O-LevelMathematicsGeometry4024/22 · May/June 2021 · Q109 Marks

A, B and C are points on the circumference of a circle, centre O. OBD is a straight line and angle ABD = 130°. Find angle ACB, giving a reason for each step of your working. [Figure 10.A]

(a)A, B and C are points on the circumference of a circle, centre O. OBD is a straight line and angle ABD = 130°. Find angle ACB, giving a reason for each step of your working. [Figure 10.A][3]
(b)(i)P and Q are points on the circumference of a different circle, centre O. PR and QR are tangents to the circle at P and Q respectively. OP = 8 cm and PÔQ = 130°. Find PR. [Figure 10.B][2]
(b)(ii)Calculate the percentage of quadrilateral OPRQ that is shaded.[4]

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Mark scheme

Marking points from the official mark scheme (May/June 2021). The mark or mark type is shown in brackets.

[3] (a) ABO = 50° angles on straight line AOB = 80° angles in isosceles triangle ACB = 40° angles at centre twice angle at circumference B1FT for one correct angle with reason B1 for ACB = 40° nfww
[2] (b)(i) 17.2 or 17.15 to 17.16 M1 for tan (130/2) = PR/8 oe
[4] (b)(ii) 47 to 47.4 M1 for [2×]1/2×their17.2×8 or [2x]1/2×8x8 / cos 65 * sin 65 or 1/2×8^2 sin130+1/2their17.2^2 sin 50 oe AND M2 for (their137 - 130/360 × π×8^2) [× 100] oe or (their137 - 130/360 × π×8^2) / their137 ×100 oe or M1 for 130/360 ×π×8^2 oe

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