| Exam Board | CAIE |
|---|---|
| Module | P1 (Pure Mathematics 1) |
| Year | 2013 |
| Session | November |
| Marks | 6 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Radians, Arc Length and Sector Area |
| Type | Simultaneous equations with arc/area |
| Difficulty | Standard +0.3 This question requires understanding that the slant height becomes the sector radius and the cone's circumference becomes the arc length, then applying standard arc length and sector area formulas. While it involves 3D to 2D visualization and Pythagoras, the mathematical steps are straightforward applications of memorized formulas with no novel problem-solving required. |
| Spec | 1.05d Radians: arc length s=r*theta and sector area A=1/2 r^2 theta |
| Answer | Marks | Guidance |
|---|---|---|
| (i) slant length = 10 cm, circumference of base = \(12\pi\), arc length = \(10\theta\) (\(= 12\pi\)), \(\to \theta = 1.2\pi\) or 3.77 radians. | B1, B1, B1✓, B1 [4] | Use of \(r\theta\), \(\theta\) calculated, not 6 or 8. |
| (ii) \(\frac{1}{3}r^2\theta = 188.5\) cm³ or \(60\pi\). | M1 A1✓ [2] | Use of \(\frac{1}{3}r^2\theta\) with radians and \(r =\) calculated '10', not 6 or 8. |
(i) slant length = 10 cm, circumference of base = $12\pi$, arc length = $10\theta$ ($= 12\pi$), $\to \theta = 1.2\pi$ or 3.77 radians. | B1, B1, B1✓, B1 [4] | Use of $r\theta$, $\theta$ calculated, not 6 or 8.
(ii) $\frac{1}{3}r^2\theta = 188.5$ cm³ or $60\pi$. | M1 A1✓ [2] | Use of $\frac{1}{3}r^2\theta$ with radians and $r =$ calculated '10', not 6 or 8.2
\begin{figure}[h]
\begin{center}
\includegraphics[alt={},max width=\textwidth]{d5f66324-e1fc-40e1-98e7-625187e24d3d-2_579_556_600_301}
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\caption{Fig. 1}
\end{center}
\end{figure}
\begin{figure}[h]
\begin{center}
\includegraphics[alt={},max width=\textwidth]{d5f66324-e1fc-40e1-98e7-625187e24d3d-2_579_876_605_973}
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\caption{Fig. 2}
\end{center}
\end{figure}
Fig. 1 shows a hollow cone with no base, made of paper. The radius of the cone is 6 cm and the height is 8 cm . The paper is cut from $A$ to $O$ and opened out to form the sector shown in Fig. 2. The circular bottom edge of the cone in Fig. 1 becomes the arc of the sector in Fig. 2. The angle of the sector is $\theta$ radians. Calculate\\
(i) the value of $\theta$,\\
(ii) the area of paper needed to make the cone.
\hfill \mbox{\textit{CAIE P1 2013 Q2 [6]}}