Bottles of Lanta contain approximately 300 ml of juice. The volume of juice, in millilitres, in a bottle is 300 + X, where X is a random variable with probability density function given by f(x) = { (3/4000)(100-x²) for −10 ≤ x ≤ 10, 0 otherwise.

Mark scheme

Marking points from the official mark scheme (Feb/Mar 2020). The mark or mark type is shown in brackets.

1. [M1] ∫(from 5 to 10) (3/4000)(100-x²)dx; Attempt integration of f(x), ignore limits. Condone omission of 3/4000
2. [M1] = (3/4000) * [100x - x³/3] (from 5 to 10); Correct limits 5 and 10. OE SOI
3. [A1] = (3/4000) * ( (1000 - 1000/3) - (500 - 125/3) ); For fully correct working seen including substitution of limits; = 0.156 (3 sf) or 5/32
4. [M1] ∫(from p to 10) (3/4000)(100-x²)dx = 1/4; Attempt integration of f(x) with any limits and = 1/4 or = 3/4 seen. Condone omission of 3/4000
5. [A1] (3/4000) * [100x - x³/3] (from p to 10) = 1/4; Correct integration with correct limits seen (or implied for limits p and 10) and = 1/4 OE; Condone omission of 3/4000
6. [M1] (3/4000) * ( (1000 - 1000/3) - (100p - p³/3) ) = 1/4; Attempt substitution correct limits in their integration of f(x). Accept limits 0 to p if clearly seen, accept limits -10 and p. Substitution must be seen.
7. [A1] e.g. 2000 – 300p + p³/3 = 1000; AG No errors seen
8. [B1] Curve is symmetrical about x = 0; May be implied by sketch. No contradictions or integrate f(x) between –q and +q and equate to 0.5 leading to q³-300q + 1000=0 oe
9. [B1] q = 3.47