Question 4 - A-Level Maths

CAIE P2 2014 June — Question 4 6 marks

Exam Board	CAIE
Module	P2 (Pure Mathematics 2)
Year	2014
Session	June
Marks	6
Paper	Download PDF ↗
Mark scheme	Download PDF ↗
Topic	Parametric differentiation
Type	Find tangent equation at parameter
Difficulty	Moderate -0.3 This is a straightforward parametric differentiation question requiring standard techniques: find dx/dt and dy/dt, compute dy/dx = (dy/dt)/(dx/dt), evaluate at t=0 to get the gradient, find the point coordinates at t=0, then write the tangent equation. All steps are routine with no conceptual challenges, making it slightly easier than average.
Spec	1.03g Parametric equations: of curves and conversion to cartesian 1.07m Tangents and normals: gradient and equations 1.07s Parametric and implicit differentiation

4 The parametric equations of a curve are $$x = 2 \ln ( t + 1 ) , \quad y = 4 \mathrm { e } ^ { t }$$ Find the equation of the tangent to the curve at the point for which $t = 0$. Give your answer in the form $a x + b y + c = 0$, where $a , b$ and $c$ are integers.

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Answer	Marks
Obtain $\frac{dx}{dt} = \frac{2}{t+1}$	B1
Obtain $\frac{dy}{dt} = 4e^t$	B1
Use $\frac{dy}{dx} = \frac{dy}{dt} \div \frac{dx}{dt}$ with $t = 0$ to find gradient	M1
Obtain $2$	A1
Form equation of tangent through $(0, 4)$ with numerical gradient obtained from attempt to differentiate	M1
Obtain $2x - y + 4 = 0$ or equivalent of required form	A1 [6]

Obtain $\frac{dx}{dt} = \frac{2}{t+1}$ | B1 |

Obtain $\frac{dy}{dt} = 4e^t$ | B1 |

Use $\frac{dy}{dx} = \frac{dy}{dt} \div \frac{dx}{dt}$ with $t = 0$ to find gradient | M1 |

Obtain $2$ | A1 |
Form equation of tangent through $(0, 4)$ with numerical gradient obtained from attempt to differentiate | M1 |

Obtain $2x - y + 4 = 0$ or equivalent of required form | A1 [6] |

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4 The parametric equations of a curve are

$$x = 2 \ln ( t + 1 ) , \quad y = 4 \mathrm { e } ^ { t }$$

Find the equation of the tangent to the curve at the point for which $t = 0$. Give your answer in the form $a x + b y + c = 0$, where $a , b$ and $c$ are integers.

\hfill \mbox{\textit{CAIE P2 2014 Q4 [6]}}

This paper (8 questions)

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Q1 4 Q2 4 Q3 5 Q4 6 Q5 6 Q6 7 Q7 9 Q8 9

Answer	Marks
Obtain \(\frac{dx}{dt} = \frac{2}{t+1}\)	B1
Obtain \(\frac{dy}{dt} = 4e^t\)	B1
Use \(\frac{dy}{dx} = \frac{dy}{dt} \div \frac{dx}{dt}\) with \(t = 0\) to find gradient	M1
Obtain \(2\)	A1
Form equation of tangent through \((0, 4)\) with numerical gradient obtained from attempt to differentiate	M1
Obtain \(2x - y + 4 = 0\) or equivalent of required form	A1 [6]