SPM 2021 Add Maths Trial Paper (Selangor) – Paper 2 (Question 5)


Question 5:
Diagram 4(a) shows two concentric circles with the centre O. ABC is a tangent to the smaller circle at B. Five triangles OAC form a regular pentagon inscribed the bigger circle as in Diagram 4(b).


By using π = 3.142,
(a) show that the length of OA = 9.889 cm, given that OD = 8 cm. [1 mark]

(b) calculate
(i) the difference between the length, in cm, of arc AC and arc⁡ DE.
(ii) the area, in cm2, of the blue coloured region in Diagram 4(b).
[6 marks]


Solution:
(a)

From diagram 4( b ), AOC= 360 o 5 = 72 o From diagram 4( a ), OD=OB=8 cm AOB= 1 2 ( 72 o )= 36 o cosAOB= OB OA cos 36 o = 8 OA OA= 8 cos 36 o OA=9.889 cm (shown)


(b)(i)

AOC= 72 o × π 180 o = 2 5 π rad Difference of arc AC and arc DE = r 1 θ r 2 θ =( 9.889 )( 2 5 π )( 8 )( 2 5 π ) =2.374 cm


(b)(ii)

Blue coloured region in Diagram 4( b ) = area of circle area of 5 triangles =π r 2 5( 1 2 r 2 sinθ ) =π ( 9.889 ) 2 5[ 1 2 ( 9.889 ) 2 sin 72 o ] =307.224232.515 =74.709  cm 2

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