C: Coordinate geometry in the (x, y) plane: Use parametric equations in modelling in a variety of contexts.

The idea

Parametric equations describe a curve by expressing x and y separately in terms of a third variable, often time t. In modelling, this is powerful because many real situations naturally involve a parameter: time, angle, or another changing quantity. For example, the path of a projectile can be modelled by x = u cosθ · t and y = u sinθ · t − ½gt², where t is time. To use parametric equations in modelling, you first identify the parameter and write equations for x and y. You may need to convert to Cartesian form by eliminating the parameter, often using substitution or trigonometric identities. You also interpret features such as initial position, maximum height, or range by substituting specific parameter values. The skill is assessed through problems set in context, where you must set up, manipulate, and interpret parametric models, sometimes linking to calculus for gradients or areas.

What earns the marks

• Correctly identify the parameter and write parametric equations for x and y that match the given context, ensuring variables and constants are clearly defined.
• Eliminate the parameter to obtain the Cartesian equation of the curve, using algebraic substitution or trigonometric identities such as sin²θ + cos²θ = 1.
• Interpret the parametric model by finding coordinates, gradients, or areas at specific parameter values, and relate these to the real-world context.
• Use differentiation with respect to the parameter to find dy/dx, applying the chain rule dy/dx = (dy/dt)/(dx/dt), and use this to find tangents or normals.
• Solve problems involving intersections, ranges, or maximum/minimum values by substituting or optimising with respect to the parameter.

Common mistakes and how to put them right

• Forgetting to eliminate the parameter when a Cartesian equation is required.

  Put it right: Check the question wording and practise elimination techniques.

• Misapplying trigonometric identities when converting parametric equations involving sine and cosine.

  Put it right: Revise Pythagorean identities and double-angle formulae.

• Ignoring the domain of the parameter, leading to incomplete curves or incorrect interpretations.

  Put it right: State the range of the parameter and consider its effect on x and y.

Exam tips

• Read the context carefully to decide what the parameter represents and what its sensible range is; this often guides the solution.
• When eliminating the parameter, show clear algebraic steps; even if you make a slip, method marks may be available.
• Use a sketch to visualise the curve and check that your Cartesian equation and any calculated points are consistent with the model.