The random variable X has probability density function given by f(x) = { kx² / a² for 0 ≤ x ≤ a, { 0 otherwise, where k and a are positive constants. It is given that E(X) = 1.

Mark scheme

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

1. [M1] Attempt to integrate f(x) and =1 ignore limits.
2. [A1] Correct integration and correct limits. [kx³/3a²] between 0 and a.
3. [A1] AG. Must see an intermediate step and answer. Convincingly obtained no errors seen.
4. [M1] Attempt to integrate xf(x) with limits 0 and a. Condone k instead of 3/a or missing 'k'.
5. [A1] Correct expression after correct integration and = 1. [k x^4 / 4a^2] from 0 to a. Condone k instead of 3/a.
6. [A1] a = 3/4 or 1.33 (3 sf)
7. [M1] Attempt to integrate f(x) with limits 0 to m and =0.5 (accept in terms of a and/or k. Condone missing k. FT their a.)
8. [A1FT] Correct expression after correct integration with a and k correctly substituted (at some point) not necessarily simplified. FT their a.
9. [A1] m = (32/27)^(1/3) or 1.06 (3 sf) OE. CWO.