Differentiation of Parametric Equations (Cambridge (CIE) AS Maths: Pure 2): Exam Questions

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

 $x=e^{2t}$    $y=3t^2+1$

The graph of  y  against  x  passes through the point P (1 , 1).

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

(ii) Find the gradient at the point P.

(iii) What does the value of the gradient tell you about point P?

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

$x=12t$   $y=9t^2-9t+4$    $0\le t\le 1$

as shown in the diagram below.

x and y are, respectively, the horizontal and vertical displacements in metres from the origin, O, and t is the time in seconds.  Point A indicates the initial position of the wrecking ball, at time $t=0$.

Find the height of the wrecking ball after 0.3 seconds.

Find the minimum height of the wrecking ball during its motion.

Find the horizontal distances from point A at the times when the wrecking ball is at a height of 2.9 m, giving your answers accurate to 1 decimal place.

The graph of the curve C shown below is defined by the parametric equations

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

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

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

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

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

(ii) Hence show 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

$x=6t^2+2$    $y=\frac{1}{t}$    $t>0$

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

(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 curve C has parametric equations

$x=t^2$    $y=2\sin t$    $0\le t\le 2\pi$

Show that, in terms of $t$, 

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

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

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

 $x=\sin 2t$    $y=e^t$

Verify that the graph of x against y passes through the point (0 , 1) and find the gradient at that point.

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

$x=8t-4$    $y=16t^2-16t+5$    $0\le t\le 1$

as shown in the diagram below.

x and y are, respectively, the horizontal and vertical displacements in metres from the origin, O, and t is the time in seconds.  Point A indicates the initial position of the wrecking ball, at time$t=0$.

Find a Cartesian equation of the curve in the form  $y=\mathrm{f}\left(x\right),$  and state the domain of $\mathrm{f}\left(x\right)$.

Find the difference between the maximum and minimum heights of the wrecking ball during its motion.

The crane is positioned such that point A is 7 m horizontally from the wall the wrecking ball is to destroy.

Find the height at which the wrecking ball will strike the wall.

The graph of the curve C shown below is defined by the parametric equations

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

Find an expression for  $\frac{\mathrm{d}y}{\mathrm{d}x}$  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 curve C has parametric equations

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

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

(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 curve$C$ has parametric equations

$x=t^2-4$    $y=3t$

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

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

Find the exact coordinates of point P

A 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 diagram below.

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

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

$x=10t$    $y=4.9t^2-4.9t+2$    $0\le t\le 1$

as shown in the diagram below.

x and y are, respectively, the horizontal and vertical displacements in metres from the origin, O, and t is the time in seconds.  Point A indicates the initial position of the wrecking ball.

(i) Write down the height of the wrecking ball when it is at point A.

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

The destruction of a building requires the wrecking ball to strike it at a height of 1.4 m whilst on the upward part of its path.

Find the horizontal distance from point A at which the ball hits the building.

The graph of the ellipse E shown below is defined by the parametric equations

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

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

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 curve C has parametric equations

$x=3t$    $y=t+\frac{1}{t}$     $t>0$

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

The graph of the curve defined by the parametric equations

$x=e^{2t}$    $y=e^{-3t}$

is shown below.

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

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

The diagram below shows a sketch of the curve defined by the parametric equations

$x=4t$     $y=e^{t^2}$

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

Show that the values of t at points 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.