Find the quotient and remainder when x² is divided by 1 + 4x².

Mark scheme

Marking points from the official mark scheme (May/June 2025). The mark or mark type is shown in brackets.

1. [B1] Obtain quotient 1/4 Could be found by using long division or by writing x^2 = q(1+4x^2) + r and comparing coefficients: 1 = 4q, 0 = q + r.
2. [B1] Obtain remainder -1/4 Allow B1B1 if implied by correct division and no further working, but do not ISW. Allow for a correct statement of the identity, but not for an incorrect statement of the remainder.
3. [M1] Commence integration by parts and reach Ax^2 tan^-1 2x + ∫B x^2 / (C + Dx^2) dx OE
4. [A1] Obtain 1/2 x^2 tan^-1 2x - ∫x^2/(1+4x^2) dx OE
5. [DM1] Integrate ∫x^2/(1+4x^2) dx to obtain an expression of the form p tan^-1 2x + qx
6. [A1] Complete integration and obtain 1/2 x^2 tan^-1 2x + 1/8 tan^-1 2x - 1/4 x FT their constant quotient and remainder from (a), and 1/4 tan^-1 2x - 1/4 x from their kx^2/(1+4x^2)
7. [DM1] Substitute limits correctly in an expression of the form Fx^2 tan^-1 2x+Gtan^-1 2x+Hx Need some evidence that they have considered the lower limit, e.g. sight of 0 in the working. No need to evaluate trigonometry. If in stages, then the 0 needs to be seen for each part.
8. [A1] Obtain answer π/8 - 1/8 or exact one- or two-term equivalent with trigonometry evaluated WWW ISW