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A-Level Pure 2 · Edexcel IAL WMA12 & CAIE 9709 · Topic pack

Topic pack · Circular measure แพ็กฝึกเฉพาะหัวข้อ · เรเดียน ส่วนโค้ง เซกเตอร์

10 questions 10 ข้อ 55 marks 55 คะแนน 70 min 70 นาที Arcs, sectors & segments, easy → hard

Ten questions on one topic, ordered easy → hard. The last three are full exam difficulty. Open the mark schemes only after a real attempt. สิบข้อหัวข้อเดียว เรียงง่าย → ยาก สามข้อสุดท้ายคือระดับข้อสอบจริง เปิดมาร์คสกีมหลังลองทำจริงเท่านั้น

⏱ 70:00 Scoreคะแนนรวม: –
Q1. 3 marks

(a) Convert $135^\circ$ and $240^\circ$ to radians, giving exact answers in terms of $\pi$. (b) Convert $\dfrac{5\pi}{6}$ radians to degrees.

Mark schemeshow ▾
(a) B1 $135^\circ=135\times\tfrac{\pi}{180}=\tfrac{3\pi}{4}$  B1 $240^\circ=\tfrac{4\pi}{3}$.
(b) B1 $\tfrac{5\pi}{6}\times\tfrac{180}{\pi}=150^\circ$.
Q2. 4 marks

A sector of a circle has radius 8 cm and angle 1.5 radians. Find (a) the arc length of the sector, (b) the perimeter of the sector.

Mark schemeshow ▾
(a) M1 $s=r\theta=8\times1.5$  A1 $s=12$ cm.
(b) M1 perimeter $=s+2r=12+16$  A1 $=28$ cm.
Q3. 4 marks

A sector of a circle of radius 6 cm has area 24 cm². Find (a) the angle of the sector in radians, (b) the arc length of the sector.

Mark schemeshow ▾
(a) M1 $24=\tfrac12\times 6^2\times\theta=18\theta$  A1 $\theta=\tfrac{4}{3}$ rad.
(b) M1 $s=r\theta=6\times\tfrac43$  A1 $s=8$ cm.
Q4. 5 marks

A sector of a circle has perimeter 26 cm and arc length 10 cm. (a) Find the radius and the angle of the sector. (b) Find the area of the sector.

Mark schemeshow ▾
(a) M1 $2r+10=26$  A1 $r=8$ cm  A1 $\theta=\dfrac{s}{r}=\dfrac{10}{8}=1.25$ rad.
(b) M1 $A=\tfrac12 r^2\theta=\tfrac12\times64\times1.25$  A1 $A=40$ cm².
Q5. 5 marks

A chord $PQ$ of a circle, centre $O$ and radius 10 cm, subtends an angle of $\dfrac{\pi}{3}$ radians at $O$. Find (a) the area of the minor segment cut off by $PQ$, to 3 significant figures, (b) the exact length of the chord $PQ$.

Mark schemeshow ▾
(a) M1 segment $=\tfrac12 r^2(\theta-\sin\theta)=\tfrac12\times100\left(\tfrac{\pi}{3}-\sin\tfrac{\pi}{3}\right)$  A1 $=50\left(\tfrac{\pi}{3}-\tfrac{\sqrt3}{2}\right)$  A1 $=9.06$ cm² (exact $9.05861\ldots$).
(b) M1 chord $=2r\sin\tfrac{\theta}{2}=2\times10\times\sin\tfrac{\pi}{6}$  A1 $PQ=10$ cm.
Q6. 6 marks

$OAB$ is a sector of a circle, centre $O$, radius 9 cm, with angle $AOB=0.8$ radians. Find, to 3 significant figures where needed, (a) the area of the sector $OAB$, (b) the area of the triangle $OAB$, (c) the area of the segment between the chord $AB$ and the arc $AB$.

