| Exam Board | CAIE |
|---|---|
| Module | P1 (Pure Mathematics 1) |
| Year | 2010 |
| Session | November |
| Marks | 8 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Chain Rule |
| Type | Optimization with constraint |
| Difficulty | Standard +0.3 This is a standard optimization problem with a constraint. Part (i) requires setting up a perimeter equation (straightforward). Part (ii) involves substituting to get area as a function of x (routine algebra). Parts (iii-iv) use basic differentiation to find and classify a stationary point. All steps are textbook-standard with no novel insight required, making it slightly easier than average. |
| Spec | 1.07n Stationary points: find maxima, minima using derivatives1.07o Increasing/decreasing: functions using sign of dy/dx1.07p Points of inflection: using second derivative |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| \(2x + 2y + \frac{\pi x}{2} = 60\) | \(M1\) | Linking 60 with sum of at least 4 sides and use of radians |
| \(\rightarrow y = 30 - x - \frac{\pi x}{4}\) | \(A1\) | co |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| \(A = xy + \frac{\pi x^2}{4}\) | ||
| \(= x\!\left(30 - x - \frac{\pi x}{4}\right) + \frac{\pi x^2}{4}\) | \(M1\) | Subs "\(y\)" into area equation and use \(\frac{1}{2}r^2\theta\) |
| \(= 30x - x^2\) | \(A1\) | co |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| \(\frac{dA}{dx} = 30 - 2x\) | Knowing to differentiate | |
| \(= 0\) when \(x = 15\) cm | \(M1\ A1\) | Sets differential to \(0\) + solution. co |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| Maximum | \(M1\ A1\) | Any valid method. co |
## Question 8:
### Part (i):
| Answer/Working | Marks | Guidance |
|---|---|---|
| $2x + 2y + \frac{\pi x}{2} = 60$ | $M1$ | Linking 60 with sum of at least 4 sides and use of radians |
| $\rightarrow y = 30 - x - \frac{\pi x}{4}$ | $A1$ | co |
### Part (ii):
| Answer/Working | Marks | Guidance |
|---|---|---|
| $A = xy + \frac{\pi x^2}{4}$ | | |
| $= x\!\left(30 - x - \frac{\pi x}{4}\right) + \frac{\pi x^2}{4}$ | $M1$ | Subs "$y$" into area equation and use $\frac{1}{2}r^2\theta$ |
| $= 30x - x^2$ | $A1$ | co |
### Part (iii):
| Answer/Working | Marks | Guidance |
|---|---|---|
| $\frac{dA}{dx} = 30 - 2x$ | | Knowing to differentiate |
| $= 0$ when $x = 15$ cm | $M1\ A1$ | Sets differential to $0$ + solution. co |
### Part (iv):
| Answer/Working | Marks | Guidance |
|---|---|---|
| Maximum | $M1\ A1$ | Any valid method. co |
---8\\
\includegraphics[max width=\textwidth, alt={}, center]{73c0c113-8f35-4e7f-ad5d-604602498b71-3_314_803_751_671}
The diagram shows a metal plate consisting of a rectangle with sides $x \mathrm {~cm}$ and $y \mathrm {~cm}$ and a quarter-circle of radius $x \mathrm {~cm}$. The perimeter of the plate is 60 cm .\\
(i) Express $y$ in terms of $x$.\\
(ii) Show that the area of the plate, $A \mathrm {~cm} ^ { 2 }$, is given by $A = 30 x - x ^ { 2 }$.
Given that $x$ can vary,\\
(iii) find the value of $x$ at which $A$ is stationary,\\
(iv) find this stationary value of $A$, and determine whether it is a maximum or a minimum value.
\hfill \mbox{\textit{CAIE P1 2010 Q8 [8]}}