CAIE P1 2019 June — Question 5 5 marks

Exam BoardCAIE
ModuleP1 (Pure Mathematics 1)
Year2019
SessionJune
Marks5
PaperDownload PDF ↗
Mark schemeDownload PDF ↗
TopicIndefinite & Definite Integrals
TypeBasic indefinite integration
DifficultyStandard +0.3 This is a straightforward geometry problem requiring students to set up equations for arc lengths and perimeters, then solve algebraically. It involves basic circle geometry (arc length = rθ) and simple equation solving, which is standard for P1 level with no complex problem-solving required.
Spec1.05d Radians: arc length s=r*theta and sector area A=1/2 r^2 theta
5 \includegraphics[max width=\textwidth, alt={}, center]{ed5b77ae-6eac-4e73-bc43-613433abd3e1-06_355_634_255_753} The diagram shows a semicircle with diameter \(A B\), centre \(O\) and radius \(r\). The point \(C\) lies on the circumference and angle \(A O C = \theta\) radians. The perimeter of sector \(B O C\) is twice the perimeter of sector \(A O C\). Find the value of \(\theta\) correct to 2 significant figures.
Question 5:
AnswerMarks Guidance
AnswerMark Guidance
Perimeter of \(AOC = 2r + r\theta\)B1
Angle \(COB = \pi - \theta\)B1 Could be on the diagram. Condone \(180 - \theta\)
Perimeter of \(BOC = 2r + r(\pi - \theta)\)B1 FT on angle \(COB\) if of form \((k\pi - \theta),\ k > 0\)
\((2r+)\ \pi r - r\theta = 2((2r) + r\theta)\) leading to \(\theta = \frac{\pi - 2}{3}\)M1 Sets up equation using \(r(k\pi - \theta)\) and \(\times 2\) on correct side. Condone any omissions of OA, OB and/or OC
\(\theta = 0.38\)A1 Equivalent answer in degrees scores A0 
## Question 5:

| Answer | Mark | Guidance |
|--------|------|----------|
| Perimeter of $AOC = 2r + r\theta$ | B1 | |
| Angle $COB = \pi - \theta$ | B1 | Could be on the diagram. Condone $180 - \theta$ |
| Perimeter of $BOC = 2r + r(\pi - \theta)$ | B1 | FT on angle $COB$ if of form $(k\pi - \theta),\ k > 0$ |
| $(2r+)\ \pi r - r\theta = 2((2r) + r\theta)$ leading to $\theta = \frac{\pi - 2}{3}$ | M1 | Sets up equation using $r(k\pi - \theta)$ and $\times 2$ on correct side. Condone any omissions of OA, OB and/or OC |
| $\theta = 0.38$ | A1 | Equivalent answer in degrees scores A0 |
5\\
\includegraphics[max width=\textwidth, alt={}, center]{ed5b77ae-6eac-4e73-bc43-613433abd3e1-06_355_634_255_753}

The diagram shows a semicircle with diameter $A B$, centre $O$ and radius $r$. The point $C$ lies on the circumference and angle $A O C = \theta$ radians. The perimeter of sector $B O C$ is twice the perimeter of sector $A O C$. Find the value of $\theta$ correct to 2 significant figures.\\

\hfill \mbox{\textit{CAIE P1 2019 Q5 [5]}}
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