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AS A-level Pure Mathematics · Edexcel 8MA0 · free A-level maths worksheet

Parametric equations

Converting between parametric and Cartesian forms, sketching, points of intersection, parametric differentiation, modelling with parameters.

12 original questions, 70 marks. Type your answers below for an instant mark and a score (algebra in any equivalent form, e.g. 2sqrt(3) or 2√3), or print the worksheet, mark it with the answers sheet, and use the solutions to see every step written out by hand.

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The questions

1

A curve C has parametric equations

x = 2t − 1,   y = t2 + 3,   t ∈ ℝ

  1. Find the coordinates of the point on C where t = 3
  2. Find a Cartesian equation for C in the form y = f(x)
3 marks
Show answer
a) (5, 12) b) y = (x + 1)24 + 3
2

A curve C has parametric equations

x = 1 + 4 cos θ,   y = 4 sin θ − 2,   0 ≤ θ < 2π

Show that C is a circle and state its centre and radius.

3 marks
Show answer
(x - 1)² + (y + 2)² = 16: circle, centre (1, -2), radius 4
3

A curve C has parametric equations

x = t3 − 2t,   y = t2 + 4t

  1. Find dydx in terms of t
  2. Find an equation of the tangent to C at the point where t = 2, giving your answer in the form ax + by + c = 0, where a, b and c are integers.
5 marks
Show answer
a) dydx = 2t + 43t2 - 2 b) 4x - 5y + 44 = 0
4

A curve C has parametric equations

x = 1 + 2 sin t,   y = 3 − cos 2t,   −π2 ≤ t ≤ π2

  1. Show that a Cartesian equation of C is y = 2 + 12(x − 1)2
  2. State the domain of values of x and the range of values of y for C.
  3. Explain why C does not include every point of the curve with equation y = 2 + 12(x − 1)2, x ∈ ℝ
5 marks
Show answer
a) shown b) -1 ≤ x ≤ 3, 2 ≤ y ≤ 4 c) C only has -1 ≤ x ≤ 3, so it is just part of the parabola
5

A curve C has parametric equations

x = 3t − 1,   y = t2t − 1,   t > 1

Show that a Cartesian equation of C is

y = (x + a)2bx,   x > 0

where a and b are integers to be found.

3 marks
Show answer
y = (x + 3)²/(3x), so a = 3, b = 3
6

A curve C has parametric equations

x = t2 − 4t,   y = t3 − 3t,   t ∈ ℝ

  1. Find the coordinates of the points on C where the tangent is parallel to the x-axis.
  2. Find the coordinates of the point on C where the tangent is parallel to the y-axis.
6 marks
Show answer
a) (-3, -2) and (5, 2) b) (-4, 2)
7

A curve C has parametric equations

x = t2 + 1,   y = 2t − 1,   t ∈ ℝ

The point P on C has parameter t = 2

  1. Show that the normal to C at P has equation y = 13 − 2x
  2. The normal meets C again at the point Q. Find the coordinates of Q.
7 marks
Show answer
a) shown b) Q(10, -7)
8

In this question you must show all stages of your working.

A curve C has parametric equations

x = 2 cos t,   y = sin 2t,   0 ≤ t < 2π

Find the exact coordinates of all the points where C meets the line with equation y = √32x

5 marks
Show answer
(0, 0), (1, √3/2), (-1, -√3/2)
9

A curve C has parametric equations

x = 2 cos t,   y = 1 + cos 2t,   0 ≤ t ≤ π

  1. Show that all points on C satisfy y = 12x2 and state the domain of values of x for C.
  2. The line with equation y = x + k, where k is a constant, meets C at two distinct points. Find the range of possible values of k, writing your answer in set notation.
7 marks
Show answer
a) y = 12x2, -2 ≤ x ≤ 2 b) {k : -12 < k ≤ 0}
10

A curve C has parametric equations

x = cos 2θ,   y = sec θ,   0 < θ < π2

  1. Show that dydx = k sec3 θ, where k is a constant to be found.
  2. Find an equation of the tangent to C at the point where y = 2, giving your answer in the form y = mx + c
6 marks
Show answer
a) k = -14 b) y = 1 - 2x
11

A reflector is fixed to the rim of a bicycle wheel of radius 0.3 m. As the bicycle moves along level ground, the position of the reflector is modelled by

x = 0.3(θ − sin θ),   y = 0.3(1 − cos θ),   0 < θ < 2π

where x metres is the horizontal distance moved by the reflector, y metres is its height above the ground and θ is the angle in radians that the wheel has turned through.

  1. Show that dydx = cot(θ2)
  2. Find the exact coordinates of the reflector at the moment its path has gradient 1
  3. The reflector is 0.45 m above the ground at two points in this interval. Find the gradient of the path at each of these points, giving exact answers.
  4. State one assumption made in this model.
9 marks
Show answer
a) shown b) (0.3(π2 - 1), 0.3) c) 1√3 and -1√3 d) for example: the wheel rolls without slipping on level ground
12

A curve C has parametric equations

x = t2 − 4,   y = t3 − 4t,   t ∈ ℝ

  1. Find the values of t at which C meets the x-axis, and hence show that C passes through the origin O twice.
  2. Find the gradients of the two tangents to C at O.
  3. Show that a Cartesian equation of C is y2 = x2(x + 4)
  4. Find the exact coordinates of the points on C where the tangent is parallel to the x-axis.
  5. Sketch C for −2.5 ≤ t ≤ 2.5 on the axes below.
xy
11 marks
Show answer
a) t = 0, 2, -2; t = ±2 both give (0, 0) b) 2 and -2 c) shown d) (-83, 16√39) and (-83, -16√39) e) a loop from (-4, 0) to O, with the two branches crossing at O and continuing into x > 0

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Every question here is original and written by Robinson Tuition. It follows the Edexcel AS specification and suits AQA and OCR AS maths too; this is not an exam-board paper.

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