CAIE P2 2022 November — Question 5 5 marks

Exam BoardCAIE
ModuleP2 (Pure Mathematics 2)
Year2022
SessionNovember
Marks5
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Mark schemeDownload PDF ↗
TopicImplicit equations and differentiation
TypeFind dy/dx at a point
DifficultyStandard +0.3 This is a straightforward implicit differentiation question requiring product rule and chain rule application, followed by substitution of given coordinates. While it involves exponential functions, it's a standard textbook exercise with clear steps: differentiate implicitly, rearrange for dy/dx, substitute the point. Slightly above average difficulty due to the exponential term but still routine for P2 level.
Spec1.07s Parametric and implicit differentiation
5 A curve has equation \(4 \mathrm { e } ^ { 2 x } y + y ^ { 2 } = 21\).
Find the gradient of the curve at the point \(( 0 , - 7 )\).
Question 5:
AnswerMarks Guidance
AnswerMark Guidance
Use product rule to differentiate \(4e^{2x}y\)M1
Obtain correct \(8e^{2x}y + 4e^{2x}\frac{dy}{dx}\)A1
Obtain \(\left[8e^{2x}y + 4e^{2x}\frac{dy}{dx}\right] + 2y\frac{dy}{dx} = 0\)B1
Substitute \(x = 0\) and \(y = -7\) to find value of \(\frac{dy}{dx}\)M1 Dependent on at least one term involving \(\frac{dy}{dx}\) from implicit differentiation
Obtain \(-\frac{28}{5}\)A1 OE 
## Question 5:

| Answer | Mark | Guidance |
|--------|------|----------|
| Use product rule to differentiate $4e^{2x}y$ | M1 | |
| Obtain correct $8e^{2x}y + 4e^{2x}\frac{dy}{dx}$ | A1 | |
| Obtain $\left[8e^{2x}y + 4e^{2x}\frac{dy}{dx}\right] + 2y\frac{dy}{dx} = 0$ | B1 | |
| Substitute $x = 0$ and $y = -7$ to find value of $\frac{dy}{dx}$ | M1 | Dependent on at least one term involving $\frac{dy}{dx}$ from implicit differentiation |
| Obtain $-\frac{28}{5}$ | A1 | OE |

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5 A curve has equation $4 \mathrm { e } ^ { 2 x } y + y ^ { 2 } = 21$.\\
Find the gradient of the curve at the point $( 0 , - 7 )$.\\

\hfill \mbox{\textit{CAIE P2 2022 Q5 [5]}}
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