June 2025 Paper 1 Q12
12 \(A\), \(B\) and \(C\) are points on a circle, centre \(O\).
Prove that \(\quad y = 180 - 5x\) [4 marks]
| Answer | Mark | Comments |
|---|---|---|
| Alternative method 1: reflex angle AOC from angle at circumference | ||
| \(2(y + 3x)\) or \(2y + 6x\) | M1 | using \(2 \times ABC\) |
| \(4x + 2(y + 3x) = 360\) | M1dep | oe implies first M mark |
| \(2y + 10x = 360\) or \(2y = 360 - 10x\) Leading to \(y = 180 - 5x\) | A1 | M2 must be scored either of the first two statements is sufficient or a correct equivalent statement that improves on M2 |
| Angle at centre \(=\) 2 \(\times\) angle at circumference and angles at a point \(=\) 360 stated and used correctly | B1 | |
| Alternative method 2: 2x on circumference | ||
| Lines drawn to make angle at circumference and stated or seen on the diagram as \(2x\) or \(180 - (y + 3x)\) | M1 | If stated in working then lines not necessary on diagram |
| \(y + 3x + 2x = 180\) or \(4x = 2(180 - (y + 3x))\) | M1dep | oe implies first M mark |
| \(y = 180 - 5x\) | A1 | M2 must be scored |
| Angle at centre \(=\) 2 \(\times\) angle at circumference and opposite angles of a cyclic quad total 180 stated and used correctly | B1 | |
| Alternative method 3: reflex angle AOC from angles at the centre | ||
| \(360 - 4x\) | M1 | using angles at the centre |
| \(360 - 4x = 2(y + 3x)\) | M1dep | oe implies first M mark |
| \(2y + 10x = 360\) or \(2y = 360 - 10x\) Leading to \(y = 180 - 5x\) | A1 | M2 must be scored either of the first two statements is sufficient or a correct equivalent statement that improves on M2 |
| Angle at centre \(=\) 2 \(\times\) angle at circumference and angles at a point \(=\) 360 stated and used correctly | B1 | |
| Alternative method 4: isosceles triangles | ||
| Radius drawn from \(O\) to \(B\) to make two isosceles triangles and shows that \(OAB\) is equal to \(OBA\) and \(OCB\) is equal to \(OBC\) in working or on diagram | M1 | |
| \(y + 3x + 4x + OAB + OCB = 360\) and \(OAB + OCB = y + 3x\) | M1dep | oe implies first M mark |
| \(y + 3x + 4x + y + 3x = 360\) or \(2y = 360 - 10x\) Leading to \(y = 180 - 5x\) | A1 | M2 must be seen either of the first two statements is sufficient or a correct equivalent statement that improves on M2 |
| Angles of a quadrilateral add to 360 and isosceles triangles formed from radii have two equal angles stated and used correctly | B1 | |
Additional guidance
| A correct angle on the diagram from any Alt will score the first M mark then follow that Alt | |
| Stop marking as soon as \(y = 180 - 5x\) is substituted in and used | |
| If angles are stated in the working without being indicated on the diagram (the diagram will take precedence) they must be labelled accurately eg Reflex O or Concave AOC \(=\) | |
| Omission of labels or incorrect labels (if not seen on diagram either) will lose the A mark but M marks can still be awarded |