| Exam Board | SPS |
|---|---|
| Module | SPS FM Pure (SPS FM Pure) |
| Year | 2023 |
| Session | June |
| Marks | 7 |
| Topic | Implicit equations and differentiation |
| Type | Find stationary points |
| Difficulty | Challenging +1.8 This is an implicit differentiation problem requiring finding dy/dx = 0, solving the resulting system of equations (3x² = 3y and 3y² = 3x gives x² = y and y² = x, leading to a quartic), and verifying two solutions exist. The algebraic manipulation and system-solving elevate this beyond standard implicit differentiation exercises, though the techniques are all A-level accessible. |
| Spec | 1.05j Trigonometric identities: tan=sin/cos and sin^2+cos^2=11.08h Integration by substitution |
A curve $C$ has equation
$$x^3 + y^3 = 3xy + 48$$
Prove that $C$ has two stationary points and find their coordinates. [7]
\hfill \mbox{\textit{SPS SPS FM Pure 2023 Q14 [7]}}