| Exam Board | CAIE |
|---|---|
| Module | P1 (Pure Mathematics 1) |
| Year | 2017 |
| Session | June |
| 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 geometry problem requiring basic trigonometry (inverse cosine to find the angle) and standard sector/triangle area formulas. Part (i) is a simple right-angle triangle calculation, and part (ii) involves computing two sectors minus two triangles—routine application of formulas with no novel insight required. Slightly easier than average due to the clear setup and standard techniques. |
| Spec | 1.05d Radians: arc length s=r*theta and sector area A=1/2 r^2 theta1.08e Area between curve and x-axis: using definite integrals |
| Answer | Marks | Guidance |
|---|---|---|
| 7(i) | sinABC =8/10 → ABC =0.927 ( 3 ) | B1 |
| Total: | 1 |
| Answer | Marks | Guidance |
|---|---|---|
| 7(ii) | AB=6 ( Pythagoras ) → ∆BCD=8×6=48.0 | M1A1 |
| Answer | Marks | Guidance |
|---|---|---|
| Area sector 2 | *M1 | Expect 92.7(3). 46.4 gets M1 |
| Area segment = 92.7(3) – 48 | *A1 | Expect 44.7(3). Might not appear until final calculation. |
| Area semi-circle ‒ segment = ½×π×82 −their ( 92.7−48 ) | DM1 | ( ) |
| Answer | Marks | Guidance |
|---|---|---|
| Shaded area = 55.8 – 56.0 | A1 | |
| Total: | 6 | |
| Question | Answer | Marks |
Question 7:
--- 7(i) ---
7(i) | sinABC =8/10 → ABC =0.927 ( 3 ) | B1 | Or cos = 6/10 or tan = 8/6. Accept 0.295π.
Total: | 1
--- 7(ii) ---
7(ii) | AB=6 ( Pythagoras ) → ∆BCD=8×6=48.0 | M1A1 | OR 8×10sin0.6435 or ½×10×10sin((2)×0.927)=48. 24or 40or80 gets M1A0
BCD=½×102×( )×their0.9273
Area sector 2 | *M1 | Expect 92.7(3). 46.4 gets M1
Area segment = 92.7(3) – 48 | *A1 | Expect 44.7(3). Might not appear until final calculation.
Area semi-circle ‒ segment = ½×π×82 −their ( 92.7−48 ) | DM1 | ( )
Dep. on previous M1A1 OR π×82 − ½×π×82 +their44.7 .
Shaded area = 55.8 – 56.0 | A1
Total: | 6
Question | Answer | Marks | Guidance\includegraphics{figure_7}
The diagram shows two circles with centres $A$ and $B$ having radii 8 cm and 10 cm respectively. The two circles intersect at $C$ and $D$ where $CAD$ is a straight line and $AB$ is perpendicular to $CD$.
\begin{enumerate}[label=(\roman*)]
\item Find angle $ABC$ in radians. [1]
\item Find the area of the shaded region. [6]
\end{enumerate}
\hfill \mbox{\textit{CAIE P1 2017 Q7 [7]}}