Standard +0.3 This is a straightforward application of sector and semicircle area formulas. Students must find angle CAD using the cosine rule or right triangle geometry, then calculate two areas and subtract. While it requires multiple steps (6 marks), the techniques are standard and the geometric setup is clearly defined with no novel insight required, making it slightly easier than average.
\includegraphics{figure_3}
In the diagram, \(CXD\) is a semicircle of radius \(7\) cm with centre \(A\) and diameter \(CD\). The straight line \(YAX\) is perpendicular to \(CD\), and the arc \(CYD\) is part of a circle with centre \(B\) and radius \(8\) cm. Find the total area of the region enclosed by the two arcs. [6]
Expect 68.1(9). Angle must be in radians (or their
61/360 × 2 × 82)
Or sector DBY
Answer
Marks
Guidance
∆BCD=7× 82 −72 or ½×82×sin ( 2×their1.0654 ) soi
M1
Expect 27.1(1). Award M1 for ABC or ABD
Semi-circle CXD = ½π×72 = 76.9 ( 7 )
M1
M1M1 for segment area formula used correctly
Total area = their68.19 ‒ their27.11 + their76.97 = 118.0–118.1
M1A1
Cannot gain M1 without attempt to find angle CBA or
CBD
6
Answer
Marks
Guidance
Question
Answer
Marks
Question 3:
3 | 7 −17
Angle CBA = sin−1 = 1.0654 or CBD=cos−1 =2.13
8 32 | B1 | Accept 61.0°, 66° or 122°
Sector BCYD = ½×82×2×their1.0654 ( rad ) soi
or sector CBY=½×82×their1.0654 ( rad ) | M1 | Expect 68.1(9). Angle must be in radians (or their
61/360 × 2 × 82)
Or sector DBY
∆BCD=7× 82 −72 or ½×82×sin ( 2×their1.0654 ) soi | M1 | Expect 27.1(1). Award M1 for ABC or ABD
Semi-circle CXD = ½π×72 = 76.9 ( 7 ) | M1 | M1M1 for segment area formula used correctly
Total area = their68.19 ‒ their27.11 + their76.97 = 118.0–118.1 | M1A1 | Cannot gain M1 without attempt to find angle CBA or
CBD
6
Question | Answer | Marks | Guidance
\includegraphics{figure_3}
In the diagram, $CXD$ is a semicircle of radius $7$ cm with centre $A$ and diameter $CD$. The straight line $YAX$ is perpendicular to $CD$, and the arc $CYD$ is part of a circle with centre $B$ and radius $8$ cm. Find the total area of the region enclosed by the two arcs. [6]
\hfill \mbox{\textit{CAIE P1 2019 Q3 [6]}}