Higher June 2025 Paper 2R Q15

EdexcelCurrent spec3 marksCircle Theorems

15 \(A\), \(B\) and \(C\) are points on a circle, centre \(O\)

Circle centre O with triangle ABC inscribed and lines OA and OC; angle BAO = 36 degrees and angle BCO = 21 degrees

Diagram NOT accurately drawn

Angle \(BAO = 36^\circ\)
Angle \(BCO = 21^\circ\)

Work out the size of angle \(ACO\)

(3)

Mark scheme

Scheme	Marks
\(ABC = 21 + 36\;(= 57)\) or angle \(ABO = 36\) and angle \(CBO = 21\) and \(21 + 36\;(=57)\) or (reflex) \(AOC = 360 - (21 + 21 + 36 + 36)\;(= 246)\) or (obtuse) \(AOC = 2 \times (21 + 36)\;(=114)\) or \(BAX = 90 - 36\;(= 54)\) with the tangent at \(A\) drawn oe or \(BCY = 90 - 21\;(= 69)\) with the tangent at \(C\) drawn oe or angle \(ADB = 180 - 90 - 36\;(= 54)\) OR eg \(x\) = angle \(ACO\) and \(y = ABC\) and \(x + x + 2y = 180\) oe and \(y + (x + 36) + (x + 21) = 180\) oe or eg \(180 - 2x = 2(180 - 2x - 21 - 36)\)	M1
\((ACO =)\;\dfrac{180 - 2 \times \text{``}{57}\text{''}}{2}\) or \((ACO =)\;\dfrac{180 - (360 - \text{``}{246}\text{''})}{2}\) or \((ACO =)\;\dfrac{180 - \text{``}{57}\text{''} - 21 - 36}{2}\) or \((ACO =)\;\text{``}{54}\text{''} - 21\) or \((ACO =)\;\text{``}{69}\text{''} - 36\) or \((ACO =)\;90 - \text{``}{57}\text{''}\) OR eg \(180 - 2x = 2(180 - 2x - 21 - 36) \Rightarrow x = \ldots\)	M1
Correct answer only scores full marks (unless from obviously incorrect working) Answer: 33	A1
	(3)
	(3 marks)

Notes

M1: for a method to find one of the angles on the scheme; for this mark, values and calculations must be linked to the correct angle by notation or by being marked on the diagram. May also be awarded for a correct method to set up and solve an equation to find one of these angles; the variable must be clearly defined

where \(X\) is a point on the tangent at \(A\)
where \(Y\) is a point on the tangent at \(C\)
where \(AD\) is a diameter
OR
for any correct pair of simultaneous equations with clearly defined variables one of which must be angle \(ACO\)

or any correct equation in terms of \(ACO\) only

M1: for a complete method to find angle \(ACO\)
OR
forms a correct equation in terms of \(ACO\) only and solves to get a value (condone arithmetic errors)

implies the 1^st M mark (provided no incorrect working seen)