| Exam Board | CAIE |
|---|---|
| Module | P1 (Pure Mathematics 1) |
| Year | 2015 |
| Session | June |
| Marks | 7 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Completing the square and sketching |
| Type | Applied quadratic optimization |
| Difficulty | Moderate -0.3 This is a straightforward optimization problem using completing the square. Part (i) requires setting up a perimeter equation (2r + arc length = 24) and substituting into the area formula—standard but multi-step. Part (ii) is routine completing the square. Part (iii) reads the maximum directly from completed square form and finds the angle using basic arc length formula. Slightly easier than average due to being highly structured with clear signposting through three parts. |
| Spec | 1.02e Complete the square: quadratic polynomials and turning points1.05d Radians: arc length s=r*theta and sector area A=1/2 r^2 theta |
A piece of wire of length 24 cm is bent to form the perimeter of a sector of a circle of radius $r$ cm.
\begin{enumerate}[label=(\roman*)]
\item Show that the area of the sector, $A$ cm$^2$, is given by $A = 12r - r^2$. [3]
\item Express $A$ in the form $a - (r - b)^2$, where $a$ and $b$ are constants. [2]
\item Given that $r$ can vary, state the greatest value of $A$ and find the corresponding angle of the sector. [2]
\end{enumerate}
\hfill \mbox{\textit{CAIE P1 2015 Q5 [7]}}