A random variable X has probability density function given by f(x) = { (1/18)(9-x²) for 0 ≤ x ≤ 3, 0 otherwise. The median of X is m.

Mark scheme

Marking points from the official mark scheme (Oct/Nov 2021). The mark or mark type is shown in brackets.

1. [M1] ∫[0 to 1.2] (1/18)(9-x²) dx (Attempt to integrate f(x), ignore limits. Must see an increase of power.)
2. [A1] (1/18)[9x - x³/3][0 to 1.2] (Correct integration and correct limits.)
3. [A1] 71/125 or 0.568 (SC unsupported answer scores B2 only.)
4. [M1] ∫[0 to 3] x(1/18)(9-x²) dx (Attempt to integrate xf(x), ignore limits. Must see an increase of power.)
5. [A1] (1/18)[9x²/2 - x⁴/4][0 to 3] (Correct integration and correct limits.)
6. [A1] 9/8 or 1.125 (SC unsupported answer scores B2 only.)
7. [M1] ∫[0 to m] (1/18)(9-x²) dx = 0.5 (Attempt to integrate f(x) with correct limits and = 0.5. OE. Accept limits m to 3. Allow x instead of m.)
8. [A1] (1/18)[9m - m³/3] - 0.5 = 0 (Any correct cubic equation in m or x.)
9. [A1] m³ - 27m + 27 = 0 (AG. Correctly obtain this equation. No errors seen.)