A certain curve is such that its gradient at a point (x, y) is proportional to y / (x√x). The curve passes through the points with coordinates (1, 1) and (4, e).

Mark scheme

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

1. [B1] (a) State dy/dx = k y / (x√x), or equivalent
2. [M1] (a) Separate variables correctly and attempt integration of at least one side
3. [A1] (a) Obtain term ln y, or equivalent
4. [A1] (a) Obtain term -2k/√x, or equivalent
5. [M1] (a) Use given coordinates to find k or a constant of integration c in a solution containing terms of the form a ln y and b/√x, where ab ≠ 0
6. [A1 + A1] (a) Obtain k = 1 and c = 2
7. [A1] (a) Obtain final answer y = exp(-2/√x + 2), or equivalent
8. [B1FT] (b) State that y approaches e² (FT their c in part (a) of the correct form)