SPM.2.SL.TZ0.2

pestleMathematicsAASLPaper 2SPM· sl-3-4-circle-radians-arcs-sectorssource ↗

The following diagram shows part of a circle with centre O and radius 4 cm.

Chord AB has a length of 5 cm and AÔB = θ.

Find the value of θ, giving your answer in radians.

[3]

Find the area of the shaded region.

[3]

Markscheme / solution

METHOD 1

attempt to use the cosine rule (M1)

cos θ = $\frac{{{4^2} + {4^2} - {5^2}}}{{2 \times 4 \times 4}}$ (or equivalent) A1

θ = 1.35 A1

METHOD 2

attempt to split triangle AOB into two congruent right triangles (M1)

sin $\left( {\frac{\theta }{2}} \right) = \frac{{2.5}}{4}$ A1

θ = 1.35 A1

[3 marks]

attempt to find the area of the shaded region (M1)

$\frac{1}{2} \times 4 \times 4 \times \left( {2\pi - 1.35 \ldots } \right)$ A1

= 39.5 (cm²) A1

[3 marks]

Examiners’ report

[N/A]