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Parametric Equations Practice Problems with Solutions - A Level Year 13

A Level Year 13 | Math | Parametric Equations
Name: _________________________ Date: _________________________ Score: _______ / 40

The curve x = t, y = t² has Cartesian equation y = x², and x = 2t, y = t² gives y = x² ÷ 4, so this Year 13 sheet opens by eliminating the parameter. The key adds x² + y² = 1 as a circle and 1 ÷ t as a gradient.

Key Takeaways

Section A: Multiple Choice Questions (1 Mark Each)

Choose the correct option for each question.
1. The curve x = t, y = t² has Cartesian equation:
(a) y = x²(b) y = 2x(c) y = x(d) x = y²
2. The curve x = 2t, y = t² has Cartesian equation:
(a) y = x² ÷ 4(b) y = 4x²(c) y = x²(d) y = 2x²
3. For the curve x = t², y = 2t, the value of dy/dx is:
(a) 1 ÷ t(b) 2t(c) t(d) 1 ÷ (2t)
4. The curve x = cos θ, y = sin θ has Cartesian equation:
(a) x² + y² = 1(b) x + y = 1(c) x² + y² = 2(d) y = x
5. For the curve x = t², y = 2t, the gradient at t = 1 is:
(a) 1(b) 2(c) ½(d) 3
6. The curve x = 3 cos θ, y = 3 sin θ has Cartesian equation:
(a) x² + y² = 9(b) x² + y² = 3(c) x + y = 9(d) x² + y² = 1

Section B: Short Answer Type Questions (2 Marks Each)

Show all steps clearly.
7. The curve is given by x = t² + 1, y = t. Find its Cartesian equation.
8. The curve is given by x = 2t, y = t² − 1. Find its Cartesian equation.
9. Find the value of dy/dx at t = 2 for the curve x = t², y = t³.
10. The curve is given by x = 3t, y = 9t². Find its Cartesian equation.
11. The curve is given by x = 2 + 3t, y = 1 − t. Find its Cartesian equation.
12. For the curve x = 2t, y = t², find dy/dx in terms of t.
13. Find the gradient of the tangent to the curve x = 2t, y = 4t − t² at t = 1.
14. The curve x = 4 cos θ, y = 4 sin θ is a circle. State its radius.

Section C: Numericals & Word Problems (3 Marks Each)

Apply the concepts to solve the problems. Show all working.
15. A projectile follows the path x = 20t, y = 20t − 5t². Eliminate t to find the Cartesian equation of its path.
16. A particle moves so that x = 2t and y = 4t². Find the value of dy/dx when t = 3.
17. Find the coordinates of the point where the curve x = 2t, y = t² crosses the y-axis.
18. The curve is given by x = t², y = 2t. Find the equation of the tangent at the point where t = 2.
19. Convert the parametric equations x = 2t + 1, y = 6t − 1 into a single Cartesian equation.
20. A circle is defined by x = 3 cos θ, y = 3 sin θ. State its centre and radius.

Answer Key

1. a) y = x²
2. a) y = x² ÷ 4
3. a) 1 ÷ t
4. a) x² + y² = 1
5. a) 1
6. a) x² + y² = 9
7. Refer to solution guide.
8. Refer to solution guide.
9. Refer to solution guide.
10. Refer to solution guide.
11. Refer to solution guide.
12. Refer to solution guide.
13. Refer to solution guide.
14. Refer to solution guide.
15. Refer to solution guide.
16. Refer to solution guide.
17. Refer to solution guide.
18. Refer to solution guide.
19. Refer to solution guide.
20. Refer to solution guide.
End of Worksheet

People Also Ask

What is the Cartesian equation of x = t, y = t²?

y = x², option (a) in this worksheet (Q1).

What is the Cartesian equation of x = 2t, y = t²?

y = x² ÷ 4, option (a) in this worksheet (Q2).

What is the Cartesian equation of a unit circle?

x² + y² = 1, option (a) in this worksheet (Q4).

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