[Figure 1] Figure 1 shows a sketch of part of the curve with equation y = x(x-2)(x-4). The curve crosses the x-axis at the origin O and at the points A and B. (a) Find the coordinates of A and B. (b) Find the area of the finite region enclosed by the curve and the x-axis between O and A.

Mark scheme

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

1. (a) M1 for setting y=0 and identifying at least one root from x(x-2)(x-4)=0
2. (a) A1 for A(2, 0) and B(4, 0) (or vice versa, provided labels are consistent with typical graphs)
3. (b) M1 for expanding y = x(x-2)(x-4) to a cubic polynomial form
4. (b) A1 for y = x(x² - 6x + 8) = x³ - 6x² + 8x
5. (b) M1 for attempting to integrate xⁿ → xⁿ⁺¹ for at least one term (powers must increase)
6. (b) A1 for correct integration (x⁴)/4 - (6x³)/3 + (8x²)/2 (or (x⁴)/4 - 2x³ + 4x²)
7. (b) M1 for correct use of limits 0 and 2 on their integrated expression
8. (b) A1 for evaluating at limits: ((2⁴)/4 - 2(2)³ + 4(2)²) - (0) = (4 - 16 + 16) = 4
9. (b) B1 for stating the area is 4 (units squared, or implies square units)