Radian measure, arc length and exact valuesEdexcel A-Level Maths: Subtopic test

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

   (a)

   Find the length of the arc $AB$.

   [1 mark]

   • A$2\pi$ cm
   • B$4\pi$ cm
   • C$12\pi$ cm
   • D$720$ cm

   (b)

   Find the area of the sector $OAB$.

   [1 mark]

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

   (c)

   Find the exact perimeter of the sector $OAB$.

   [2 marks]

   Total for question 1: 4 marks

2. 2
   Consider the angle $\theta=\frac{5\pi}{6}$ radians, which lies in the second quadrant.

   (a)

   Find the exact value of $\sin\theta$.

   [1 mark]

   • A$-\frac12$
   • B$\frac{\sqrt3}{2}$
   • C$\frac12$
   • D$-\frac{\sqrt3}{2}$

   (b)

   Find the exact value of $\cos\theta$.

   [1 mark]

   • A$\frac{\sqrt3}{2}$
   • B$-\frac12$
   • C$\frac12$
   • D$-\frac{\sqrt3}{2}$

   (c)

   Hence show that $\tan\theta=-\frac{\sqrt3}{3}$.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   A sector of a circle of radius $5$ cm has perimeter $16$ cm. The angle of the sector is $\theta$ radians.

   (a)

   Find the value of $\theta$.

   [3 marks]

   (b)

   Find the area of the segment bounded by the chord $AB$ and the arc $AB$. Give your answer to 3 significant figures.

   [4 marks]

   Total for question 3: 7 marks

4. 4
   A circle has centre $O$ and radius $8$ cm. The points $A$ and $B$ lie on the circle with $\angle AOB=\frac{\pi}{3}$. The minor segment is the region between the chord $AB$ and the minor arc $AB$.

   (a)

   Show that the area of the minor segment is $\frac{32\pi}{3}-16\sqrt3$ cm$^2$.

   [6 marks]

   (b)

   (i) Find the exact length of the chord $AB$.
   (ii) Find the exact perimeter of the minor segment.
   (iii) Find the area of the minor segment as a percentage of the area of the whole circle, to 3 significant figures.

   [6 marks]

   Total for question 4: 12 marks

End of questions