3.7 Radian measureIB Maths: Applications and Interpretation HL: Subtopic test

1. 1
   A sector $OAB$ of a circle has centre $O$, radius 6 cm and angle $A\hat{O}B=\frac{2\pi}{3}$.

   (a)

   Find the length of arc $AB$.

   [1 mark]

   • A$4\pi$ cm
   • B$12\pi$ cm
   • C$\frac{2\pi}{3}$ cm
   • D$2\pi$ cm

   (b)

   Find the area of sector $OAB$.

   [1 mark]

   • A$4\pi$ cm$^2$
   • B$12\pi$ cm$^2$
   • C$24\pi$ cm$^2$
   • D$2\pi$ cm$^2$

   (c)

   Find the perimeter of the sector, correct to 3 significant figures.

   [2 marks]

   Total for question 1: 4 marks

2. 2
   An angle $\theta$ is measured as $150^\circ$.

   (a)

   Express $\theta$ in radians.

   [1 mark]

   • A$\frac{5\pi}{3}$
   • B$\frac{5\pi}{12}$
   • C$\frac{5\pi}{6}$
   • D$\frac{6\pi}{5}$

   (b)

   Another angle is 2.4 radians. Find its size in degrees, correct to 3 significant figures.

   [1 mark]

   • A$0.0419^\circ$
   • B$432^\circ$
   • C$7.54^\circ$
   • D$138^\circ$

   (c)

   A circular sector has angle $\theta$ and radius 12 cm. Find the exact length of its arc.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   A sector $OPQ$ of a circle with centre $O$ and radius $r$ cm has arc length 15 cm and area 75 cm$^2$.

   (a)

   Find the value of $r$ and the size of angle $P\hat{O}Q$ in radians.

   [3 marks]

   (b)

   Use your GDC, in radian mode, to find the area of the region between the chord $PQ$ and the arc $PQ$.

   [4 marks]

   Total for question 3: 7 marks

4. 4
   A windscreen wiper has an arm $OB$ of length 50 cm. A rubber blade covers the outer 35 cm of the arm, from $A$ to $B$, where $OA=15$ cm. When the wiper moves, the arm turns about $O$.

   (a)

   On one setting the arm turns through 2.2 radians.
   (i) Find the distance moved by $B$.
   (ii) Find the area cleared by the blade.
   (iii) Convert 2.2 radians to degrees, to the nearest degree.

   [6 marks]

   (b)

   On a different setting the area cleared by the blade is 1800 cm$^2$.
   (i) Find the angle through which the arm turns, in radians.
   (ii) Find the distance moved by $B$ in one sweep.
   (iii) One sweep takes 0.6 seconds. Find the average speed of $B$ in m s$^{-1}$.

   [6 marks]

   Total for question 4: 12 marks

End of questions