CAIE P2 2021 November — Question 5 6 marks

Exam BoardCAIE
ModuleP2 (Pure Mathematics 2)
Year2021
SessionNovember
Marks6
PaperDownload PDF ↗
Mark schemeDownload PDF ↗
TopicImplicit equations and differentiation
TypeFind dy/dx at a point
DifficultyStandard +0.3 This is a straightforward implicit differentiation question requiring the product rule and chain rule, followed by substitution to find dy/dx, then taking the negative reciprocal for the normal. The trigonometric component (cos 3y) and exact values add minor complexity, but the method is standard and well-practiced at A-level, making it slightly easier than average.
Spec1.07m Tangents and normals: gradient and equations1.07s Parametric and implicit differentiation
5 A curve has equation \(x ^ { 2 } + 4 x \cos 3 y = 6\).
Find the exact value of the gradient of the normal to the curve at the point \(\left( \sqrt { 2 } , \frac { 1 } { 12 } \pi \right)\).
Question 5:
AnswerMarks Guidance
AnswerMark Guidance
Use product rule to differentiate \(4x\cos 3y\) to obtain \(k_1\cos 3y+k_2 x\sin 3y\frac{\mathrm{d}y}{\mathrm{d}x}\)*M1 \(k_1k_2\neq 0\)
Obtain \(4\cos 3y-12x\sin 3y\frac{\mathrm{d}y}{\mathrm{d}x}\)A1 Allow unsimplified
Obtain \(2x+4\cos 3y-12x\sin 3y\frac{\mathrm{d}y}{\mathrm{d}x}=0\)A1 Allow unsimplified
Substitute \(x\)- and \(y\)-values to find value of first derivativeDM1
Apply \(-1/\frac{\mathrm{d}y}{\mathrm{d}x}\) to find gradient of normalM1
Obtain \(-\frac{3}{\sqrt{2}}\) or \(-\frac{3}{2}\sqrt{2}\) or exact equivalentA1
Total6 
## Question 5:

| Answer | Mark | Guidance |
|--------|------|----------|
| Use product rule to differentiate $4x\cos 3y$ to obtain $k_1\cos 3y+k_2 x\sin 3y\frac{\mathrm{d}y}{\mathrm{d}x}$ | *M1 | $k_1k_2\neq 0$ |
| Obtain $4\cos 3y-12x\sin 3y\frac{\mathrm{d}y}{\mathrm{d}x}$ | A1 | Allow unsimplified |
| Obtain $2x+4\cos 3y-12x\sin 3y\frac{\mathrm{d}y}{\mathrm{d}x}=0$ | A1 | Allow unsimplified |
| Substitute $x$- and $y$-values to find value of first derivative | DM1 | |
| Apply $-1/\frac{\mathrm{d}y}{\mathrm{d}x}$ to find gradient of normal | M1 | |
| Obtain $-\frac{3}{\sqrt{2}}$ or $-\frac{3}{2}\sqrt{2}$ or exact equivalent | A1 | |
| **Total** | **6** | |

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5 A curve has equation $x ^ { 2 } + 4 x \cos 3 y = 6$.\\
Find the exact value of the gradient of the normal to the curve at the point $\left( \sqrt { 2 } , \frac { 1 } { 12 } \pi \right)$.\\

\hfill \mbox{\textit{CAIE P2 2021 Q5 [6]}}
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Q1 8 Q2 7 Q3 6 Q4 5 Q5 6 Q6 8 Q7 10