| Exam Board | CAIE |
|---|---|
| Module | P1 (Pure Mathematics 1) |
| Year | 2021 |
| Session | November |
| Marks | 7 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Radians, Arc Length and Sector Area |
| Type | Multiple circles or sectors |
| Difficulty | Standard +0.3 This is a straightforward application of arc length formulas and basic trigonometry. Part (a) requires using Pythagoras or trigonometry to find an angle (showing CX=9, then using inverse trig), and part (b) applies the arc length formula s=rθ twice. The geometry is clearly defined with perpendicular lines, making it easier than average but requiring multiple standard steps. |
| Spec | 1.05b Sine and cosine rules: including ambiguous case1.05d Radians: arc length s=r*theta and sector area A=1/2 r^2 theta |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| Angle \(XYC = \sin^{-1}\left(\frac{9}{11}\right) = 0.9582\) or \(\sin XYC = \frac{9}{11}\) leading to \(XYC = 0.9582\) | B1 | AG. OE using cosine rule. |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \(XY = \sqrt{11^2 - 9^2} = \sqrt{40}\) or using \(0.9582\) and trigonometry | \*M1 A1 | |
| \(AB = 9 + 11 - \text{their } XY\) | B1 FT | OE e.g. \(20 - 2\sqrt{10}\), \(2 + 9 - 2\sqrt{10} + 11 - 2\sqrt{10}\) |
| Arc \(AC = 11 \times 0.9582\) | M1 | |
| Arc \(BC = 9 \times \frac{\pi}{2}\) | M1 | |
| Perimeter \(= [13.6(8) + 10.5(4) + 14.1(4) =] \ 38.4\) | A1 | AWRT. Answer must be evaluated as a single decimal. |
## Question 5(a):
| Answer | Marks | Guidance |
|--------|-------|----------|
| Angle $XYC = \sin^{-1}\left(\frac{9}{11}\right) = 0.9582$ or $\sin XYC = \frac{9}{11}$ leading to $XYC = 0.9582$ | B1 | AG. OE using cosine rule. |
## Question 5(b):
| Answer | Marks | Guidance |
|--------|-------|----------|
| $XY = \sqrt{11^2 - 9^2} = \sqrt{40}$ or using $0.9582$ and trigonometry | \*M1 A1 | |
| $AB = 9 + 11 - \text{their } XY$ | B1 FT | OE e.g. $20 - 2\sqrt{10}$, $2 + 9 - 2\sqrt{10} + 11 - 2\sqrt{10}$ |
| Arc $AC = 11 \times 0.9582$ | M1 | |
| Arc $BC = 9 \times \frac{\pi}{2}$ | M1 | |
| Perimeter $= [13.6(8) + 10.5(4) + 14.1(4) =] \ 38.4$ | A1 | AWRT. Answer must be evaluated as a single decimal. |
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\includegraphics[max width=\textwidth, alt={}, center]{cba7391d-2cf7-483c-9a21-bc9fd49860ee-06_720_686_260_733}
In the diagram, $X$ and $Y$ are points on the line $A B$ such that $B X = 9 \mathrm {~cm}$ and $A Y = 11 \mathrm {~cm}$. Arc $B C$ is part of a circle with centre $X$ and radius 9 cm , where $C X$ is perpendicular to $A B$. Arc $A C$ is part of a circle with centre $Y$ and radius 11 cm .
\begin{enumerate}[label=(\alph*)]
\item Show that angle $X Y C = 0.9582$ radians, correct to 4 significant figures.
\item Find the perimeter of $A B C$.
\end{enumerate}
\hfill \mbox{\textit{CAIE P1 2021 Q5 [7]}}