Differentiation of Parametric Equations (Cambridge (CIE) A Level Maths: Pure 3): Exam Questions

The parametric equations of a curve are

$x=e^{2t}$, $y=3t^2+1$.

Find an expression for $\frac{\text{d}y}{\text{d}x}$ in terms of $t$.

The curve passes through the point $P (1,1)$.

(i) Find the value of $t$ at the point $P$.

(ii) Find the gradient of the curve at the point $P$.

(iii) State what the value of the gradient tells you about the point $P$.

The parametric equations of a curve are

$x=12t$, $y=9t^2-9t+4$,

for $0\le t\le 1$.

Find the $y$-coordinate of the point on the curve where $t=0.3$.

Find the minimum value of $y$ on the curve.

Find the values of $x$ when $y=2.9$, giving your answers correct to 1 decimal place.

The diagram shows the curve $C$ with parametric equations

$x=3 \text{sin}3\theta$, $y=6 \text{cos}2\theta$,

for $-\frac{\pi }{2}\le \theta \le \frac{\pi }{2}$.

(i) Write down the value of $\frac{\text{d}y}{\text{d}\theta }$ at the point $(0,6)$.

(ii) Write down the value of $\frac{\text{d}x}{\text{d}\theta }$ at the points $(-3,3)$ and $(3,3)$.

Find an expression for $\frac{\text{d}y}{\text{d}x}$ in terms of $\theta$.

(i) Find the values of $x$, $y$ and $\frac{\text{d}y}{\text{d}x}$ at the point where $\theta =\frac{\pi }{12}$.

(ii) Hence show that the equation of the tangent to $C$ at the point where $\theta =\frac{\pi }{12}$ is

$2\sqrt{2}x+3y-\left(9\sqrt{3}+6\right)=0$

The parametric equations of a curve $C$ are

$x=6t^2+2$, $y=\frac{1}{t}$,

where $t>0$.

Find an expression, in terms of $t$, for $\frac{\text{d}y}{\text{d}x}$.

(i) Find the gradient of the tangent to $C$ at the point $(8,1)$.

(ii) Hence write down the gradient of the normal to $C$ at the point $(8,1)$.

(iii) Find the equation of the normal to $C$ at the point $(8,1)$.

The parametric equations of a curve are

$x=\text{sin}2t$, $y=e^t$.

Find an expression for $\frac{\text{d}y}{\text{d}x}$ in terms of $t$.

Verify that the curve passes through the point $(0,1)$, and find the gradient of the curve at that point.

The parametric equations of a curve are

$x=8t-4$, $y=16t^2-16t+5$,

for $0\le t\le 1$.

Find $\frac{\text{d}x}{\text{d}t}$ and $\frac{\text{d}y}{\text{d}t}$, and hence find $\frac{\text{d}y}{\text{d}x}$ in terms of $t$.

Find the difference between the greatest and least values of $y$.

Find the value of $y$ when $x=3$.

The diagram shows the curve $C$ with parametric equations

$x=2 \text{cos}3\theta$, $y=5 \text{sin}\theta$,

for $0\le \theta \le 2\pi$.

Find an expression for $\frac{\text{d}y}{\text{d}x}$ in terms of $\theta$.

(i) Show that the gradient of the tangent to $C$ at the point where $\theta =\frac{\pi }{4}$ is $-\frac{5}{6}$.

(ii) Hence find the equation of the tangent to $C$ at the point where $\theta =\frac{\pi }{4}$.

The parametric equations of a curve $C$ are

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

where $t>0$.

Find an expression, in terms of $t$, for $\frac{\text{d}y}{\text{d}x}$.

(i) Find the gradient of the tangent to $C$ at the point where $t=\frac{1}{2}$.

(ii) Hence find the equation of the normal to $C$ at the point where $t=\frac{1}{2}$.

The parametric equations of a curve are

$x=10t$, $y=4.9t^2-4.9t+2$,

for $0\le t\le 1$.

(i) Write down the value of $y$ at the point where $t=0$.

(ii) Find the minimum value of $y$ on the curve.

Find the value of $x$ at the point where $y=1.4$ and $t>0.5$.

The diagram shows the ellipse $E$ with parametric equations

$x=2 \text{cos}\left(\theta +\frac{\pi }{3}\right)$, $y=4 \text{sin}\theta$,

for $-\pi \le \theta \le \pi$.

Find an expression for $\frac{\text{d}y}{\text{d}x}$ in terms of $\theta$.

Find the equation of the tangent to $E$ at the point where $\theta =-\frac{\pi }{6}$, giving your answer in the form $y=a-bx$, where $a$ and $b$ are real numbers that should be given in exact form.

The parametric equations of a curve $C$ are

$x=t^2$, $y=2 \text{sin}t$,

where $0\le t\le 2\pi$.

Show that, in terms of $t$,

$\frac{\text{d}y}{\text{d}x}=\frac{\text{cos}t}{t}$

Show that the distance between the maximum and minimum points on $C$ is $2\sqrt{{\pi }^4+4}$ units.

The parametric equations of a curve $C$ are

$x=t^2-4$, $y=3t$.

Show that $t=2$ at the point $(0,6)$, and find the value of $\frac{\text{d}y}{\text{d}x}$ at this point.

The tangent at the point $(0,6)$ is parallel to the normal at the point $P$.

Find the exact coordinates of the point $P$.

The diagram shows the curve with parametric equations

$x=e^{2t}$, $y=e^{-3t}$.

(i) Verify that the curve passes through the point $(1,1)$.

(ii) Show that the line with equation $y=x$ is not the normal to the curve at the point $(1,1)$.

The parametric equations of a curve $C$ are

$x=9-t^2$, $y=5-t$.

The tangents to $C$ at the points $R$ and $S$ meet at the point $T$, as shown in the diagram.

Given that the $x$-coordinate of both $R$ and $S$ is 5, find the area of the triangle $RST$.

The parametric equations of a curve $C$ are

$x=3t$, $y=t+\frac{1}{t}$,

where $t>0$.

Find the equation of the normal to $C$ at the point where $C$ intersects the line $y=x$.

The diagram shows the curve with parametric equations

$x=4t$, $y=e^{t^2}$.

The tangents to the curve that pass through the origin meet the curve at the points $A$ and $B$.

Show that the values of $t$ at $A$ and $B$ are $t=-\frac{\sqrt{2}}{2}$ and $t=\frac{\sqrt{2}}{2}$.

Hence, or otherwise, show that the area of the triangle $OAB$ is $2\sqrt{2}{e}^{\frac{1}{2}}$ square units.