The time, X hours, taken by a large number of people to complete a challenge is modelled by the probability density function given by f(x) = { 2/x² a ≤ x ≤ b, 0 otherwise, where a and b are constants.

Mark scheme

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

1. Min and max times [to complete challenge]
2. ∫(from a to b) (2/x²) dx = 1
3. [[-2/x] (from a to b) = 1] –2/b + 2/a = 1
4. -a + b = ab or b = a(b + 1) / OE ⇒ a = b/(b+1) AG.
5. Attempt to integrate xf(x). Limits a and b or b/(b+1) and b (condone a and 2 for M1) See SC for use of limits 2/3 and 2
6. Correct integration and limits substituted. Condone ln 2 – ln a.
7. For correct equation in b only (i.e. using part (b)).
8. b = 2 (AG) a = 2/3 / Both obtained correctly (Note: if b=2 not shown but used can score M1 A1, A1/A0 depending on where b=2 is introduced, A0)
9. ∫(from a to m) (2/x²) dx = 0.5 or ∫(from m to b) (2/x²) dx = 0.5
10. [[-2/x] (from a to m) = 0.5 ] or [ [-2/x] (from m to b) = 0.5 ]
11. [-2/m + 2/a = 0.5] or [-2/b + 2/m = 0.5] / m=1