| Exam Board | CAIE |
|---|---|
| Module | P1 (Pure Mathematics 1) |
| Year | 2011 |
| Session | November |
| Marks | 8 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Radians, Arc Length and Sector Area |
| Type | Multiple circles or sectors |
| Difficulty | Standard +0.3 This question involves standard arc length and sector area formulas with some coordinate geometry. Part (i) requires setting up perpendicular distances (6sin(π/3) = 10sin(θ)) which is straightforward given the hint. Part (ii) applies arc length formulas to find the perimeter. While multi-step, it follows predictable patterns with clear guidance and no novel insight required, making it slightly easier than average. |
| Spec | 1.03f Circle properties: angles, chords, tangents1.05d Radians: arc length s=r*theta and sector area A=1/2 r^2 theta1.05i Inverse trig functions: arcsin, arccos, arctan domains and graphs |
**(i)**
- B1: Distance from $D$ to $AX$ = $6 \sin \frac{\pi}{3} = 6 \times \frac{\sqrt{3}}{2}$. Needs $\frac{\sqrt{3}}{2}$ not just $3\sqrt{3}$
- B1: Distance from $E$ to $AX$ = $10 \sin \theta$
- B1: Equate these $\Rightarrow \theta = \sin^{-1} \frac{3}{10}$. Correct method. ag.
**(ii)**
- B1: Arc $DX = 6 \times \frac{\pi}{3} = 2\pi$. Use of $s = r\theta$ radians. Use of decimals loses this B mark.
- M1: Arc $EX = 10 \times 0.5464 = 5.464$. Attempt at both steps needed
- M1: Horizontal steps = $6 \cos \frac{\pi}{3}$ and $10 \cos \theta$. Full method for $DE$
- M1: $DE = 10 + 6 - 6 \cos \frac{\pi}{3} - 10 \cos \theta$
- A1: Perimeter = arc $DX$ + arc $BX$ + $DE$ $\Rightarrow 16.20$. Co – must be exactly $16.20$, not more or less places.
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The diagram shows a circle $C _ { 1 }$ touching a circle $C _ { 2 }$ at a point $X$. Circle $C _ { 1 }$ has centre $A$ and radius 6 cm , and circle $C _ { 2 }$ has centre $B$ and radius 10 cm . Points $D$ and $E$ lie on $C _ { 1 }$ and $C _ { 2 }$ respectively and $D E$ is parallel to $A B$. Angle $D A X = \frac { 1 } { 3 } \pi$ radians and angle $E B X = \theta$ radians.\\
(i) By considering the perpendicular distances of $D$ and $E$ from $A B$, show that the exact value of $\theta$ is $\sin ^ { - 1 } \left( \frac { 3 \sqrt { } 3 } { 10 } \right)$.\\
(ii) Find the perimeter of the shaded region, correct to 4 significant figures.
\hfill \mbox{\textit{CAIE P1 2011 Q6 [8]}}