| Exam Board | CAIE |
|---|---|
| Module | P1 (Pure Mathematics 1) |
| Year | 2009 |
| 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 requiring the arc length formula (s = rθ), area of sector formula (A = ½r²θ), substitution to eliminate θ, and basic differentiation to find a maximum. While it involves multiple steps and the chain rule conceptually, it follows a very familiar pattern for A-level optimization questions with all formulas typically known, making it slightly easier than average. |
| Spec | 1.02v Inverse and composite functions: graphs and conditions for existence1.07i Differentiate x^n: for rational n and sums1.08h Integration by substitution |
| Answer | Marks | Guidance |
|---|---|---|
| \(A = \frac{1}{2}r^2\theta \rightarrow A = 25r - r^2\) | M1, A1, M1, A1 [4] | Must use \(s = r\theta\) and link with perimeter. co. Used with \(\theta\) as f(r). co (answer given) |
| Answer | Marks | Guidance |
|---|---|---|
| 2nd differential negative \(\rightarrow\) Maximum | B1, M1, A1, B1 [4] | co. sets differential to 0 + solution. co. Could be quoted directly from quadratic. |
**(i)** $2r + r\theta = 50$
$\theta = \frac{1}{r}(50 - 2r)$
$A = \frac{1}{2}r^2\theta \rightarrow A = 25r - r^2$ | M1, A1, M1, A1 [4] | Must use $s = r\theta$ and link with perimeter. co. Used with $\theta$ as f(r). co (answer given)
**(ii)** $\frac{dA}{dr} = 25 - 2r = 0$ when $r = 12.5$
$A = 156\frac{1}{4}$
2nd differential negative $\rightarrow$ Maximum | B1, M1, A1, B1 [4] | co. sets differential to 0 + solution. co. Could be quoted directly from quadratic.
**Total: [8]**
---7\\
\includegraphics[max width=\textwidth, alt={}, center]{b566719c-216e-41e5-8431-da77e1dad73e-3_301_485_264_829}
A piece of wire of length 50 cm is bent to form the perimeter of a sector $P O Q$ of a circle. The radius of the circle is $r \mathrm {~cm}$ and the angle $P O Q$ is $\theta$ radians (see diagram).\\
(i) Express $\theta$ in terms of $r$ and show that the area, $A \mathrm {~cm} ^ { 2 }$, of the sector is given by
$$A = 25 r - r ^ { 2 } .$$
(ii) Given that $r$ can vary, find the stationary value of $A$ and determine its nature.
\hfill \mbox{\textit{CAIE P1 2009 Q7 [8]}}