The diagram shows the curve with equation y = 4(3x+4)¹ᐟ² - 2x - 6 for values of x such that 0 ≤ x ≤ 7. The tangent to the curve at the point P (7,0) meets the y-axis at the point Q. Region A is bounded by the curve and the two axes. Region B is bounded by the curve, the line segment PQ and the y-axis. [Figure X.X]

Mark scheme

Marking points from the official mark scheme (Feb/Mar 2025). The mark or mark type is shown in brackets.

1. [B1] ∫(4(3x+4)¹ᐟ² - 2x - 6)dx = 4(3x+4)³ᐟ² / (3/2 * 3) - 2x²/2 - 6x [+C] B1 for each correct { }.
2. [B1] ∫(4(3x+4)¹ᐟ² - 2x - 6)dx = (8/9)(3x+4)³ᐟ² B1 for each correct { }.
3. [B1] ∫(4(3x+4)¹ᐟ² - 2x - 6)dx = - x² - 6x B1 for each correct { }.
4. [M1] [A =] (8/9)(21+4)³ᐟ² - 7² - 6×7) - ((8/9)(4)³ᐟ² - 0 - 0) Correct use of 7 and 0 in an expression with at least two terms with two correct powers.
5. [A1] 13
6. [B1] dy/dx = 2(3x+4)⁻¹ᐟ² * 3 - 2 = 6(3x+4)⁻¹ᐟ² - 2
7. [*M1] y - 0 = (their dy/dx with x=7)(x-7) Using x = 7, their value for dy/dx and then any form of the equation of a straight line using (7, 0).
8. [DM1] [Q is] (0, 28/5) Use x = 0 in their equation of PQ.
9. [DM1] [Area of OPQ =] 1/2 * 7 * (their 28/5)
10. [A1] [1/2 * 7 * (28/5)] - [(their 13 from (a))] = 33/5 FT Only FT, following B0M1DM1DM1, from their (a) if (their 13 from(a)) < 98/5 and the area is > 0.