OCR C4 2007 January — Question 7 8 marks

Exam BoardOCR
ModuleC4 (Core Mathematics 4)
Year2007
SessionJanuary
Marks8
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Mark schemeDownload PDF ↗
TopicImplicit equations and differentiation
TypeFind stationary points
DifficultyChallenging +1.2 This is a standard implicit differentiation problem requiring finding dy/dx, setting it to zero, and solving the resulting system. While it involves multiple steps (implicit differentiation, algebraic manipulation, simultaneous equations), the techniques are all routine C4 material with no novel insight required. The 8 marks reflect computational length rather than conceptual difficulty, placing it moderately above average.
Spec1.07s Parametric and implicit differentiation
The equation of a curve is \(2x^2 + xy + y^2 = 14\). Show that there are two stationary points on the curve and find their coordinates. [8]
AnswerMarks Guidance
\(\frac{d}{dx}(xy) = x\frac{dy}{dx} + y\)B1
\(\frac{d}{dx}(y^2) = 2y\frac{dy}{dx}\)B1
\(4x + x\frac{dy}{dx} + y + 2y\frac{dy}{dx} = 0\)B1
Put \(\frac{dy}{dx} = 0\)*M1 and no other (different) result
Obtain \(4x + y = 0\)A1 AEF
Attempt to solve simultaneously with eqn of curvedep*M1
Obtain \(x^2 = 1\) or \(y^2 = 16\) from \(4x + y = 0\)A1
\((1, -4)\) and \((-1, 4)\) and no other solutionsA1 8 marks 
$\frac{d}{dx}(xy) = x\frac{dy}{dx} + y$ | B1 |

$\frac{d}{dx}(y^2) = 2y\frac{dy}{dx}$ | B1 |

$4x + x\frac{dy}{dx} + y + 2y\frac{dy}{dx} = 0$ | B1 |

Put $\frac{dy}{dx} = 0$ | *M1 | and no other (different) result

Obtain $4x + y = 0$ | A1 | AEF

Attempt to solve simultaneously with eqn of curve | dep*M1 |

Obtain $x^2 = 1$ or $y^2 = 16$ from $4x + y = 0$ | A1 |

$(1, -4)$ and $(-1, 4)$ and no other solutions | A1 | 8 marks | Accept $(\pm 1, \mp 4)$ but not $(\pm 1, \pm 4)$ |

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The equation of a curve is $2x^2 + xy + y^2 = 14$. Show that there are two stationary points on the curve and find their coordinates. [8]

\hfill \mbox{\textit{OCR C4 2007 Q7 [8]}}
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