The normal to the curve y = ln(3x² + 2)/(x + 1), at the point A on the curve where x = 0, meets the x-axis at point B. Point C has coordinates (0, 3 ln 2). Find the gradient of the line BC in terms of ln 2.

Mark scheme

Marking points from the official mark scheme (May/June 2022). The mark or mark type is shown in brackets.

1. [B1] When x = 0, y = ln 2 soi
2. [1] B1 for 6x/(3x²+2) allow when seen
3. [M1] for attempt at a quotient or product rule for dy/dx
4. [1] A1 dy/dx = ( (x+1) (6x/(3x²+2)) - ln(3x²+2) ) / (x+1)² correct
5. [M1] When x = 0, dy/dx = -ln 2 (Dep on previous M mark for attempt to find the gradient using their dy/dx)
6. [M1] For attempt at a normal equation using their y (not 3 ln2) and -1/(their(-ln2)) , must be from an attempt at differentiation
7. [M1] For attempt to find the value of x when y = 0 using their normal equation
8. [M1 Dep] on both previous M marks (for gradient BC)
9. [A1] Gradient BC = 3/ln 2 or 3(ln2)^-1