| Exam Board | CAIE |
|---|---|
| Module | P1 (Pure Mathematics 1) |
| Year | 2010 |
| Session | November |
| Marks | 8 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Circles |
| Type | Pulley, cord, and tangent applications |
| Difficulty | Standard +0.3 This is a standard circles problem requiring Pythagoras' theorem, right-angled triangle trigonometry, and sector area formulas. While it involves multiple steps and careful geometric reasoning, all techniques are routine for P1 level with no novel insights required. The 'show that' part guides students to the key relationship, making it slightly easier than average. |
| Spec | 1.03d Circles: equation (x-a)^2+(y-b)^2=r^21.03f Circle properties: angles, chords, tangents1.05d Radians: arc length s=r*theta and sector area A=1/2 r^2 theta |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| \(RS^2 = 10^2 - 6^2 \rightarrow RS = 8\) cm | \(M1\ A1\) | Use of Pythagoras (or other). Answer given. |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| \(\sin\theta = 8/10\) oe \(\rightarrow\) angle \(RPQ = 0.9273\) radians | \(M1\ A1\) | Use of trig – even if with degrees. co in radians. (Accept 0.927) |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| Region = trapezium \(-\) 2 sectors; Area of trapezium \(= 40\) cm² | \(B1\) | co |
| Large sector \(= \frac{1}{2} \times 8^2 \times 0.9273\) | \(M1\) | Use of \(\frac{1}{2}r^2\theta\) |
| Small sector angle \(= (\pi - 0.9273)\) | ||
| Small sector \(= \frac{1}{2} \times 2^2 \times 2.214\) | \(M1\) | Use of \(\frac{1}{2}r^2\theta\) with angle \(= \pi -\) (ii) |
| \(\rightarrow 5.90\) cm² | \(A1\) | co |
## Question 9:
### Part (i):
| Answer/Working | Marks | Guidance |
|---|---|---|
| $RS^2 = 10^2 - 6^2 \rightarrow RS = 8$ cm | $M1\ A1$ | Use of Pythagoras (or other). Answer given. |
### Part (ii):
| Answer/Working | Marks | Guidance |
|---|---|---|
| $\sin\theta = 8/10$ oe $\rightarrow$ angle $RPQ = 0.9273$ radians | $M1\ A1$ | Use of trig – even if with degrees. co in radians. (Accept 0.927) |
### Part (iii):
| Answer/Working | Marks | Guidance |
|---|---|---|
| Region = trapezium $-$ 2 sectors; Area of trapezium $= 40$ cm² | $B1$ | co |
| Large sector $= \frac{1}{2} \times 8^2 \times 0.9273$ | $M1$ | Use of $\frac{1}{2}r^2\theta$ |
| Small sector angle $= (\pi - 0.9273)$ | | |
| Small sector $= \frac{1}{2} \times 2^2 \times 2.214$ | $M1$ | Use of $\frac{1}{2}r^2\theta$ with angle $= \pi -$ (ii) |
| $\rightarrow 5.90$ cm² | $A1$ | co |
---9\\
\includegraphics[max width=\textwidth, alt={}, center]{73c0c113-8f35-4e7f-ad5d-604602498b71-4_837_1020_255_559}
The diagram shows two circles, $C _ { 1 }$ and $C _ { 2 }$, touching at the point $T$. Circle $C _ { 1 }$ has centre $P$ and radius 8 cm ; circle $C _ { 2 }$ has centre $Q$ and radius 2 cm . Points $R$ and $S$ lie on $C _ { 1 }$ and $C _ { 2 }$ respectively, and $R S$ is a tangent to both circles.\\
(i) Show that $R S = 8 \mathrm {~cm}$.\\
(ii) Find angle $R P Q$ in radians correct to 4 significant figures.\\
(iii) Find the area of the shaded region.
\hfill \mbox{\textit{CAIE P1 2010 Q9 [8]}}