Mark schemeshow ▾
(a) M1 $\tfrac12\times81\times0.8$  A1 $=32.4$ cm².
(b) M1 $\tfrac12\times9\times9\times\sin 0.8$  A1 $=29.1$ cm² (exact $29.0529\ldots$).
(c) M1 segment $=32.4-29.0529$  A1 $=3.35$ cm² (exact $3.34708\ldots$).
Q7. 6 marks

A chord $AB$ of a circle of radius 6 cm subtends an angle of 1.4 radians at the centre. Find, to 3 significant figures, (a) the perimeter of the minor segment cut off by $AB$, (b) the area of the minor segment.

Mark schemeshow ▾
(a) M1 arc $=6\times1.4=8.4$ cm  M1 chord $=2\times6\times\sin 0.7$  A1 chord $=7.73$ cm (exact $7.73061\ldots$)  A1 perimeter $=8.4+7.73061=16.1$ cm (exact $16.1306\ldots$).
(b) M1 $\tfrac12\times36\times(1.4-\sin 1.4)$  A1 $=7.46$ cm² (exact $7.46191\ldots$).
Q8. 7 marks

$OAB$ is a sector of a circle, centre $O$, radius 9 cm, with angle $AOB=1.2$ radians. The points $C$ on $OA$ and $D$ on $OB$ are such that $OC=OD=5$ cm. The region $R$ is bounded by the arc $AB$, the arc $CD$ of the circle centre $O$ radius 5 cm, and the straight lines $CA$ and $DB$. Find (a) the perimeter of $R$, (b) the area of $R$.

Mark schemeshow ▾
(a) M1 arc $AB=9\times1.2$  A1 $=10.8$ cm  M1 arc $CD=5\times1.2=6$ cm and $CA=DB=9-5=4$ cm  A1 perimeter $=10.8+6+4+4=24.8$ cm.
(b) M1 area $=\tfrac12\times81\times1.2-\tfrac12\times25\times1.2$  A1 $=48.6-15$  A1 $=33.6$ cm².
Q9. 7 marks

A sector of a circle of radius $r$ cm has angle $\theta$ radians, where $\theta<2$. The area of the sector is 20 cm² and the perimeter of the sector is 18 cm. (a) Find $r$ and $\theta$. (b) Find, to 3 significant figures, the area of the segment between the chord and the arc of the sector.

Mark schemeshow ▾
(a) M1 $\tfrac12 r^2\theta=20$  M1 $2r+r\theta=18 \Rightarrow r\theta=18-2r$  M1 substitute: $\tfrac12 r(18-2r)=20$  A1 $r^2-9r+20=0 \Rightarrow (r-4)(r-5)=0$, so $r=4$ or $r=5$  A1 $r=4$ gives $\theta=\tfrac{10}{4}=2.5>2$ (reject); $r=5$, $\theta=\tfrac{8}{5}=1.6$ rad.
(b) M1 $\tfrac12\times25\times(1.6-\sin 1.6)$  A1 $=7.51$ cm² (exact $7.50533\ldots$).
Q10. 8 marks

In triangle $ABC$, $AB=AC=10$ cm and angle $BAC=0.9$ radians. The points $D$ on $AB$ and $E$ on $AC$ are such that $AD=AE=6$ cm, and $DE$ is an arc of the circle centre $A$, radius 6 cm. The region $R$ inside the triangle but outside the sector $ADE$ is shaded. Find, to 3 significant figures, (a) the area of triangle $ABC$, (b) the area of the sector $ADE$, (c) the area of $R$, (d) the perimeter of $R$.

Mark schemeshow ▾
(a) M1 $\tfrac12\times10\times10\times\sin 0.9$  A1 $=39.2$ cm² (exact $39.1663\ldots$).
(b) B1 $\tfrac12\times36\times0.9=16.2$ cm².
(c) M1 $R=39.1663-16.2$  A1 $=23.0$ cm² (exact $22.9663\ldots$).
(d) M1 $BC=2\times10\times\sin 0.45$  A1 $BC=8.70$ cm (exact $8.69931\ldots$)  A1 perimeter $=BC+DB+EC+\text{arc }DE=8.69931+4+4+5.4=22.1$ cm (exact $22.0993\ldots$).