CAIE FP1 2015 June — Question 6 9 marks

Exam BoardCAIE
ModuleFP1 (Further Pure Mathematics 1)
Year2015
SessionJune
Marks9
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TopicImplicit equations and differentiation
TypeFind second derivative d²y/dx²
DifficultyStandard +0.3 This is a standard implicit differentiation question requiring two derivatives. Finding dy/dx is routine application of the chain rule, and finding d²y/dx² follows a well-practiced technique. The verification at (3,1) and nature determination are straightforward. Slightly above average difficulty due to the second derivative step, but this is a textbook exercise in FP1.
Spec1.07s Parametric and implicit differentiation
6 A curve has equation \(x ^ { 2 } - 6 x y + 25 y ^ { 2 } = 16\). Show that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = 0\) at the point \(( 3,1 )\). By finding the value of \(\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } }\) at the point \(( 3,1 )\), determine the nature of this turning point.
Question 6:
AnswerMarks Guidance
WorkingMarks Notes
\(2x - 6\left(x\frac{dy}{dx} + y\right) + 50y\frac{dy}{dx} = 0\)M1A1
At \((3,1)\): \(6 - 18y' - 6 + 50y' = 0 \Rightarrow y' = 0\) (AG)A1 (4)
\(2 - 6(xy'' + y' + y') + 50[(y')^2 + yy''] = 0\)B1B1
At \((3,1)\): \(2 - 18y'' + 50y'' = 0 \Rightarrow y'' = -\frac{1}{16}\)M1A1
\(\Rightarrow\) maximum. (All previous marks required for final mark, i.e. CSO)A1 (5)
Total: 9 
## Question 6:

| Working | Marks | Notes |
|---------|-------|-------|
| $2x - 6\left(x\frac{dy}{dx} + y\right) + 50y\frac{dy}{dx} = 0$ | M1A1 | |
| At $(3,1)$: $6 - 18y' - 6 + 50y' = 0 \Rightarrow y' = 0$ (AG) | A1 | (4) |
| $2 - 6(xy'' + y' + y') + 50[(y')^2 + yy''] = 0$ | B1B1 | |
| At $(3,1)$: $2 - 18y'' + 50y'' = 0 \Rightarrow y'' = -\frac{1}{16}$ | M1A1 | |
| $\Rightarrow$ maximum. (All previous marks required for final mark, i.e. CSO) | A1 | (5) |
| | **Total: 9** | |

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6 A curve has equation $x ^ { 2 } - 6 x y + 25 y ^ { 2 } = 16$. Show that $\frac { \mathrm { d } y } { \mathrm {~d} x } = 0$ at the point $( 3,1 )$.

By finding the value of $\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } }$ at the point $( 3,1 )$, determine the nature of this turning point.

\hfill \mbox{\textit{CAIE FP1 2015 Q6 [9]}}
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