Higher November 2022 Paper 1 Q26
26 \(P\), \(Q\) and \(R\) are points on a circle.
\(SP\) is a tangent to the circle.
\(RQ = PQ\)
Not drawn accurately
Prove that \(\quad y = 90^\circ - x\) [4 marks]
| Answer | Mark | Comments |
|---|---|---|
| Alternative method 1 | ||
| \(RPQ = y\) | M1 | may be seen on diagram |
| \(RPQ = y\) and \(RQP = 180 - 2y\) | M1dep | may be seen on diagram |
| \(RQP = 2x\) and \(2x = 180 - 2y\) and correct rearrangement to \(y = 90 - x\) with M1M1 awarded | A1 | \(RQP = 2x\) may be implied by ‘alternate segment theorem’ |
| Correct reasons given with M1M1 scored and a correct initial equation for the A mark | B1 | (base angles of an) isosceles triangle (are equal) sum of the angles in a triangle is 180° alternate segment (theorem) |
| Alternative method 2 | ||
| \(RPQ = y\) | M1 | may be seen on diagram |
| \(RQP = 2x\) | M1 | may be seen on diagram |
| \(2x + 2y = 180\) and correct rearrangement to \(y = 90 - x\) with M1M1 awarded | A1 | |
| Correct reasons given with M1M1 scored and a correct initial equation for the A mark | B1 | (base angles of an) isosceles triangle (are equal) alternate segment (theorem) sum of the angles in a triangle is 180° |
| Alternative method 3 | ||
| \(RQP = 2x\) | M1 | may be seen on diagram |
| \(RQP = 2x\) and \(RPQ = 180 - 2x - y\) | M1dep | may be seen on diagram |
| \(y = 180 - 2x - y\) and correct rearrangement to \(y = 90 - x\) with M1M1 awarded | A1 | |
| Correct reasons given with M1M1 scored and a correct initial equation for the A mark | B1 | alternate segment theorem sum of the angles in a triangle is 180° (base angles of an) isosceles triangle (are equal) |
| Alternative method 4 | ||
| \(RPQ = y\) | M1 | may be seen on diagram |
| \(SP\) extended to \(T\) and \(QPT = y\) | M1 | may be seen on diagram any or no letter for \(T\) |
| \(2x + 2y = 180\) and correct rearrangement to \(y = 90 - x\) with M1M1 awarded | A1 | |
| Correct reasons given with M1M1 scored and a correct initial equation for the A mark | B1 | (base angles of an) isosceles triangle (are equal) alternate segment theorem angles on a straight line sum to 180° |
Additional guidance
Method marks can be scored using angle notation
eg \(RPQ = QRP\) is equivalent to \(RPQ = y\)