Find the equation of the normal to the curve y = x³ - 7x² + 12x - 5 at the point (1, 1).

Mark scheme

Marking points from the official mark scheme (Oct/Nov 2023). The mark or mark type is shown in brackets.

1. [M2] M1 for two terms of x³ - 7x² + 12x - 5 differentiated correctly for 3x² - 14x + 12
2. [A1] [At x = 1] gradient of tangent: 1
3. [M1] FT their -1 / (dy/dx)_x=1 for y - 1 = their(-1)(x - 1) oe
4. [A1] y - 1 = -1(x - 1) or y = -x + 2 oe, isw
5. [B1] FT their y = ax + b where a is a non-zero constant for x³ - 7x² + 12x - 5 = their(-x + 2)
6. [M1] Uses the correct linear factor x - 1 and the correct cubic x³ - 7x² + 13x - 7 [= 0] to find a quadratic factor with at least two terms correct
7. [A1] x² - 6x + 7
8. [M1] FT their 3-term quadratic providing it is from an attempt at finding a quadratic factor and the discriminant is not negative for Correct use of formula or completing the square on their 3-term quadratic, e.g., x = (-(-6)±√((-6)²-4[1](7))) / (2[1])
9. [A1] x = 3 ± √2