CAIE FP1 2017 June — Question 4 7 marks

Exam BoardCAIE
ModuleFP1 (Further Pure Mathematics 1)
Year2017
SessionJune
Marks7
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Mark schemeDownload PDF ↗
TopicImplicit equations and differentiation
TypeFind second derivative d²y/dx²
DifficultyStandard +0.8 This is a Further Maths implicit differentiation question requiring two differentiations and careful algebraic manipulation. While the technique is standard for FP1, finding the second derivative implicitly involves substituting the first derivative back into a second differentiation, which requires more steps and algebraic care than typical single-differentiation problems. The evaluation at a specific point adds computational complexity but is routine once the expression is found.
Spec1.07d Second derivatives: d^2y/dx^2 notation1.07s Parametric and implicit differentiation
4 A curve \(C\) has equation \(x ^ { 3 } - 3 x y + y ^ { 2 } = 4\). Find the value of \(\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } }\) at the point \(( 0,2 )\) of \(C\).
Question 4:
AnswerMarks Guidance
\(3x^2 - \left(3y + 3x\frac{dy}{dx}\right) + 2y\frac{dy}{dx} = 0\)B1B1 SOI
At \((0,2)\): \(-6 + 4\frac{dy}{dx} = 0 \Rightarrow \frac{dy}{dx} = \frac{3}{2}\)B1
\(6x - 3\frac{dy}{dx} - \left(3\frac{dy}{dx} + 3x\frac{d^2y}{dx^2}\right)\)M1A1
\(+2\left(\frac{dy}{dx}\right)^2 + 2y\frac{d^2y}{dx^2} = 0\)A1
At \((0,2)\): \(-\frac{9}{2} - \frac{9}{2} + \frac{9}{2} + 4\frac{d^2y}{dx^2} = 0 \Rightarrow \frac{d^2y}{dx^2} = \frac{9}{8}\)A1
Total: 7
## Question 4:

| $3x^2 - \left(3y + 3x\frac{dy}{dx}\right) + 2y\frac{dy}{dx} = 0$ | B1B1 | SOI |
|---|---|---|
| At $(0,2)$: $-6 + 4\frac{dy}{dx} = 0 \Rightarrow \frac{dy}{dx} = \frac{3}{2}$ | B1 | |
| $6x - 3\frac{dy}{dx} - \left(3\frac{dy}{dx} + 3x\frac{d^2y}{dx^2}\right)$ | M1A1 | |
| $+2\left(\frac{dy}{dx}\right)^2 + 2y\frac{d^2y}{dx^2} = 0$ | A1 | |
| At $(0,2)$: $-\frac{9}{2} - \frac{9}{2} + \frac{9}{2} + 4\frac{d^2y}{dx^2} = 0 \Rightarrow \frac{d^2y}{dx^2} = \frac{9}{8}$ | A1 | |

**Total: 7**

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4 A curve $C$ has equation $x ^ { 3 } - 3 x y + y ^ { 2 } = 4$. Find the value of $\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } }$ at the point $( 0,2 )$ of $C$.\\

\hfill \mbox{\textit{CAIE FP1 2017 Q4 [7]}}
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