Edexcel C3 2009 January — Question 4 6 marks

Exam BoardEdexcel
ModuleC3 (Core Mathematics 3)
Year2009
SessionJanuary
Marks6
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Mark schemeDownload PDF ↗
TopicImplicit equations and differentiation
TypeFind tangent equation at point
DifficultyStandard +0.3 This is a straightforward implicit differentiation question requiring students to differentiate cos(2y+π) with respect to x using the chain rule, evaluate dy/dx at the given point, and write the tangent equation. While it involves implicit differentiation (a C3 topic), the steps are routine and mechanical with no conceptual challenges beyond applying standard techniques.
Spec1.03g Parametric equations: of curves and conversion to cartesian1.07m Tangents and normals: gradient and equations1.07s Parametric and implicit differentiation
4. Find the equation of the tangent to the curve \(x = \cos ( 2 y + \pi )\) at \(\left( 0 , \frac { \pi } { 4 } \right)\). Give your answer in the form \(y = a x + b\), where \(a\) and \(b\) are constants to be found.
AnswerMarks Guidance
\(x = \cos(2y + \pi)\)Given
\(\frac{dx}{dy} = -2\sin(2y + \pi)\)M1 A1
\(\frac{dy}{dx} = \frac{1}{-2\sin(2y + \pi)}\)A1ft, Follow through their \(\frac{dx}{dy}\) before or after substitution
At \(y = \frac{\pi}{4}\): \(\frac{dy}{dx} = \frac{1}{2\sin\frac{3\pi}{2}} = \frac{1}{2 \times (-1)} = -\frac{1}{2}\)B1
\(y - \frac{\pi}{4} = \frac{1}{2}x\)M1
\(y = \frac{1}{2}x + \frac{\pi}{4}\)A1 (6) [6] 
$x = \cos(2y + \pi)$ | Given

$\frac{dx}{dy} = -2\sin(2y + \pi)$ | M1 A1

$\frac{dy}{dx} = \frac{1}{-2\sin(2y + \pi)}$ | A1ft, Follow through their $\frac{dx}{dy}$ before or after substitution

At $y = \frac{\pi}{4}$: $\frac{dy}{dx} = \frac{1}{2\sin\frac{3\pi}{2}} = \frac{1}{2 \times (-1)} = -\frac{1}{2}$ | B1

$y - \frac{\pi}{4} = \frac{1}{2}x$ | M1

$y = \frac{1}{2}x + \frac{\pi}{4}$ | A1 | (6) [6]

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4. Find the equation of the tangent to the curve $x = \cos ( 2 y + \pi )$ at $\left( 0 , \frac { \pi } { 4 } \right)$.

Give your answer in the form $y = a x + b$, where $a$ and $b$ are constants to be found.\\

\hfill \mbox{\textit{Edexcel C3 2009 Q4 [6]}}
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