The straight line y = x+5 meets the curve 2x² + 3y² = k at a single point P.

Mark scheme

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

1. [*M1] Attempt substitution for y in quadratic equation Or substitution for x ...
2. [A1] Obtain 5x² +30x+75-k[=0] or 5y²−20y+50-k[=0] OE e.g. x² +6x+15-k/5 (all terms gathered together).
3. [DM1] Use b²-4ac =0 with their a, b and c '= 0' may be implied in subsequent working or the answer.
4. [A1] Obtain 900-20(75-k)=0 or equivalent and hence k=30 ... obtaining 400-20(50-k)=0 and k=30.
5. [M1] Substitute their value of k in equation from part (a) and attempt solution Expect 5x² +30x+45[=0] or 5y²−20y+20[=0].
6. [A1] Obtain coordinates (-3, 2) SC B1 only (-3, 2) without attempt at quadratic solution.