Calculus & Modelling with Parametric Equations (Edexcel A Level Maths: Pure): Exam Questions

The curve $C$ with parametric equations

$x=5\sin \theta y={\theta }^2-\pi \le \theta \le \pi$

is shown in the figure below.

Find the exact coordinates of the point $A$.

(i) Write down the value of $\frac{dy}{d\theta }$ at the origin.

(ii) Write down the value(s) of  $\frac{dx}{d\theta }$  at the points where $x=-5$and$x=5$.

Find the exact gradient of the point on the curve where  $\theta =\frac{\pi }{3}$

The curve $C$ has parametric equations

$\begin{array}{ccccc}x=\sin 2\theta & & y={\mathrm{cosec}}^3\theta & & 0<\theta <\frac{\pi }{2}\end{array}$

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

Hence find the exact value of the gradient of the tangent to $C$ at the point where $y=8$

Figure 6 shows a sketch of the curve $C$ with parametric equations

$x=2\tan t+1y=2{\sec }^{2}t+3-\frac{\pi }{4}\le t\le \frac{\pi }{3}$

The line $l$ is the normal to $C$ at the point $P$ where $t=\frac{\pi }{4}$

Using parametric differentiation, show that an equation for $l$ is

$y=-\frac{1}{2}x+\frac{17}{2}$

A particle travels along a curve with parametric equations

$x=6ty=8t^2-8t+30\le t\le 1$

where the coordinates $\left(x,y\right)$ give the position of the particle after time $t$ seconds.

Find the coordinates of the position of the particle after 0.2 seconds.

Find an expression for $\frac{dy}{dx}$ in terms of $t$.

Find the coordinates of the position of the particle when it is at the minimum point on the curve.

Find an expression for $\frac{dy}{dx}$ in terms of $t$ for the curve with parametric equations

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

The graph of $y$ against $x$ passes through the point $P$ with coordinates $\left(1,1\right)$.

Show that $P$ is a stationary point.

The graph shows the curve with parametric equations

$x=t^{3}y=2t^2-1$

• The point where $t=t_1$has coordinates $\left(1,1\right)$

• The point where $t=t_2$ has coordinates $\left(8,7\right)$

Find the values of $t_1$ and $t_2$.

Hence find the exact area of the shaded region.

The graph of the curve $C$ with parametric equations

 $x=3\sin 3\theta y=6\cos 2\theta -\frac{\pi }{2}\le \theta \le \frac{\pi }{2}$

is shown in the figure below.

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

(ii) Write down the value(s) of $\frac{\mathrm{d}x}{\mathrm{d}\theta }$ at the points $\left(-3,3\right)$ and $\left(3,3\right)$

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

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 curve $C$ has parametric equations

 $x=6t^2+2y=\frac{1}{t}t>0$

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

Hence find the equation of the normal to $C$ at the point with coordinate $\left(8,1\right)$.

Give your answer in the form $y=mx+c$.

A company logo, in the shape of the symbol for infinity ($\infty$), is printed on a flag, as shown below.

The curve has parametric equations

 $x=3\cos ty=\sin 2t-\pi \le t\le \pi$

where $x$ and $y$ are measured in metres.

(i) Find the values of $x$at the points where $t=-\pi$ and $t=-\frac{\pi }{2}$

(ii) Find the coordinates of the point on the curve where $t=-\frac{3\pi }{4}$

(i) Show that the total area of the logo is given by

$4{\int }_{-\pi }^{-\frac{\pi }{2}}\left(-6\cos t{\sin }^{2}t\right)\mathrm{d}t$

(ii) Hence find the total area of the logo.

A curve is defined by the parametric equations

$x=2\sin t,y=3\cos 2t,0\le t\le \pi$

Show that $\frac{\mathrm{d}y}{\mathrm{d}x}=-6\sin t$, showing all steps of your working.

Find the equation of the tangent to the curve at the point where $t=\frac{\pi }{6}$.

The curve $C$ has parametric equations

$x=t^2+6t-16y=6\ln \left(t+3\right)t>-3$

The curve $C$ cuts the $y$-axis at the point $P$.

Show that the equation of the tangent to $C$ at $P$ can be written in the form

$ax+by=c\ln 5$

where $a$, $b$ and $c$ are integers to be found.

The curve shown in Figure 3 has parametric equations

$x=6\sin ty=5\sin 2t0\le t\le \frac{\pi }{2}$

The region $R$, shown shaded in Figure 3, is bounded by the curve and the $x$-axis.

