X is a random variable with probability density function given by f(x) = { 1/π (1+cos πx) 0 ≤ x ≤ 1, { 0 otherwise.

Mark scheme

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

1. [M1] ∫[0 to 1/2] (1+cos πx)dx Attempt to integrate f(x). Ignore limits.
2. [A1] = (1/π)[x + (sin πx)/π] [from 0 to 1/2] Correct integration and limits.
3. [A1] = 1/(2π) + 1/π² AG. Must see intermediate step. Convincingly obtained. No errors seen.
4. [M1*] ∫[0 to 1] (x+x cos πx)dx Attempt to integrate xf(x). Ignore limits.
5. [M1*] Attempt to integrate by parts to reach expression with at least two terms correct (must be more than two terms). May be implied by next line.
6. [A1] = [x^2/2 + x sin πx / π + cos πx / π^2] [from 0 to 1] Correct integration and limits.
7. [DM1] Substitute correct limits, dep M1M1.
8. [A1] = 1/(2π) - 2/π^3 AG. Legitimately obtained. No errors seen