A random variable X has probability density function f defined by f(x) = { a/x² - 18/x³ for 2 ≤ x ≤ 3, 0 otherwise, where a is a constant.

Mark scheme

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

1. [M1] ∫ (a/x² - 18/x³) dx = 1 Attempt integrate f(x), ignore limits and ‘= 1'.
2. [A1] [-a/x + 9/x²] from 2 to 3 = 1 OE Correct integration and limits.
3. [A1] [-a/3 + 9/9] - [-a/2 + 9/4] = 1 a = 27/2 (AG) Must see correct substitution of limits. Correct working no errors seen.
4. [M1] ∫ x(a/x² - 18/x³) dx Attempt to integrate xf(x), ignore limits.
5. [A1] [a ln x + 18/x] from 2 to 3 or [a ln 2x + 18/x] from 2 to 3 OE e.g. using In 2x.
6. [A1] = 27/2 ln 3 + 6 - (27/2 ln 2 + 9) = 27/2 ln 3/2 - 3 AG Must see correct substitution of limits. Correct working no errors seen.