Show that the area of $R$ is given by ${\int }_0^{\frac{\pi }{2}}60\sin t{\cos }^{2}t\mathrm{d}t$

Hence show, by algebraic integration, that the area of $R$ is exactly 20

The curve $C$ has parametric equations

 $x=t^{2}y=2\sin t0\le t<2\pi$

Show that the distance between the maximum point and the minimum point on $C$ is 

$2\sqrt{{\pi }^4+4}$

The graph of the curve $C$ with parametric equations

 $x=2\cos 3\theta y=5\sin \theta 0\le \theta <2\pi$

is shown in the figure below.

Find the equation of the tangent to $C$ at the point where  $\theta =\frac{\pi }{4}$ .

Give your answer in the form $y=mx+c$.

The curve $C$ has parametric equations

$x=\frac{1}{t^2}y=t+\frac{1}{t}t>0$

Find the equation of the normal to $C$ at the point where $t=\frac{1}{2}$.

Given your answer in the form $y=mx+c$.

The ellipse $E$, shown in the figure below, has parametric equations

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

Find the equation of the tangent to $E$ at the point where $\theta =-\frac{\pi }{6}$.

Given your answer in the form $y=a-bx$, where $a$ and $b$ are exact real numbers to be found.

The graph of the curve with parametric equations

$x={\mathrm{e}}^{2t}y={\mathrm{e}}^{-3t}$

is shown in the figure below.

(i) Show that the graph passes through the point with coordinates $\left(1,1\right)$.

(ii) Prove that the straight line with equation $y=x$ is not the normal to the curve at the point $\left(1,1\right)$.

A crane swings a wrecking ball along a two-dimensional path modelled by the parametric equations

 $x=10ty=4.9t^2-4.9t+20\le t\le 1$

as shown in the figure below.

The variables $x$ and $y$ are the horizontal and vertical displacements, in metres, from the origin, $O$, and $t$ is the time in seconds.

The wrecking ball is initially released from the point $A$.

(i) Find the vertical height of the wrecking ball when it is at the point $A$.

(ii) Find the shortest distance between the wrecking ball and the horizontal ground during its motion.

The crane is positioned such that the wrecking ball hits a building at a vertical height of 1.4 metres above the ground, on the upwards part of the swing.

Find the horizontal distance from $A$ to the building.

Figure 6 shows a sketch of the curve $C$ with parametric equations

$x=8{\sin }^{2}ty=2\sin 2t+3\sin t0\le t\le \frac{\pi }{2}$

The region $R$, shown shaded in Figure 6, is bounded by $C$, the $x$-axis and the line with equation $x=4$

Show that the area of $R$ is given by

${\int }_0^a\left(8-8\cos 4t+48{\sin }^{2}t\cos t\right)dt$

where $a$ is a constant to be found.

Hence, using algebraic integration, find the exact area of $R$.

A curve is defined parametrically by

$x=3t,y=t^2{e}^{-2t},t\ge 0$

Find the exact coordinates of the stationary point of the curve where $t>0$.

The finite region $R$ is bounded by the curve, the $x$-axis and the lines $x=0$ and $x=3$.

Use algebraic integration to find the exact area of $R$.

The curve $C$ has parametric equations

$x=9-t^{2}y=5-t$

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

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

The curve $C$ has parametric equations

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

The tangent at the point $\left(0,6\right)$ on $C$ is parallel to the normal at the point $P$ on $C$.

Find the exact coordinates of the point $P$.

The curve $C$ has parametric equations

 $x=3ty=t+\frac{1}{t}t>0$

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

Give your answer in the form $y=mx+c$.

The graph of the curve $C$ with parametric equations

 $x=4ty={\mathrm{e}}^{t^2}$

is shown in the figure below.

The two tangents to $C$ that pass through the origin, $O$, touch $C$ at the points $A$ and $B$ (not shown on the diagram).

Find the values of $t$ at $A$ and $B$.

Hence show that the area of triangle $OAB$ is 

$2\sqrt{2}{\mathrm{e}}^{\frac{1}{2}}$

A model car travels around a track that follows the curve with parametric equations

$x=\cos ty=\sin 3t0\le t\le 20\pi$

where $x$ and $y$ are the horizontal and vertical displacements, in metres, from the origin $O$, at time t seconds.

 (i) Write down the coordinates of the starting position of the model car.

(ii) Indicate on the graph the direction in which the model car travels.

(iii) How many laps of the track does the model car complete?

A second track is to be constructed within the central area of the original track, indicated by the shaded region.

The design for the second track requires a minimum area of 1.25 m².

Use algebraic integration to determine whether there is sufficient room for the second track to be built within the central area of the original track.

In your calculations, you may use without proof the result that

$\int \sin t\sin 3t\mathrm{d}t=\cos t{\sin }^{3}t+c$

where $c$ is a constant.