The circle C has equation (x - 2)^2 + (y + 1)^2 = 25.

Mark scheme

Marking points from the official mark scheme. The mark or mark type is shown in brackets.

1. "(a)"
2. B1 for centre (2, -1).
3. B1 for radius 5 (accept √25).
4. "(b)"
5. M1 for substituting x=5 and y=3 into the equation of C: (5-2)^2 + (3+1)^2.
6. A1 for (3)^2 + (4)^2 = 9 + 16 = 25, clearly concluding P lies on C.
7. "(c)"
8. M1 for finding the gradient of the radius CP. (Gradient = (y2-y1)/(x2-x1) using P(5,3) and C(2,-1)).
9. A1 for gradient of radius = 4/3.
10. M1 for finding the gradient of the tangent (negative reciprocal of gradient of radius).
11. A1 for gradient of tangent = -3/4.
12. M1 for using y - y1 = m(x - x1) with their tangent gradient and point P(5, 3) or (y-3) = (-3/4)(x-5).
13. A1 for equation of tangent, e.g., 4y + 3x = 27 or y = -3/4x + 27/4 (or equivalent integer form).