A curve has the equation y = 3/(2x²-5). Find the equation of the normal to the curve at the point $(2,1)$, giving your answer in the form ax+by+c = 0, where a, b and c are integers.

Mark scheme

Marking points from the official mark scheme (Feb/Mar 2024). The mark or mark type is shown in brackets.

1. [M1] Differentiate to obtain form kx(2x² – 5)⁻²
2. [A1] Obtain correct -12x(2x² – 5)⁻² OE
3. [A1] Substitute (2, 1) to obtain gradient -24/3 = -8/3 OE e.g. −2.67.
4. [*M1] Apply negative reciprocal to their numerical gradient to obtain gradient of normal Must have been some attempt at differentiation. Expect 3/8.
5. [DM1] Attempt equation of normal using their gradient of the normal and (2, 1) Expect y−1=(3/8)(x-2).
6. [A1] Obtain 3x-8y+2=0 (allow multiples) Or equivalent of requested form e.g. 8y-3x-2=0 .