June 2024 Paper 2 Q22
22 \(A\), \(B\) and \(C\) are points on a circle.
\(PAC\) is a straight line.
\(PB\) is a tangent to the circle.
\(AC = BC\)
Prove that \(\quad x = 90 - \dfrac{3}{2}y\) [5 marks]
| Answer | Mark | Comments |
|---|---|---|
Fully correct proof with
| B5 | B4 a correct unsimplified equation in \(x\) and \(y\) and rearrangement to \(x = 90 - \dfrac{3}{2}y\) with no rearrangement errors seen B3 a correct unsimplified equation in \(x\) and \(y\) eg \(x + y = \dfrac{180 - y}{2}\) or \(y = 180 - (2x + 2y)\) or \(x + \dfrac{180 - y}{2} + y + y = 180\) or \(180 - (x + y) + \dfrac{180 - y}{2} = 180\) B2 two expressions for angles correct in terms of \(x\) and/or \(y\) eg two of angle \(ACB = y\) angle \(PAB = 180 - (x + y)\) angle \(CAB = x + y\) angle \(CAB = \dfrac{180 - y}{2}\) angle \(CBQ = x + y\) angle \(CBQ = \dfrac{180 - y}{2}\) B1 expression for one angle correct in terms of \(x\) and/or \(y\) |
Additional guidance
| Up to B2 may be awarded for correct work with no answer or incorrect answer, even if this is seen amongst multiple attempts | |
| \(Q\) is the end of \(PB\) extended (\(Q\) may be eg \(D\)) | |
| angle \(PCB\) = angle \(ACB\) | |
| Condone eg \(PCB\) for angle \(PCB\) | |
| Accept unambiguous indication of angles eg1 angle \(C = y\) in working is unambiguous eg2 angle \(A = 180 - (x + y)\) in working is ambiguous unless shown in correct position on diagram | |
| Missing brackets must be recovered | |
| Up to B2 may be awarded from the diagram | |
| B2 can be awarded for two different expressions for the same angle | |
| B2 can be awarded for the same expression for angles \(CAB\) and \(ABC\) | |
| Most unsimplified equations will be from one of (i) equating two expressions for angle \(CAB\) (ii) equating two expressions for angle \(ACB\) (iii) angle sum of triangle \(PBC = 180\) (iv) angle \(PAB\) + angle \(CAB = 180\) (v) angle sum of triangle \(ABC = 180\) (vi) angle \(ABP\) + angle \(ABC\) + angle \(CBQ = 180\) |