A-level · Coordinate geometry

Modelling with parametric equationsExam-style questions

Using parametric equations to model real situations where x and y each depend on time — projectiles and rotating points — reading positions, maxima and ranges from the model, and finding the Cartesian equation of the path.

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19 questions · A-level level · mark scheme and worked solution on each · no sign-up to try

DifficultyAll Level 3 Level 4 Level 5

1
Figure 1 shows the profile of part of a roller coaster track. The track runs from $P$ down to its lowest point and then climbs to $Q$ .

Figure 1This part of the track is modelled by the parametric equations

$x = 2t^2 + t \qquad y = 2t^2 - 16\ln t + 10$

for $0.5 \leq t \leq 3.5$ , where $x$ metres is the horizontal distance from a fixed point $O$ on level ground and $y$ metres is the height above the ground.

The ends of this part of the track, $P$ and $Q$ , are the points where $t = 0.5$ and $t = 3.5$ respectively.
(a) To allow for energy lost to friction, the design requires $Q$ to be at least $6$ metres lower than $P$ .

Show that the model satisfies this condition.

(3 marks)
(b) (i) Find an expression for $\dfrac{\mathrm{d}y}{\mathrm{d}x}$ in terms of $t$

(ii) The track is held up at its lowest point, $R$ , by a vertical steel column, $RS$ , which stands on the ground at $S$ , as shown in Figure 2.

Figure 2Find the length of $RS$

(iii) Calculate, to the nearest degree, the acute angle between the track and the horizontal at $Q$

(9 marks)
Level 5 · 12 marksTry it