The graph of the probability density function f of a random variable X is symmetrical about the line x = 2. It is given that P(2 < X < 5) = 117/256.

Mark scheme

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

1. [M1] Use of symmetry about x = 2 to relate P(X > -1) to P(X < 5). (e.g. 1/2 + 117/256).
2. [A1] Any correct numerical expression seen leading to 245/256 AG.
3. [M1] Attempt to integrate f(x) = k(12+4x-x²) with any limits. (Integral: [k(12x + 2x² - x³/3)])
4. [M1] Correct application of limits and equating to probability. (Use of limits 2 and 5 and equating their integration attempt to 117/256. Or limits -2 and 6 equated to 1. Or limits -1 and 6 equated to 245/256. Or limits -1 to 5 equated to 234/256. Oe. No mixed methods.)
5. [A1] k = 3/256 or 0.0117.
6. [B1] Domain of X found as -1 ≤ x ≤ 2 (x = -1 and x = 2 seen or implied).
7. [B1] Mean = 0.5.
8. [*M1] Attempt to integrate x²g(x) with correct limits. (Integral: (9/7) * [2x³/3 + x⁴/4 - x⁵/5] from -1 to 2)
9. [DM1] Subtract their mean² from their ∫x²g(x)dx (both must be numerical).
10. [A1] = 0.45 or 9/20.