A-Level · Mathematics · AQA · Mark scheme decoded
AQA A-Level Mathematics: Parametric Equations in Modelling — mark scheme explained
The short answer
Parametric equations are a powerful tool in coordinate geometry, allowing us to describe curves and paths that might be difficult or impossible to express using a single Cartesian equation.
The question
Convert the parametric equations x = t 3 and y = t 2 to Cartesian form. [Paraphrased for study — not reproduced from any exam paper.]
Mark scheme, decoded
What each mark is really for — in plain English — and the wording trap that loses it.
- S1
Solve the second equation for t: y = t 2 → t = ±√y
- S2
Substitute t = √y into the first equation: x = (√y) 3 = y 3/2
- S3
Square both sides to obtain a Cartesian relation valid for both roots: x 2 = y 3
Model answer
Worked through, with each step tagged to the mark it earns.
- S1
Solve the second equation for t: y = t 2 → t = ±√y
- S2
Substitute t = √y into the first equation: x = (√y) 3 = y 3/2
- S3
Square both sides to obtain a Cartesian relation valid for both roots: x 2 = y 3
Final answer: x 2 = y 3
Common mistakes
- Forgetting to eliminate the parameter t when converting parametric equations to Cartesian form. — Always solve one of the parametric equations for t and substitute it into the other equation to eliminate t.
- Incorrectly calculating the derivative dy/dx for parametric equations. — Double-check your calculations and ensure you are using the correct formula: dy/dx = (dy/dt) / (dx/dt).
- Using the wrong point when finding the equation of a tangent or normal. — Always verify the coordinates by substituting the given value of t into both x and y parametric equations.
- Forgetting to find the slope of the normal as the negative reciprocal of the tangent's slope. — Remember that the slope of the normal is always the negative reciprocal of the slope of the tangent: m normal = -1 / (dy/dx).
- Incorrectly applying trigonometric values in projectile motion problems. — Always double-check your trigonometric values, especially for common angles like 30°, 45°, and 60°.
- Failing to consider both positive and negative roots when eliminating the parameter t. — When solving for t, always consider both positive and negative roots if applicable.
- Using the wrong formula for the maximum height in projectile motion problems. — Always remember that the maximum height is found by setting the vertical velocity (dy/dt) to zero and solving for t, then substituting back into the y equation.
Where the marks go
- Full worked solution (all marking points)4 marks