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Date November 2015 Marks available 2 Reference code 15N.2.sl.TZ0.1
Level SL only Paper 2 Time zone TZ0
Command term Find Question number 1 Adapted from N/A

Question

The following diagram shows a circle with centre \(O\) and radius \(3\) cm.

Points A, B, and C lie on the circle, and \({\rm{A\hat OC}} = 1.3{\text{ radians}}\).

Find the length of arc \(ABC\).

[2]
a.

Find the area of the shaded region.

[4]
b.

Markscheme

correct substitution     (A1)

eg\(\;\;\;l = 1.3 \times 3\)

\(l\)  = \(3.9\) (cm)     A1     N2

[2 marks]

a.

METHOD 1

valid approach     (M1)

eg\(\;\;\;\)finding reflex angle, \(2\pi  - {\rm{C\hat OA}}\)

correct angle     (A1)

eg\(\;\;\;2\pi  - 1.3,{\text{ }}4.98318\)

correct substitution     (A1)

eg\(\;\;\;\frac{1}{2}(2\pi  - 1.3){3^2}\)

\(22.4243\)

\({\text{area}} = 9\pi  - 5.85{\text{ (exact), }}22.4{\text{ }} {\text{ }}({\text{c}}{{\text{m}}^2})\)     A1     N3

METHOD 2

correct area of small sector     (A1)

eg\(\;\;\;\frac{1}{2}(1.3){3^2},{\text{ }}5.85\)

valid approach     (M1)

eg\(\;\;\;\)circle − small sector, \(\pi {r^2} - \frac{1}{2}\theta {r^2}\)

correct substitution     (A1)

eg\(\;\;\;\pi ({3^2}) - \frac{1}{2}(1.3){3^2}\)

\(22.4243\)

\({\text{area}} = 9\pi  - 5.85{\text{ }}({\text{exact}}),{\text{ }}22.4{\text{ }}{\text{ }}({\text{c}}{{\text{m}}^2})\)     A1     N3

[4 marks]

Total [6 marks]

b.

Examiners report

[N/A]
a.
[N/A]
b.

Syllabus sections

Topic 3 - Circular functions and trigonometry » 3.1 » The circle: radian measure of angles; length of an arc; area of a sector.
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