CAIE P2 2020 June — Question 3 5 marks

Exam BoardCAIE
ModuleP2 (Pure Mathematics 2)
Year2020
SessionJune
Marks5
PaperDownload PDF ↗
Mark schemeDownload PDF ↗
TopicParametric differentiation
TypeFind tangent equation at parameter
DifficultyModerate -0.3 This is a straightforward parametric differentiation question requiring students to find dy/dx using the chain rule (dy/dt รท dx/dt), evaluate at t=0 to find the gradient, find the point coordinates, then write the tangent equation. It's slightly easier than average because it involves standard exponential differentiation with no algebraic complications, though it does require multiple routine steps.
Spec1.03g Parametric equations: of curves and conversion to cartesian1.07s Parametric and implicit differentiation
3 A curve has parametric equations $$x = \mathrm { e } ^ { t } - 2 \mathrm { e } ^ { - t } , \quad y = 3 \mathrm { e } ^ { 2 t } + 1$$ Find the equation of the tangent to the curve at the point for which \(t = 0\).
Question 3:
AnswerMarks Guidance
AnswerMark Guidance
State \(\frac{dx}{dt} = e^t + 2e^{-t}\), \(\frac{dy}{dt} = 6e^{2t}\)B1
Use \(\frac{dy}{dx} = \frac{dy}{dt} / \frac{dx}{dt}\) either in terms of \(t\) or after substitution of \(t = 0\)\*M1
Obtain gradient of tangent is 2A1
Attempt equation of tangent with numerical gradient and coordinatesDM1
Obtain \(y = 2x + 6\) or equivalentA1 
## Question 3:

| Answer | Mark | Guidance |
|--------|------|----------|
| State $\frac{dx}{dt} = e^t + 2e^{-t}$, $\frac{dy}{dt} = 6e^{2t}$ | B1 | |
| Use $\frac{dy}{dx} = \frac{dy}{dt} / \frac{dx}{dt}$ either in terms of $t$ or after substitution of $t = 0$ | \*M1 | |
| Obtain gradient of tangent is 2 | A1 | |
| Attempt equation of tangent with numerical gradient and coordinates | DM1 | |
| Obtain $y = 2x + 6$ or equivalent | A1 | |

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3 A curve has parametric equations

$$x = \mathrm { e } ^ { t } - 2 \mathrm { e } ^ { - t } , \quad y = 3 \mathrm { e } ^ { 2 t } + 1$$

Find the equation of the tangent to the curve at the point for which $t = 0$.\\

\hfill \mbox{\textit{CAIE P2 2020 Q3 [5]}}
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