Further Pure Math (Sector)

 1. (2011/june/Paper02/q6) 


Figure 1 shows a circle, centre $O$, with radius $8 \mathrm{~cm}$. The arc $A B$ has length $6 \mathrm{~cm}$.

(a) Find, in radians, the size of angle $A O B$.

(b) Find the area of the sector $A O B$.

(c) Find, to 3 significant figures, the area of the shaded segment. (3)


2. (2012/jan/paper02/q4)


Figure 2 shows an arc $A B$ of a circle with centre $O .$ The arc subtends an angle of $1.2$ radians at $O$ and the area of the sector $A O B$ is $15 \mathrm{~cm}^{2}$.

Find

(a) the radius of the circle,

(b) the length of the arc $A B$,

(c) the area of the shaded segment, giving your answer to 3 significant figures. (3)


3. (2013/jan/paper02/q1)


Figure 1 shows the sector, $A O B$ of a circle with centre $O$ and radius $8 \mathrm{~cm}$. A circle of radius $2 \mathrm{~cm}$ touches the lines $O A$ and $O B$ and the arc $A B$. Angle $A O B$ is $2 \theta$ radians, $0<\theta<\frac{\pi}{4}$

(a) Find, to 4 significant figures, the value of $\theta$

(b) Find, to 3 significant figures, the area of the region shaded in Figure 1 .


4. (2013/june/paper01/q1)

A circle has centre $O$ and radius $12 \mathrm{~cm}$. The sector $A O B$ of the circle has area $126 \mathrm{~cm}^{2}$.

Find the length of the arc $A B$.


5. (2014/jan/paper02/q4)


Figure 1 shows a sector of a circle of radius $10 \mathrm{~cm}$ and centre $O$. The area of triangle $O A B$ is $20 \mathrm{~cm}^{2}$ and the size of angle $A O B$ is $\theta$ radians.

Find, to 3 significant figures,

(a) the value of $\theta$, $(2)$

(b) the length of the $\operatorname{arc} A B$, (2)

(c) the area of the shaded segment. (3)


6. (2014/june/paper02/q1)


Figure 1 shows the sector $O A B$ of a circle. The circle has centre $O$ and radius $6 \mathrm{~cm}$. The area of the sector is $12 \mathrm{~cm}^{2}$.

(a) Find, in radians, the size of angle $A O B$. (2)

(b) Find, in $\mathrm{cm}$, the length of the arc $A B$. (2)


7. (2015/jan/paper01/q4)


Figure 2 shows the sector $A O B$ of a circle of radius $5 \mathrm{~cm}$. The centre of the circle is $O$ and the angle $A O B$ is $1.8$ radians.

(a) Find the length of the arc $A B$.

(b) Find the area of the sector $A O B$.


8. (2015/june/paper02/q6)


Figure 1 shows a sector $O A B$ of the circle with centre $O$ and radius $10 \mathrm{~cm}$.

The points $C$ and $D$ lie on $O B$ and $O A$ respectively and $C D$ is an arc of the circle with centre $O$ and radius $6 \mathrm{~cm}$. The size of angle $A O B$ is $\theta$ radians. The shaded region is bounded by the arcs $A B$ and $C D$ and the lines $A D$ and $B C$.

The area of the shaded region is $S \mathrm{~cm}^{2}$.

(a) Show that $S=32 \theta$. (3)

The size of angle $A O B$ is increasing at a constant rate of $0.2 \mathrm{rad} / \mathrm{s}$.

(b) Find the rate of increase of $S$. $(2)$

When the area of the shaded region is $20 \mathrm{~cm}^{2}$

(c) calculate the perimeter of the shaded region,


9. (2016/jan/paper02/q2)

The sector $O A B$ of a circle, centre $O$, has area $48 \mathrm{~cm}^{2}$.

The length of the $\operatorname{arc} A B$ is $8 \mathrm{~cm}$ and the size of angle $A O B$ is $\theta$ radians.

Find

(i) the radius of sector $O A B$

(ii) the value of $\theta$ (5)


10. (2017/jan/paper01/q1)


Figure 1 shows a sector of a circle. The circle has radius $r \mathrm{~cm}$ and the sector has angle $\theta$ radians. The sector has an arc length of $18 \pi \mathrm{cm}$ and an area of $126 \pi \mathrm{cm}^{2}$.

Find

(i) the value of $r$,

(ii) the exact value of $\theta$. (5)


11. (2018/jan/paper02/q1)


Figure 1 shows the sector $A O B$ of a circle with centre $O$ and radius $12 \mathrm{~cm}$. The angle $A O B$ is $\theta$ radians and the area of the sector is $192 \mathrm{~cm}^{2}$

Calculate

(a) the value of $\theta$, (2)

(b) the length, in $\mathrm{cm}$, of the arc $A B$. (2)


12. (2018/june/paper01/q1) 


Figure 1 shows a sector $O A B$ of a circle. The circle has centre $O$ and radius $10 \mathrm{~cm}$. The area of the sector is $25 \mathrm{~cm}^{2}$ and angle $A O B=\theta$ radians.

Find

(a) the value of $\theta$,

(b) the length of the arc $A B$. (2)


13. (2019/june/paper01/q4)

Figure 1 shows a sector $O A B


$ of a circle where angle $A O B=\theta$ radians. The circle has centre $O$ and radius $15 \mathrm{~cm}$. The point $C$ divides $O A$ in the ratio $2: 1$ and the point $D$ divides $O B$ in the ratio $2: 1$

The area of the region $A B D C$, shown shaded in Figure 1 , is $100 \mathrm{~cm}^{2}$

Find

(a) the value of $\theta$ (3)

(b) the perimeter of the region $A B D C$.


14. (2019/juneR/paper01/q1)


Figure 1 shows sector $A O B$ of a circle with centre $O$ and radius $r \mathrm{~cm}$. The angle $A O B$ is $1.5$ radians and the length of arc $A B$ is $12 \mathrm{~cm}$.

Calculate

(a) the value of $r$,

(b) the area of the sector $A O B$.



Answer

1. (a) $\quad \theta=\frac{3}{4}$ (b) $A=24$ (c) $A=2 \cdot 19$

2. (a) $r=5$ (b) $l=6$ (c) $A=3.35$

3. (a) $\theta=0.3398$ (b) $A=9.18$

4. $\ell=21 \mathrm{~cm}$

5. (a) $\theta=0.412$ (b) $4.12$ (c) $0.576$

6. (a) $\theta=\frac{2}{3}(b) 4$

7. (a) $l=9$ (b) $A=22.5$

8. (a) Show (b) $\frac{d S}{d t}=6.4$ (c) $P=18$

9. (i) $r=12$ (ii) $\theta=\frac{2}{3}$

10. (i) $r=14$ (ii) $\theta=\frac{9\pi}{7}$

11. (a) $\theta=\frac{8}{3}$ (b) $l=32$ 

12. (a) $\theta=\frac{1}{2}$ (b) $l=5$

13. (a) $\frac{8}{5}$ (b) $P=50$

14. (a) $r=8$ (b) $A=48$


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