Radian measure, arc length and sector areaAQA A-Level Maths: Subtopic test

1. 1
   A sector $OAB$ has centre $O$, radius $5$ cm and $A\hat{O}B=1.2$ radians.

   (a)

   Find the length of the arc $AB$.

   [1 mark]

   • A$3$ cm
   • B$4.17$ cm
   • C$15$ cm
   • D$6$ cm

   (b)

   Find the area of the sector $OAB$.

   [1 mark]

   • A$6$ cm$^2$
   • B$15$ cm$^2$
   • C$30$ cm$^2$
   • D$12.5$ cm$^2$

   (c)

   Convert $1.2$ radians to degrees, giving your answer to 1 decimal place.

   [2 marks]

   Total for question 1: 4 marks

2. 2
   A sector of a circle has arc length $8$ cm and area $24$ cm$^2$.

   (a)

   Find the radius of the sector.

   [1 mark]

   • A$6$ cm
   • B$3$ cm
   • C$12$ cm
   • D$4$ cm

   (b)

   Find the angle of the sector in radians.

   [1 mark]

   • A$\frac34$
   • B$2$
   • C$\frac43$
   • D$\frac83$

   (c)

   Find the perimeter of the sector.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   A circle has centre $O$ and radius $10$ cm. The chord $AB$ subtends an angle of $\frac{\pi}{3}$ radians at $O$. The minor segment is the region between the chord $AB$ and the minor arc $AB$.

   (a)

   Find the exact perimeter of the minor segment.

   [3 marks]

   (b)

   Show that the area of the minor segment is $\frac{50\pi}{3}-25\sqrt3$ cm$^2$, and hence find this area as a percentage of the area of the circle, to 3 significant figures.

   [4 marks]

   Total for question 3: 7 marks

4. 4
   A windscreen wiper arm pivots about a point $O$. The rubber blade lies along the arm between the points $20$ cm and $60$ cm from $O$. In one sweep the arm turns through $150^\circ$ in $0.8$ seconds, and the blade clears the part of a sector between these two distances.

   (a)

   (i) Write $150^\circ$ in radians, in terms of $\pi$.
   (ii) Find the area cleared by the blade in one sweep.
   (iii) Find the speed of the outer end of the blade, assuming the arm turns at a constant rate.

   [6 marks]

   (b)

   A designer proposes increasing the sweep to $180^\circ$ in the same time. She claims that this raises the area cleared and the speed of the outer end of the blade by the same percentage. Evaluate her claim.

   [6 marks]

   Total for question 4: 12 marks

End of questions