CAIE FP1 2017 Specimen — Question 1 4 marks

Exam BoardCAIE
ModuleFP1 (Further Pure Mathematics 1)
Year2017
SessionSpecimen
Marks4
PaperDownload PDF ↗
Mark schemeDownload PDF ↗
TopicParametric differentiation
TypeFind second derivative d²y/dx²
DifficultyStandard +0.8 This is a Further Maths parametric differentiation question requiring the chain rule formula d²y/dx² = (d/dt(dy/dx))/(dx/dt), involving trigonometric functions with powers. While the technique is standard for FP1, the algebraic manipulation with sec⁴t cosec t is non-trivial and requires careful simplification of multiple trigonometric identities, placing it moderately above average difficulty.
Spec1.03g Parametric equations: of curves and conversion to cartesian1.07s Parametric and implicit differentiation
1 The curve \(C\) is defined parametrically by $$x = 2 \cos ^ { 3 } t \quad \text { and } \quad y = 2 \sin ^ { 3 } t , \quad \text { for } 0 < t < \frac { 1 } { 2 } \pi$$ Show that, at the point with parameter \(t\), $$\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } = \frac { 1 } { 6 } \sec ^ { 4 } t \operatorname { cosec } t$$
Question 1:
AnswerMarks Guidance
AnswerMarks Guidance
\(\dot{x} = -6\cos^2 t \sin t\), \(\dot{y} = 6\sin^2 t \cos t\)1 B1
\(\Rightarrow \frac{dy}{dx} = -\tan t\) (OE)1 B1
\(\frac{d^2y}{dx^2} = -\sec^2 t \times \frac{-1}{6\cos^2 t \sin t} = \frac{1}{6}\sec^4 t \cosec t\) AG2 M1A1
Total4 
## Question 1:

| Answer | Marks | Guidance |
|--------|-------|----------|
| $\dot{x} = -6\cos^2 t \sin t$, $\dot{y} = 6\sin^2 t \cos t$ | 1 | B1 |
| $\Rightarrow \frac{dy}{dx} = -\tan t$ (OE) | 1 | B1 |
| $\frac{d^2y}{dx^2} = -\sec^2 t \times \frac{-1}{6\cos^2 t \sin t} = \frac{1}{6}\sec^4 t \cosec t$ AG | 2 | M1A1 |
| **Total** | **4** | |

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1 The curve $C$ is defined parametrically by

$$x = 2 \cos ^ { 3 } t \quad \text { and } \quad y = 2 \sin ^ { 3 } t , \quad \text { for } 0 < t < \frac { 1 } { 2 } \pi$$

Show that, at the point with parameter $t$,

$$\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } = \frac { 1 } { 6 } \sec ^ { 4 } t \operatorname { cosec } t$$

\hfill \mbox{\textit{CAIE FP1 2017 Q1 [4]}}
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