CAIE P1 2024 June — Question 8 10 marks

Exam BoardCAIE
ModuleP1 (Pure Mathematics 1)
Year2024
SessionJune
Marks10
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Mark schemeDownload PDF ↗
TopicRadians, Arc Length and Sector Area
TypeSector with attached triangle
DifficultyModerate -0.8 This is a straightforward application of standard arc length and sector area formulas with angle π/4. Part (a)(i) is simple arithmetic verification, (a)(ii) requires calculating two sector areas plus a rectangle, and (b) involves setting up and solving a linear equation from the perimeter condition. All steps are routine with no conceptual challenges beyond basic formula recall.
Spec1.05d Radians: arc length s=r*theta and sector area A=1/2 r^2 theta
\includegraphics{figure_8} The diagram shows a symmetrical plate \(ABCDEF\). The line \(ABCD\) is straight and the length of \(BC\) is 2cm. Each of the two sectors \(ABF\) and \(DCE\) is of radius \(r\)cm and each of the angles \(ABF\) and \(DCE\) is equal to \(\frac{1}{4}\pi\) radians.
  1. It is given that \(r = 0.4\)cm.
    1. Show that the length \(EF = 2.4\)cm. [2]
    2. Find the area of the plate. Give your answer correct to 3 significant figures. [4]
  2. It is given instead that the perimeter of the plate is 6cm. Find the value of \(r\). Give your answer correct to 3 significant figures. [4]
Question 8:
AnswerMarks
8(a)(i)π
C XC ˆ E 
6
π
CEˆX 
3
X E
XE π XE π
 sin or cos [XE = 0.2]
AnswerMarks Guidance
0.4 6 0.4 3M1 A correct trig expression involving XE.
Do not condone a mixture of degrees and radians.
AnswerMarks Guidance
Length EF = 220.2 = 2.4A1 AG
2
AnswerMarks Guidance
QuestionAnswer Marks
AnswerMarks
8(a)(ii)π π
CX 0.4cos or 0.4sin or 0.42 0.22
AnswerMarks Guidance
6 3B1 OE, SOI
3
Expect or 0.3464.
5
1 π
Sector  0.42 
AnswerMarks Guidance
2 3B1 SOI
2π 60
Expect 0.0838 or . Allow use of π0.42 .
75 360
Either Area of their (rectangle + two triangles + two sectors)
AnswerMarks Guidance
Or Area of their (trapezium + two sectors)M1 Either implied by a correct answer or areas clearly
labelled.
Expect 0.6928 + 0.06928 + 0.1676 or
2 3 3 4π
  .
5 25 75
11 3 4π
Or 0.7621 + 0.1676 or  .
25 75
AnswerMarks Guidance
0.930A1 AWRT
11 3 4π
Condone  .
25 75
4
AnswerMarks Guidance
QuestionAnswer Marks
AnswerMarks Guidance
8(b)[Length AD =] 22r B1
product.
π
[Arc length =] r
AnswerMarks Guidance
3B1 π
May be implied by r 2.
3
Must be seen alone or part of a list.
π π
EF22rsin or 22rcos or 2 + r
AnswerMarks Guidance
6 3B1 Must be seen alone or part of a list and not part of a
product.
2πr
[4+ 3r + 6 leading to] 0.393
AnswerMarks Guidance
3B1 AWRT
6
Condone .
2π9
NB: Using EF = 2.4 gives 0.391.
4
AnswerMarks Guidance
QuestionAnswer Marks 
Question 8:
--- 8(a)(i) ---
8(a)(i) | π
C XC ˆ E 
6
π
CEˆX 
3
X E
XE π XE π
 sin or cos [XE = 0.2]
0.4 6 0.4 3 | M1 | A correct trig expression involving XE.
Do not condone a mixture of degrees and radians.
Length EF = 220.2 = 2.4 | A1 | AG
2
Question | Answer | Marks | Guidance
--- 8(a)(ii) ---
8(a)(ii) | π π
CX 0.4cos or 0.4sin or 0.42 0.22
6 3 | B1 | OE, SOI
3
Expect or 0.3464.
5
1 π
Sector  0.42 
2 3 | B1 | SOI
2π 60
Expect 0.0838 or . Allow use of π0.42 .
75 360
Either Area of their (rectangle + two triangles + two sectors)
Or Area of their (trapezium + two sectors) | M1 | Either implied by a correct answer or areas clearly
labelled.
Expect 0.6928 + 0.06928 + 0.1676 or
2 3 3 4π
  .
5 25 75
11 3 4π
Or 0.7621 + 0.1676 or  .
25 75
0.930 | A1 | AWRT
11 3 4π
Condone  .
25 75
4
Question | Answer | Marks | Guidance
--- 8(b) ---
8(b) | [Length AD =] 22r | B1 | Must be seen alone or part of a list and not part of a
product.
π
[Arc length =] r
3 | B1 | π
May be implied by r 2.
3
Must be seen alone or part of a list.
π π
EF22rsin or 22rcos or 2 + r
6 3 | B1 | Must be seen alone or part of a list and not part of a
product.
2πr
[4+ 3r + 6 leading to] 0.393
3 | B1 | AWRT
6
Condone .
2π9
NB: Using EF = 2.4 gives 0.391.
4
Question | Answer | Marks | Guidance
\includegraphics{figure_8}

The diagram shows a symmetrical plate $ABCDEF$. The line $ABCD$ is straight and the length of $BC$ is 2cm. Each of the two sectors $ABF$ and $DCE$ is of radius $r$cm and each of the angles $ABF$ and $DCE$ is equal to $\frac{1}{4}\pi$ radians.

\begin{enumerate}[label=(\alph*)]
\item It is given that $r = 0.4$cm.
\begin{enumerate}[label=(\roman*)]
\item Show that the length $EF = 2.4$cm. [2]

\item Find the area of the plate. Give your answer correct to 3 significant figures. [4]
\end{enumerate}

\item It is given instead that the perimeter of the plate is 6cm.

Find the value of $r$. Give your answer correct to 3 significant figures. [4]
\end{enumerate}

\hfill \mbox{\textit{CAIE P1 2024 Q8 [10]}}
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