The curve C has equation y = 3x² - 4x + 1. The line L has equation y = x - 1. (a) Find the coordinates of the points of intersection of C and L. (b) Find the exact area of the region bounded by C and L.

Mark scheme

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

1. (a)
2. M1 for setting 3x² - 4x + 1 = x - 1 to find the x-coordinates of intersection.
3. A1 for 3x² - 5x + 2 = 0 (or equivalent correct quadratic).
4. A1 for (x=1, y=0) and (x=2/3, y=-1/3). (Both pairs required for A1, accept fractions or decimals).
5. (b)
6. M1 for setting up the integral for the area, ∫_(x₁)^(x₂) (y_(L) - y_(C)) dx or ∫_(x₁)^(x₂) (y_(C) - y_(L)) dx. (Limits must be the x-coordinates found in (a)).
7. A1 for correct integrand (x-1) - (3x² - 4x + 1) = -3x² + 5x - 2 (or 3x² - 5x + 2 if integrated in reverse).
8. M1 for integrating their quadratic expression with respect to x.
9. A1 for [ -x³ + 5/2x² - 2x ]_(2/3)¹. (Correct integrated expression)
10. M1 for correctly substituting both limits into their integrated expression and subtracting.
11. A1 for ( -1 + 5/2 - 2 ) - ( -(2/3)³ + 5/2(2/3)² - 2(2/3) ) = (-1/2) - (-8/27 + 10/9 - 4/3).
12. A1 for correct exact area 1/54. (cao)