OCR MEI C4 — Question 6 7 marks

Exam BoardOCR MEI
ModuleC4 (Core Mathematics 4)
Marks7
PaperDownload PDF ↗
Mark schemeDownload PDF ↗
TopicParametric differentiation
TypeFind gradient at given parameter
DifficultyStandard +0.3 This is a straightforward parametric differentiation question requiring the chain rule (dy/dx = dy/dt รท dx/dt) and finding t from given coordinates. The algebra is routine with quotient rule application, making it slightly easier than average for C4 level.
Spec1.03g Parametric equations: of curves and conversion to cartesian1.07s Parametric and implicit differentiation
6 A curve has parametric equations $$x = a t ^ { 3 } , \quad y = \frac { a } { 1 + t ^ { 2 } }$$ where \(a\) is a constant.
Show that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { - 2 } { 3 t \left( 1 + t ^ { 2 } \right) ^ { 2 } }\).
Hence find the gradient of the curve at the point \(\left( a , \frac { 1 } { 2 } a \right)\).
Question 6:
AnswerMarks Guidance
Answer/WorkingMarks Guidance
\(\frac{dy}{dt} = -a(1+t^2)^{-2} \cdot 2t\)M1 A1 \((1+t^2)^{-2} \times kt\) for method
\(\frac{dx}{dt} = 3at^2\)B1
\(\frac{dy}{dx} = \frac{dy/dt}{dx/dt} = \frac{-2at}{3at^2(1+t^2)^2}\)M1 ft
\(= \frac{-2}{3t(1+t^2)^2}\)E1
At \((a, \frac{1}{2}a)\), \(t=1\)M1 finding \(t\)
gradient \(= \frac{-2}{3 \times 2^2} = -\frac{1}{6}\)A1
[7] 
## Question 6:

| Answer/Working | Marks | Guidance |
|---|---|---|
| $\frac{dy}{dt} = -a(1+t^2)^{-2} \cdot 2t$ | M1 A1 | $(1+t^2)^{-2} \times kt$ for method |
| $\frac{dx}{dt} = 3at^2$ | B1 | |
| $\frac{dy}{dx} = \frac{dy/dt}{dx/dt} = \frac{-2at}{3at^2(1+t^2)^2}$ | M1 | ft |
| $= \frac{-2}{3t(1+t^2)^2}$ | E1 | |
| At $(a, \frac{1}{2}a)$, $t=1$ | M1 | finding $t$ |
| gradient $= \frac{-2}{3 \times 2^2} = -\frac{1}{6}$ | A1 | |
| **[7]** | | |

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6 A curve has parametric equations

$$x = a t ^ { 3 } , \quad y = \frac { a } { 1 + t ^ { 2 } }$$

where $a$ is a constant.\\
Show that $\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { - 2 } { 3 t \left( 1 + t ^ { 2 } \right) ^ { 2 } }$.\\
Hence find the gradient of the curve at the point $\left( a , \frac { 1 } { 2 } a \right)$.

\hfill \mbox{\textit{OCR MEI C4  Q6 [7]}}
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