A-LevelMathematicsCircular measure9709/12 · Oct/Nov 2021 · Q76 Marks

In the diagram the lengths of AB and AC are both 15 cm. The point P is the foot of the perpendicular from C to AB. The length CP = 9cm. An arc of a circle with centre B passes through C and meets AB at Q. [Figure X.X]

(a)Show that angle ABC = 1.25 radians, correct to 3 significant figures.[2]
(b)Calculate the area of the shaded region which is bounded by the arc CQ and the lines CP and PQ.[4]

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Mark scheme

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

[M1] EITHER By using trigonometry: BAC = 0.6435... and ABC = (π - 0.6435) / 2 OR By Pythagoras: AP = 12 ⇒ BP = 3 so tan ABC = 9/3 OR Using APBC and either the sine or cosine rule sin ABC = 9/√10 or cos ABC = √10/10
[A1] 1.25 radians (3 sf) AG. Final answer must be 1.25, more accurate value 1.24904... with no rounding to 3sf seen as the final answer gets M1A0. If decimals are used all values must be given to at least 4sf for A1.
[M1] Using correct method(s) to find BC.
[M1] Using tan⁻¹3 or 1.25 and their BC, but not 9 or 15, in correct area of sector formula.
[B1] Area of triangle PBC = 13.4 to 13.6 or 1/2 × 9 × 3
[A1] 42.7 or 42.8 AWRT

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