The constant a is such that ∫₀ᵃ xe⁻²ˣ dx = 1/8.

Mark scheme

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

1. [*M1] Commence integration and reach pxe⁻²ˣ + q∫e⁻²ˣdx OE
2. [A1] Obtain -½xe⁻²ˣ + ½∫e⁻²ˣdx OE
3. [A1] Complete integration and obtain -½xe⁻²ˣ - ¼e⁻²ˣ
4. [DM1] Use limits correctly and equate to 1/8, having integrated twice
5. [A1] Obtain a = ½ln(4a + 2) correctly AG
6. [M1] Calculate the values of a relevant expression or pair of expressions at a = 0.5 and a = 1
7. [A1] Justify the given statement with correct calculated values e.g. 0.5 < 0.69..., 1 > 0.89... 0.193 > 0, -1.105 < 0 0.066 < 0.125, 0.148 > 0.125 if put limits in the integral. Condone if they use calculator for the definite integral.
8. [M1] Use the iterative process an+1 = ½ln(4an + 2) correctly at least once.
9. [A1] Obtain final answer 0.84
10. [A1] Show sufficient iterations to at least 4 d.p. to justify 0.84 to 2 d.p. or show that there is a sign change in (0.835, 0.845) e.g. 0.75, 0.8047, 0.8261, 0.8343, 0.8373, 0.8385 1, 0.8959, 0.8599, 0.8469, 0.8420, 0.8402 .