The variables x and t satisfy the differential equation e³ᵗ dx/dt = cos² 2x, for t ≥ 0. It is given that x = 0 when t = 0.

Mark scheme

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

1. [B1] Correct separation of variables [ ∫ sec² 2x dx = ∫ e⁻³ᵗ dt ]
2. [B1] Obtain term ∫ e⁻³ᵗ dt = -1/3 e⁻³ᵗ
3. [M1] Obtain term of the form k tan 2x for ∫ sec² 2x dx
4. [A1] Obtain term 1/2 tan 2x
5. [M1] Use x = 0, t = 0 to evaluate a constant, or as limits in a solution containing terms of the form a tan 2x and b e⁻³ᵗ, where ab ≠ 0
6. [A1] Obtain correct solution in any form (e.g. 1/2 tan 2x = -1/3 e⁻³ᵗ + 1/3)
7. [A1] Obtain final answer x = 1/2 tan⁻¹(2/3(1 - e⁻³ᵗ))
8. [B1] State that x approaches 1/2 tan⁻¹(2/3)