June 2019 Paper 1 Q17
17 \(A\), \(B\) and \(C\) are points on the circumference of a circle, centre \(O\).
\(ECD\) is a tangent to the circle at \(C\).
Angle \(AOB = 2x + 46^\circ\)
Angle \(OBC = 37^\circ\)
Angle \(ACD = 3x\)
Work out the value of \(x\). [4 marks]
| Answer | Mark | Comments |
|---|---|---|
| Any angle in terms of \(x\) either in working or on diagram. These could include: \(CBA = 3x\); \(ACB = x + 23\); \(OBA = 3x - 37\); \(OBA = \dfrac{180 - (2x + 46)}{2}\ (= 67 - x)\); \(BAC = BCE = 157 - 4x\); \(BOA\) (reflex) \(= 314 - 2x\) | M1 | oe but this must be an explicit expression for an angle mark positively (look for any correct angle) \(67 - x\) will be awarded M1 if seen on diagram but if in working would need \(OBA\) stating. Apply to other angles |
| A further angle that could be a different expression for one of the angles in the first mark or an angle that doesn’t include \(x\): eg \(OCE = 90\); \(BCP = 90\); \(OCB = 37\); \(BCE = 53\); \(PCD = 37\) | M1dep | look for isosceles triangles being formed by inserting a radius from O to C. P is the point on the circumference where BO is extended. could be a further angle that has the same expression as the first |
| A correct equation formed for \(x\) that would lead to the solution; eg \((3x - 37) + (3x - 37) + (2x + 46) = 180\) or \(3x + (x + 23) + 53 = 180\) many of these will lead to \(4x = 104\) | M1 | oe |
| \((x =)\ 26\) | A1 |
Additional guidance
| Look for angles on diagram | |
| \((x =)\ 37.5\) could be a common error. Mark as per mark scheme | |
| Labelling angle ABO as angle \(y\) and then writing \(2y + 2x + 46 = 180\) would not be enough for the first M mark. If they went on to write \(y = 67 - x\) this would gain the first M mark |