Standard +0.8 This requires implicit differentiation to find dy/dx, setting it to zero for stationary points, then solving the resulting system of equations (one quadratic, one from the original curve). The algebraic manipulation is non-trivial and requires careful handling of the simultaneous equations, making it moderately harder than average C4 questions.
Differentiate equation (=0 can be implied), solve for \(\frac{dy}{dx}\) and put \(\frac{dy}{dx}=0\)
M1
Produce only \(2x + 4y = 0\) (AEF acceptable)
*A1
Without any error seen
Eliminate \(x\) or \(y\) from curve equation & equation(s) just produced
M1
Produce either \(x^2 = 36\) or \(y^2 = 9\)
dep*A1
Disregard other solutions
\((\pm 6, \mp 3)\) AEF, as the only answer ISW
dep*A1
Sign aspect must be clear
Total: 7 marks
# Question 5:
| Working/Answer | Mark | Guidance |
|---|---|---|
| $\frac{d}{dx}(xy) = x\frac{dy}{dx} + y$ | B1 | Implied by e.g. $4x\frac{dy}{dx}+y$ |
| $\frac{d}{dx}(y^2) = 2y\frac{dy}{dx}$ | B1 | |
| Differentiate equation (=0 can be implied), solve for $\frac{dy}{dx}$ and put $\frac{dy}{dx}=0$ | M1 | |
| Produce only $2x + 4y = 0$ (AEF acceptable) | *A1 | Without any error seen |
| Eliminate $x$ or $y$ from curve equation & equation(s) just produced | M1 | |
| Produce either $x^2 = 36$ or $y^2 = 9$ | dep*A1 | Disregard other solutions |
| $(\pm 6, \mp 3)$ AEF, as the only answer ISW | dep*A1 | Sign aspect must be clear |
**Total: 7 marks**
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5 Find the coordinates of the two stationary points on the curve with equation
$$x ^ { 2 } + 4 x y + 2 y ^ { 2 } + 18 = 0$$
\hfill \mbox{\textit{OCR C4 2010 Q5 [7]}}