The normal to the curve `y = tan(3x + π/2)` at the point P with coordinates `(p, -1)`, where `0 < p < π/6`, meets the x-axis at the point A and the y-axis at the point B. Find the exact coordinates of the mid-point of AB.

Mark scheme

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

1. [10] M1 for a complete method to find the value of p (for -1 = tan(3p + π/2))
2. B1 (for p = π/12)
3. M1 for a sec²(3x + π/2)
4. A1 all correct (for dy/dx = 3 sec²(3x + π/2))
5. M1 For attempt to find the gradient using their p from differentiation (for When x = π/12, dy/dx = 6)
6. M1 For attempt at normal equation using their p and 1 / (their value for dy/dx) (for Equation of normal: y+1 = -1/6 (x - π/12))
7. M1 For attempt to find B using their normal equation (must be from differentiation) (for When x = 0, y = -1/72 π)
8. M1 For attempt to find A using their normal equation (must be from differentiation) (for When y = 0, x = π/12 - 6)
9. A1 for x value (must be exact) (for Mid-point (π/24 - 3, -π/144 - 2))
10. A1 for y value (must be exact)