The graph of the function f is a straight line segment from (0, 0) to (2, 1). [Figure for part (a)] The graph of the function g is a semicircle, centre (0, 0), entirely above the x-axis. [Figure for part (b)] The time, X minutes, taken by a large number of students to complete a test has probability density function h, as shown in the diagram. [Figure for part (c)]

Mark scheme

Marking points from the official mark scheme (May/June 2023). The mark or mark type is shown in brackets.

1. [B1] 1/2 × 2 × 1 or ∫xdx = 1, which is the correct area under a pdf. (Calculation and result.)
2. [B1] f(x) ≥ 0 (Condone f(x) > 0 or ‘Line is above x-axis’ OE.)
3. [M1] 1/2 πr² = 1 (Area of semi-circle equated to 1 OE. Missing factor of ½ gets M1A0.)
4. [A1] r = √(2/π) or 0.798 (3sf)
5. [B1] Area to the left of 15 is greater than 0.5 (OE, e.g. 'The distribution of X is skewed to the right / positively skewed, suggesting the median will be less than the mid-point of the interval.' or 'The distribution of X is skewed to the right / positively skewed' or 'It is a decreasing function suggesting the median will be less than the mid-point of the interval'.)
6. [M1] Integration of xh(x) attempted. Ignore limits.
7. [A1] [40lnx - x^2/20] evaluated from 10 to 20 (Correct integration and limits (can be implied by final answer).)
8. [A1] = 40ln 2 - 15 or 12.7 (3sf)