Intermediate November 2017 Unit 1 Q11
11. \(PQ\) and \(PR\) are tangents to a circle with centre \(O\).
\(R\hat{P}Q = 30^\circ\).
Diagram not drawn to scale
Find the size of \(O\hat{Q}R\).
You must indicate any angles you calculate.
You must give a reason for each stage of your working. [5]
\(O\hat{Q}R\) = ……………… °
| Scheme | Marks | Comments |
|---|---|---|
| Note : Both E1 marks are awarded for a suitable/valid attempt at statement (not an implied reason from a calculation). Both E marks are dependent on attempt at related work. Look for angles seen on the diagram. For this question allow angles shown in diagram to take precedence over answer space. | ||
| (RQP or QRP =) \(\dfrac{180 - 30}{2}\) | M1 | |
| = 75(°) | A1 | |
| Tangents (from external point) are equal (in length) OR a geometric consequence based on this fact (e.g. ‘QPR is isosceles’ or ‘ PQOR is a kite’). | E1 | Accept any suitable attempt at a valid statement. Allow PQ = PR. Also allow unambiguous indication on the diagram. ‘Angles in a triangle’ not sufficient. |
| (OQR = 90 – 75 =) 15(°) | B1 | F.T. ’their derived 75’ provided acute. |
| Tangent and radius (at any point) are perpendicular | E1 | Accept any suitable attempt at a valid statement. Also allow unambiguous indication on the diagram. |
Alternative method 1
| Scheme | Marks | Comments |
|---|---|---|
| (ROQ = 360−90−90−30 =) 150(°) | B1 | |
| Tangent and radius (at any point) are perpendicular. | E1 | |
| OQR = \(\dfrac{180 - 150}{2}\) | M1 | FT ‘their derived 150’ |
| = 15(°) | A1 | |
| Radii form an isosceles triangle. | E1 |
Alternative method 2 (with line OP drawn)
| Scheme | Marks | Comments |
|---|---|---|
| (POQ or RQP=) 180 −90 – 15 | M1 | |
| = 75(°) | A1 | |
| Tangents (from external point) are equal (in length) OR a geometric consequence based on this fact (e.g. ‘QPR is isosceles’ or ‘ PQOR is a kite’). | E1 | |
| (OQR = 90 – 75 =) 15(°) | B1 | F.T. ’their derived 75’ provided acute. |
| Tangent and radius (at any point) are perpendicular. | E1 |
[Note: Do not ‘mix and match’ marks from alternative methods.]
Corrected by Westie's Workshop
- The board's mark scheme printed “(e.g. ‘QPR is isosceles’ or ‘ PQOR is a kite’.”. Changed to “(e.g. ‘QPR is isosceles’ or ‘ PQOR is a kite’).” because the bracket opened before “e.g.” was never closed (this happens twice in the scheme).
Original: see the board's mark scheme PDF (link above).