CAIE P1 2023 November — Question 4 6 marks

Exam BoardCAIE
ModuleP1 (Pure Mathematics 1)
Year2023
SessionNovember
Marks6
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Mark schemeDownload PDF ↗
TopicRadians, Arc Length and Sector Area
TypeLogo and design problems
DifficultyModerate -0.5 This is a standard Reuleaux triangle problem requiring basic arc length and sector area formulas. Part (a) uses arc length with angle π/3 for three identical arcs. Part (b) combines the triangle area with three identical circular segments. While it requires careful geometric visualization and multiple steps (6 marks total), the techniques are routine applications of standard formulas with no novel problem-solving insight needed.
Spec1.05d Radians: arc length s=r*theta and sector area A=1/2 r^2 theta
\includegraphics{figure_4} The diagram shows the shape of a coin. The three arcs \(AB\), \(BC\) and \(CA\) are parts of circles with centres \(C\), \(A\) and \(B\) respectively. \(ABC\) is an equilateral triangle with sides of length 2 cm.
  1. Find the perimeter of the coin. [2]
  2. Find the area of the face \(ABC\) of the coin, giving the answer in terms of \(\pi\) and \(\sqrt{3}\). [4]
Question 4:
AnswerMarks
4(a)π 60
 Arc length =  2 or 2π2
AnswerMarks Guidance
3 360B1 Finding one correct arc length – may be implied by correct
final answer.
 =
AnswerMarks Guidance
Perimeter 2π or 6.28B1 AWRT
2
AnswerMarks Guidance
QuestionAnswer Marks
AnswerMarks
4(b)1 π 60  2π 
[Area of one sector =] 22 or π22 = or 2.09
 
AnswerMarks Guidance
2 3 360  3 B1 SOI AWRT
1 π
[Area of triangle =] 22sin  or other valid method
2 3
= 3 or1 .73
AnswerMarks Guidance
 B1 AWRT
Allow use of 60
2π 
 Area of coin=3 segments+triangle  3 − 3+ 3 [= 2.82]
AnswerMarks Guidance
 3 M1 OE
 2π 
Or 3 sectors – 2 triangles 3 −2 3 or
 3 
2π 2π 
Sector + 2 segments  +2 − 3
 3  3 
AnswerMarks Guidance
2π−2 3 or 2(π- 3)A1 Must be one of these simplified versions but equivalent
decimal answers can score B1B1M1
4
AnswerMarks Guidance
QuestionAnswer Marks 
Question 4:
--- 4(a) ---
4(a) | π 60
 Arc length =  2 or 2π2
3 360 | B1 | Finding one correct arc length – may be implied by correct
final answer.
 =
Perimeter 2π or 6.28 | B1 | AWRT
2
Question | Answer | Marks | Guidance
--- 4(b) ---
4(b) | 1 π 60  2π 
[Area of one sector =] 22 or π22 = or 2.09
 
2 3 360  3  | B1 | SOI AWRT
1 π
[Area of triangle =] 22sin  or other valid method
2 3
= 3 or1 .73
  | B1 | AWRT
Allow use of 60
2π 
 Area of coin=3 segments+triangle  3 − 3+ 3 [= 2.82]
 3  | M1 | OE
 2π 
Or 3 sectors – 2 triangles 3 −2 3 or
 3 
2π 2π 
Sector + 2 segments  +2 − 3
 3  3 
2π−2 3 or 2(π- 3) | A1 | Must be one of these simplified versions but equivalent
decimal answers can score B1B1M1
4
Question | Answer | Marks | Guidance
\includegraphics{figure_4}

The diagram shows the shape of a coin. The three arcs $AB$, $BC$ and $CA$ are parts of circles with centres $C$, $A$ and $B$ respectively. $ABC$ is an equilateral triangle with sides of length 2 cm.

\begin{enumerate}[label=(\alph*)]
\item Find the perimeter of the coin. [2]

\item Find the area of the face $ABC$ of the coin, giving the answer in terms of $\pi$ and $\sqrt{3}$. [4]
\end{enumerate}

\hfill \mbox{\textit{CAIE P1 2023 Q4 [6]}}
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