It is given that ∫(from 0 to a) x cos(x/3) dx = 3, where the constant a is such that 0 < a < (3/2)π.

Mark scheme

Marking points from the official mark scheme (Oct/Nov 2019). The mark or mark type is shown in brackets.

1. [*M1] Commence integration by parts, reaching ax sin(x/3) - ∫b sin(x/3) dx
2. [A1] Obtain 3x sin(x/3) - ∫3 sin(x/3) dx
3. [A1] Complete integration and obtain 3x sin(x/3) + 9 cos(x/3)
4. [DM1] Substitute limits correctly and equate result to 3 in an integral of the form p x sin(x/3) + q cos(x/3)
5. [A1] Obtain a = (4 - 3 cos(a/3)) / sin(a/3) correctly
6. [M1] Calculate values at a = 2.5 and a = 3 of a relevant expression or pair of expressions.
7. [A1] Complete the argument correctly with correct calculated values
8. [M1] Use the iterative process a_(n+1) = (4 - 3 cos(a_n/3)) / sin(a_n/3) correctly at least once
9. [A1] Show sufficient iterations to at least 5 d.p. to justify 2.736 to 3d.p., or show a sign change in the interval (2.7355, 2.7365)
10. [A1] Obtain final answer 2.736