Edexcel IAL P4 Parametric Equations Questions
Parametric equations of curves; converting between parametric and Cartesian forms; parametric differentiation; tangents and normals.
- P4 · Pure Mathematics 4
- Calculator allowed
- 15 practice questions
What the questions test
The 15 P4 parametric equations practice questions cover:
Worked example 1
A curve $C$ has parametric equations $x = t^2$, $y = t^3 - 3t$.
(a) Find $\dfrac{dy}{dx}$ in terms of $t$. (3)
(b) Find the coordinates of the points on $C$ where the tangent is horizontal. (3)
(c) Find the Cartesian equation of the tangent at $t = 2$. (2)
Mark scheme
(a) $\tfrac{dx}{dt} = 2t$, $\tfrac{dy}{dt} = 3t^2 - 3$ M1 A1. $\tfrac{dy}{dx} = \tfrac{3t^2 - 3}{2t} = \tfrac{3(t^2 - 1)}{2t}$ A1.
(b) Horizontal iff $\tfrac{dy}{dt} = 0$: $3t^2 = 3 \Rightarrow t = \pm 1$ M1. $t = 1$: $(1, -2)$; $t = -1$: $(1, 2)$ A1 A1.
(c) At $t = 2$: $(x, y) = (4, 2)$; gradient $= \tfrac{3 \cdot 3}{4} = \tfrac{9}{4}$; equation: $y - 2 = \tfrac{9}{4}(x - 4)$, i.e. $y = \tfrac{9}{4}x - 7$ M1 A1.
Worked example 2
A curve has parametric equations $x = t + 1$, $y = t^3 - 12t$.
(a) Find the values of $t$ at the stationary points on the curve. (3)
(b) Find the coordinates of the stationary points. (2)
(c) Determine the nature of each stationary point. (2)
Mark scheme
(a) $\tfrac{dy}{dx} = \tfrac{3t^2 - 12}{1} = 3t^2 - 12$ M1 A1. Set $= 0$: $t^2 = 4 \Rightarrow t = \pm 2$ A1.
(b) $t = 2$: $(3, -16)$; $t = -2$: $(-1, 16)$ A1 A1.
(c) $\tfrac{d^2y}{dx^2} = 6t \cdot \tfrac{dt}{dx} = 6t$ (since $\tfrac{dt}{dx} = 1$) M1. $t = 2$: $12 > 0$ ⇒ minimum; $t = -2$: $-12 < 0$ ⇒ maximum A1.
FAQ
What parametric equations is in Edexcel IAL P4?
Parametric equations of curves; converting between parametric and Cartesian forms; parametric differentiation; tangents and normals.
Is P4 a calculator paper?
Yes, a calculator is allowed in P4, but method marks are only given for working you write down.
How are P4 parametric equations questions marked?
With M marks for a correct method, A marks for accurate answers that depend on the M mark before them, and B marks for independent results, as in the worked examples on this page.