1. 1
   The sector $OAB$ is part of a circle with centre $O$ and radius 6 cm. The angle $A\hat{O}B$ is $\frac{5\pi}{6}$ radians.

   (a)

   Find the length of the arc $AB$.

   [1 mark]

   • A$\frac{5\pi}{36}$ cm
   • B$15\pi$ cm
   • C$900$ cm
   • D$5\pi$ cm

   (b)

   Find the perimeter of the sector $OAB$.

   [1 mark]

   • A$5\pi$ cm
   • B$6+5\pi$ cm
   • C$12+5\pi$ cm
   • D$12+15\pi$ cm

   (c)

   Find the area of the sector $OAB$, giving your answer as an exact multiple of $\pi$.

   [2 marks]

   Total for question 1: 4 marks

2. 2
   The string of a pendulum is 80 cm long. The pendulum swings from one extreme position to the other, and the string turns through an angle of 0.3 radians.

   (a)

   Find the length of the arc travelled by the end of the pendulum in one swing.

   [1 mark]

   • A$24$ cm
   • B$960$ cm
   • C$267$ cm
   • D$1375$ cm

   (b)

   Find the area swept out by the string in one swing.

   [1 mark]

   • A$1920$ cm$^2$
   • B$960$ cm$^2$
   • C$12$ cm$^2$
   • D$24$ cm$^2$

   (c)

   Using a calculator, find the angle through which the string turns, in degrees, correct to 1 decimal place.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   A circle has centre $O$ and radius 10 cm. The points $A$ and $B$ lie on the circle and $A\hat{O}B=\frac{2\pi}{3}$. The minor segment is the region between the chord $[AB]$ and the minor arc $AB$.

   (a)

   Find the exact area of the minor segment.

   [3 marks]

   (b)

   Find the exact perimeter of the minor segment.

   [4 marks]

   Total for question 3: 7 marks

4. 4
   A garden sprinkler at a point $O$ waters a region of lawn in the shape of a sector of a circle, centre $O$, with radius $r$ metres and angle $\theta$ radians, where $0<\theta<2\pi$. The watered region has area $50$ m$^2$ and perimeter $30$ m (two radii and the arc).

   (a)

   Show that $r^2-15r+50=0$, and hence find the two possible pairs of values of $r$ and $\theta$.

   [6 marks]

   (b)

   The gardener chooses the setting that waters lawn furthest from $O$. The ends of the arc are $A$ and $B$, and a straight path runs along the chord $[AB]$.
   (i) Write down the values of $r$ and $\theta$ for this setting, and convert $\theta$ to degrees.
   (ii) Using a calculator, find the area of the watered region that lies between the path and the arc.

   [6 marks]

   Total for question 4: 12 marks

End of questions