Further Parametric Equations (Edexcel International A Level (IAL) Maths: Pure 4): Exam Questions

Find the Cartesian equation of the curve C, defined by the parametric equations

x = t-1 and y = 2 ln t

(i) Find in terms of x.

(ii) Find the gradient of C at the point where t = 1.

Hence find the equation of the tangent to C at the point where t = 1.

A sketch of the graph defined by the parametric equations

x = 8t and y=t²+1

is shown below.

The point where t = t₁ has x-coordinate 8.

The point where t = t₂ has x-coordinate 16.

Find the values of t₁ and t₂.

(i) Show that the shaded area can be found using the integral

(ii) Hence find the shaded area.

A particle travels along a path defined by the parametric equations

x = 6t and y=8t²-8t + 3, 0 ≤ t ≤ 1,

where (x ,y) are the coordinates of the particle at time t seconds.

Find the coordinates of the particle after 0.2 seconds.

(i) Find and .

(ii) Hence find in terms of t.

Find the coordinates of the particle when it is at its minimum point.

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

x = 5 sin and y =

Find the exact coordinates of point A.

(i) Write down the value of at the origin.

(ii) Write down the value of at the points where x = -5 and x = 5.

(i) Find and

(ii) Hence find in terms of .

(iii) Find the gradient at the point where

The curve C has parametric equations

x = 5t²-1 and y = 3t, t>0.

(i) Find and

(ii) Hence find in terms of t.

(i) Find the gradient of the tangent to C at the point (4,3).

(ii) Hence find the equation of the tangent to C at the point (4,3).

The curve C has parametric equations

x = 2t³ and y = 4t -1, t>0.

(i) Find and

(ii) Hence find in terms of t.

(i) Find the gradient of the tangent to C at the point (16,7).

(ii) Hence find the gradient of the normal to C at the point (16,7).

(iii) Find the equation of the normal to C at the point (16,7).

A company logo is in the shape of a semi-ellipse as shown in the diagram below.

The graph of the logo is defined by the parametric equations

x = 3 + 2 cost and y = -3sin t,

where x and y are measured in centimetres.

Verify that the values of t, labelled t₁ and t₂ on the diagram above where y = 0, are t₁ = and t₂ = 2

(i) Find .

(ii) Show that the shaded area is given by

(iii) Hence using your calculator or otherwise, find the area of the logo.

The diagram below shows part of the curve C with parametric equations

x =t²+1 and y = 2t -4 t≥0

The point on the graph where t = t₁ has x-coordinate 5.

The point on the graph where t = t₂ has x-coordinate 10.

Show that t₁ = 2 and t₂ = 3, and find the coordinates of the corresponding points on C.

The region R shown in the diagram is bounded by C, the x-axis, and the line x = 10.
Region R is rotated through 360° about the x-axis to form a solid of revolution.

(i) Show that the volume of the solid of revolution is given by the integral

(ii) Hence find the exact volume of the solid generated.