CAIE P2 2022 June — Question 4 7 marks

Exam BoardCAIE
ModuleP2 (Pure Mathematics 2)
Year2022
SessionJune
Marks7
PaperDownload PDF ↗
Mark schemeDownload PDF ↗
TopicImplicit equations and differentiation
TypeFind normal equation at point
DifficultyStandard +0.3 This is a straightforward implicit differentiation question requiring students to differentiate, substitute a point to find the gradient, then find the perpendicular gradient and write the normal equation. While it involves multiple steps, each is routine and the question provides the point to use, making it slightly easier than average.
Spec1.07m Tangents and normals: gradient and equations1.07s Parametric and implicit differentiation
4 A curve has equation \(x ^ { 2 } y + 2 y ^ { 3 } = 48\).
Find the equation of the normal to the curve at the point ( 4,2 ), giving your answer in the form \(a x + b y + c = 0\) where \(a , b\) and \(c\) are integers.
Question 4:
AnswerMarks Guidance
AnswerMarks Guidance
Use product rule to differentiate \(x^2 y\)\*M1 There must be evidence of implicit differentiation
Obtain correct \(2xy + x^2\frac{dy}{dx}\)A1
Obtain \(\left[2xy + x^2\frac{dy}{dx}\right] + 6y^2\frac{dy}{dx} = 0\)\*B1
Substitute \(x=4\), \(y=2\) to find value of \(\frac{dy}{dx}\)DM1 dependent on at least one term involving \(\frac{dy}{dx}\)
Obtain \(-\frac{2}{5}\)A1 SOI, OE
Attempt equation of normal passing through \((4, 2)\)M1 with numerical gradient correctly obtained from the negative reciprocal of *their* derivative
Obtain \(5x - 2y - 16 = 0\)A1 
## Question 4:

| Answer | Marks | Guidance |
|--------|-------|----------|
| Use product rule to differentiate $x^2 y$ | \*M1 | There must be evidence of implicit differentiation |
| Obtain correct $2xy + x^2\frac{dy}{dx}$ | A1 | |
| Obtain $\left[2xy + x^2\frac{dy}{dx}\right] + 6y^2\frac{dy}{dx} = 0$ | \*B1 | |
| Substitute $x=4$, $y=2$ to find value of $\frac{dy}{dx}$ | DM1 | dependent on at least one term involving $\frac{dy}{dx}$ |
| Obtain $-\frac{2}{5}$ | A1 | SOI, OE |
| Attempt equation of normal passing through $(4, 2)$ | M1 | with numerical gradient correctly obtained from the negative reciprocal of *their* derivative |
| Obtain $5x - 2y - 16 = 0$ | A1 | |

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4 A curve has equation $x ^ { 2 } y + 2 y ^ { 3 } = 48$.\\
Find the equation of the normal to the curve at the point ( 4,2 ), giving your answer in the form $a x + b y + c = 0$ where $a , b$ and $c$ are integers.\\

\hfill \mbox{\textit{CAIE P2 2022 Q4 [7]}}
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