| Exam Board | Pre-U |
|---|---|
| Module | Pre-U 9794/3 (Pre-U Mathematics Paper 3) |
| Year | 2016 |
| Session | Specimen |
| Marks | 6 |
| Topic | Variable acceleration (1D) |
| Type | Finding when particle at rest |
| Difficulty | Moderate -0.3 This is a straightforward mechanics question requiring factorization of a cubic to find when v=0, and integration of velocity to find displacement. Both are standard A-level techniques with no conceptual challenges—slightly easier than average due to the clean factorization and routine calculus. |
| Spec | 1.08d Evaluate definite integrals: between limits3.02a Kinematics language: position, displacement, velocity, acceleration3.02f Non-uniform acceleration: using differentiation and integration |
**Question 7**
(i) $v = t(t-4)(t-5)$ [M1]
$t = 4$ and $5$ [A1]
(ii) $x = \frac{t^4}{4} - 3t^3 + 10t^2 + c$ [M1]
All terms correct including "$+ c$" [A1]
When $x = 0$, $t = 0$ therefore $c = 0$ [A1]
When $t = 2$, $x = 4 - 24 + 40 = 20$ [A1]
**Total: 6 marks**7 A particle travels along a straight line. Its velocity $v \mathrm {~ms} ^ { - 1 }$ after $t$ seconds is given by
$$v = t ^ { 3 } - 9 t ^ { 2 } + 20 t$$
When $t = 0$, the particle is at rest at $P$.\\
(i) Find the times, other than $t = 0$, at which the particle is at rest.\\
(ii) Find the displacement of the particle from $P$ when $t = 2$.
\hfill \mbox{\textit{Pre-U Pre-U 9794/3 2016 Q7 [6]}}