| Exam Board | CAIE |
|---|---|
| Module | P1 (Pure Mathematics 1) |
| Year | 2018 |
| Session | November |
| Marks | 7 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Radians, Arc Length and Sector Area |
| Type | Proving angle relationships |
| Difficulty | Standard +0.3 This is a straightforward application of cosine rule to find an angle, followed by standard arc length calculations. Part (i) is routine triangle work with a calculator, and part (ii) requires only the arc length formula s=rθ applied twice. The 'show that' format in (i) removes problem-solving demand, and no geometric insight is needed beyond recognizing which radii and angles to use. |
| Spec | 1.05b Sine and cosine rules: including ambiguous case1.05c Area of triangle: using 1/2 ab sin(C)1.05d Radians: arc length s=r*theta and sector area A=1/2 r^2 theta1.05g Exact trigonometric values: for standard angles |
| Answer | Marks |
|---|---|
| 8(i) | −1 7 |
| Answer | Marks | Guidance |
|---|---|---|
| 4 3 | M1 | Correct method for ABˆC, expect 1.696cawrt |
| Answer | Marks | Guidance |
|---|---|---|
| CBˆ Y = π – ABˆC or 2×CA ˆ B | M1 | For attempt at CBˆY = π – ABˆC or CBˆY = 2 × CA ˆ B |
| Answer | Marks | Guidance |
|---|---|---|
| Find CY from ∆ ACY using Pythagoras or similar ∆s | M1 | Expect 4 7 |
| Answer | Marks | Guidance |
|---|---|---|
| | M1 | Correct use of cosine rule |
| CBˆ Y = 1.445c AG | A1 | Numerical values for angles in radians, if given, need to be |
| Answer | Marks | Guidance |
|---|---|---|
| 8(ii) | Arc CY = 8 × 1.445 | B1 |
| BA ˆ C = ½(π – ABˆ C) or cos−1(¾) | *M1 | For a valid attempt at BA ˆ C, may be from (i). Expect 0.7227c |
| Arc XC = 12 × (their BA ˆ C) | DM1 | Expect 8.673 |
| Perimeter = 11.56 + 8.673 + 4 = 24.2 cm awrt www | A1 | Omission of ‘+4’ only penalised here. |
| Answer | Marks | Guidance |
|---|---|---|
| Question | Answer | Marks |
Question 8:
--- 8(i) ---
8(i) | −1 7
ABˆ C using cosine rule giving cos-1( ) or 2sin−1(¾) or 2cos−1
8 2
7 7
or BA ˆ C = cos-1(¾) or BA ˆ C = sin-1 or BA ˆ C = tan−1
4 3 | M1 | Correct method for ABˆC, expect 1.696cawrt
Or for BA ˆ C, expect 0.723cawrt
CBˆ Y = π – ABˆC or 2×CA ˆ B | M1 | For attempt at CBˆY = π – ABˆC or CBˆY = 2 × CA ˆ B
OR
Find CY from ∆ ACY using Pythagoras or similar ∆s | M1 | Expect 4 7
82 +82 − ( their CY )2
CBˆ Y = cos−1
2×8×8
| M1 | Correct use of cosine rule
CBˆ Y = 1.445c AG | A1 | Numerical values for angles in radians, if given, need to be
correct to 3 decimal places. Method marks can be awarded
for working in degrees.
Need 82.8° awrt converted to radians for A1.
Identification of angles must be consistent for A1.
3
--- 8(ii) ---
8(ii) | Arc CY = 8 × 1.445 | B1 | Use of s=8θ for arc CY, Expect 11.56
BA ˆ C = ½(π – ABˆ C) or cos−1(¾) | *M1 | For a valid attempt at BA ˆ C, may be from (i). Expect 0.7227c
Arc XC = 12 × (their BA ˆ C) | DM1 | Expect 8.673
Perimeter = 11.56 + 8.673 + 4 = 24.2 cm awrt www | A1 | Omission of ‘+4’ only penalised here.
4
Question | Answer | Marks | Guidance\includegraphics{figure_8}
The diagram shows an isosceles triangle $ACB$ in which $AB = BC = 8$ cm and $AC = 12$ cm. The arc $XC$ is part of a circle with centre $A$ and radius $12$ cm, and the arc $YC$ is part of a circle with centre $B$ and radius $8$ cm. The points $A$, $B$, $X$ and $Y$ lie on a straight line.
\begin{enumerate}[label=(\roman*)]
\item Show that angle $CBY = 1.445$ radians, correct to $4$ significant figures. [3]
\item Find the perimeter of the shaded region. [4]
\end{enumerate}
\hfill \mbox{\textit{CAIE P1 2018 Q8 [7]}}