
| $\frac{du}{dx} = 18x - 9x^2$ or $\frac{du}{dy} = 27 - 12y + y^2$ | DM1A1 | Knowing to differentiate |
| $= 0$ when $x = 2$ or $y = 3 \rightarrow u = 12$ | DM1 A1 | Setting differential to 0 |
| $\frac{d^2u}{dx^2} = 18 - 18x$ — ve | DM1 A1 [7] | Any valid method |

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6 The non-zero variables $x , y$ and $u$ are such that $u = x ^ { 2 } y$. Given that $y + 3 x = 9$, find the stationary value of $u$ and determine whether this is a maximum or a minimum value.

\hfill \mbox{\textit{CAIE P1 2013 Q6 [7]}}
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