A random variable X has probability density function given by f(x) = {1/16 (4x-x²) for 2 ≤ x ≤ 4, 0 otherwise.

Mark scheme

Marking points from the official mark scheme (May/June 2022). The mark or mark type is shown in brackets.

1. [M1] Attempt to integrate xf(x) Ignore limits (must see a power increase for attempted integration)
2. [M1] Attempt integrate xf(x) with correct limits (must see a power increase for attempted integration)
3. [A1] Correct substitution of correct limits (at least 2 terms seen) and answer seen. No errors seen i.e. NO recovery of errors and no non-exact decimals (e.g. 21.33) seen
4. [*M1] Attempt to integrate x² f(x) with correct limits (integration must be seen not implied. Must see a power increase for attempted integration)
5. [DM1] their ∫x²f(x)dx – (11/4)², with ∫x²f(x)dx evaluated, not necessarily simplified
6. [A1] 19/80 or 0.2375 (or 0.238 (3 sf)) SC If M0 then score B1 for 39/5 and B1 for 19/80
7. [M1] Attempt to integrate correct integral and limits must see a power increase for attempted integration Oe (Integrate 3 to 4) OR ALTERNATIVE METHOD integrate from m to 4 or 2 to m and equate to 0.5 to obtain cubic (m³ -6m² + 24 = 0 oe) (NB Integrating from m to 3 and equating to 0.5M0)
8. [M1] Their ∫_2^3 f(x)dx - 1/2 oe (1/2 – 5/16)
9. [A1] 3/16 or 0.1875 Condone 0.187 or 0.188