HSC Worked Solutions
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09
5c
The diagram shows a circle with centre O and radius
2 centimetres. The points A and B lie on the circumference
of the circle and ∠AOB = .
(i)
There are two possible values of for which
the area of  AOB is 3 square centimetres.
2

. Find the other value.
3

Suppose that  = .
3
One value is
(ii)
(Not to scale)
Find the area of the sector AOB.
Find the exact length of the perimeter
of the minor segment bounded by the
chord AB and the arc AB.
1
(2) To find length of chord AB,
Area of  AOB = ab sinC
2
use cosine rule:
(1)
(2)
i.
3 =
1
× 2 × 2 × sin 
2
=4+4–4
=4
x =2
 chord AB is 2 cm long
3 = 2 sin 
3
2

2
 =
or
3
3
sin  =
 The other value is
(ii)
(1)
x2 = 22 + 22 – 2(2)(2) cos
2
.
3
1 2
r 
2
1

=
× 22 ×
2
3
2
=
3
2
units2
 area is
3
1
2

3
State Mean:
0.96/2
0.69/1
1.03/2
To find length of arc AB, use
l = r
= 2×
Area =
=
 arc is

3
2
3
2
units
3
 Perimeter is (2 +
2
) units.
3
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Board of Studies: Notes from the Marking Centre
Source: http://www.boardofstudies.nsw.edu.au/hsc_exams/
HSC examination papers © Board of Studies NSW for and on behalf of the Crown in right of State of New South Wales
HSC Worked Solutions
projectmaths.com.au
HSC examination papers © Board of Studies NSW for and on behalf of the Crown in right of State of New South Wales