Let f(x) = 2e²ˣ / (e²ˣ - 3eˣ + 2)

Mark scheme

Marking points from the official mark scheme (Oct/Nov 2024). The mark or mark type is shown in brackets.

1. [M1] Use correct quotient rule NB the question asks for f'(x) so need complete form
2. [A1] Obtain correct derivative in any form, e.g. 4e²ˣ(e²ˣ - 3eˣ + 2) – 2e²ˣ(2e²ˣ – 3eˣ) / (e²ˣ - 3eˣ + 2)²
3. [*M1] Equate their derivative to zero
4. [DM1] Solve for x to obtain x = lna
5. [A1] Obtain x = ln(4/3) and y = -16
6. [B1] State or imply du/dx = eˣ
7. [B1] Obtain ∫(2u / (u² - 3u + 2)) du or equivalent
8. [B1ft] State or imply partial fractions of the form A/(u-1) + B/(u-2)
9. [M1] Use a correct method for finding a constant
10. [A1] Obtain correct -2/(u-1) + 4/(u-2)
11. [*M1] Integrate to obtain a ln(u-1) + b ln(u-2) or equivalent
12. [A1ft] Obtain correct -2 ln(u-1) + 4 ln(u-2) or equivalent
13. [DM1] Correctly use limits u = 5 and 3 in an expression of the form a ln(u – 1) + b ln(u – 2) or x = ln 5 and ln 3 in an expression of the form a ln(eˣ – 1) + b ln(eˣ – 2)
14. [A1] Obtain ln(81/4)