Question 1 - A-Level Maths

Edexcel C4 — Question 1 8 marks

Exam Board	Edexcel
Module	C4 (Core Mathematics 4)
Marks	8
Paper	Download PDF ↗
Mark scheme	Download PDF ↗
Topic	Implicit equations and differentiation
Type	Tangent with given gradient
Difficulty	Standard +0.8 This implicit differentiation problem requires finding dy/dx, setting it to zero, then solving the resulting system of equations including the original curve equation. While the techniques are standard C4 content, the algebraic manipulation and simultaneous equation solving (likely involving substitution into a quadratic) make this more demanding than routine differentiation questions.
Spec	1.07s Parametric and implicit differentiation

A curve has the equation $$2x^2 + xy - y^2 + 18 = 0.$$ Find the coordinates of the points where the tangent to the curve is parallel to the $x$-axis. [8]

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Answer	Marks	Guidance
$\frac{dy}{dx} = 0 \Rightarrow 4x + y = 0, \quad y = -4x$	M1 A1
Sub: $2x^2 - 4x^2 - 16x^2 + 18 = 0$	M1
$x^2 = 1, \quad x = \pm 1 \Rightarrow (-1, 4), (1, -4)$	A2	(8 marks total)

$\frac{dy}{dx} = 0 \Rightarrow 4x + y = 0, \quad y = -4x$ | M1 A1 |

Sub: $2x^2 - 4x^2 - 16x^2 + 18 = 0$ | M1 |

$x^2 = 1, \quad x = \pm 1 \Rightarrow (-1, 4), (1, -4)$ | A2 | (8 marks total) |

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A curve has the equation
$$2x^2 + xy - y^2 + 18 = 0.$$
Find the coordinates of the points where the tangent to the curve is parallel to the $x$-axis. [8]

\hfill \mbox{\textit{Edexcel C4  Q1 [8]}}

This paper (7 questions)

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Q1 8 Q2 8 Q3 9 Q4 9 Q5 11 Q6 13 Q7 17

Answer	Marks	Guidance
\(\frac{dy}{dx} = 0 \Rightarrow 4x + y = 0, \quad y = -4x\)	M1 A1
Sub: \(2x^2 - 4x^2 - 16x^2 + 18 = 0\)	M1
\(x^2 = 1, \quad x = \pm 1 \Rightarrow (-1, 4), (1, -4)\)	A2	(8 marks total)