11 The tangent to the curve y = (3x-1)¹/² at the point where x = 3 meets the coordinate axes at the points A and B. The point with coordinates (a, a) lies on the perpendicular bisector of the line AB. Find the exact value of a.

Mark scheme

Marking points from the official mark scheme (May/June 2024). The mark or mark type is shown in brackets.

1. [10] B1 (When x = 3, y = 2)
2. [10] M1 Allow for k(3x-1)^(-1/2) (dy/dx = 3/2 (3x-1)^(-1/2))
3. [10] A1 (When x = 3, dy/dx = 1/4)
4. [10] M1 Allow using their y and their dy/dx (Tangent equation: y-2 = 1/4(x-3))
5. [10] A1 For both (Intercepts on the axes: x = -5, y = 5/4)
6. [10] M1 Dep on M mark for equation of tangent FT on their intercepts (Midpoint of AB: (-5/2, 5/8))
7. [10] M1 FT on their dy/dx or from their x and y intercept (Grad of perp bisector = -4)
8. [10] M1 Allow using their – 4 and their midpoint (Perp bisector equation: y-5/8 = -4(x+5/2))
9. [10] M1 Dep on previous M mark for use of (a, a) in their perp bisector equation
10. [10] A1 a = -15/8 or -1.875