Higher January 2022 Paper 1R Q23
23 Two particles, \(P\) and \(Q\), move along a straight line.
The fixed point \(O\) lies on this line.
The displacement of \(P\) from \(O\) at time \(t\) seconds is \(s\) metres, where
\[s = t^3 - 4t^2 + 5t \quad \text{for } t \gt 1\]The displacement of \(Q\) from \(O\) at time \(t\) seconds is \(x\) metres, where
\[x = t^2 - 4t + 4 \quad \text{for } t \gt 1\]Find the range of values of \(t\) where \(t \gt 1\) for which both particles are moving in the same direction along the straight line.
(6)
| Scheme | Marks |
|---|---|
| \(3t^2 - 2 \times 4t + 5\) or \(3t^2 - 8t + 5\) | M1 |
| \(3t^2 - 2 \times 4t + 5 = 0\) or \(3t^2 - 8t + 5 = 0\) | M1 |
| \((t =)\ \dfrac{5}{3}\) oe (and \(t = 1\)) | A1 |
| \(2t - 4 = 0\) | M1 |
| \((t =)\ 2\) | A1 |
| \((1 \lt )\ t \lt \dfrac{5}{3}\) and \(t \gt 2\) | A1 |
| (6) | |
| (6 marks) |
Notes
M1: for differentiation of \(s\) with 2 out of 3 terms correct (can be implied by subsequent working)
M1: (dep on previous M1) for equating at least a 2TQ to zero (allow inequality signs),
E.g. \(3t^2 - 8t = 0\) or \(3t^2 + 5 = 0\)
(can be implied by subsequent working)
M1: for differentiation of \(x\) to find \(at + b = 0\) (allow inequality signs) where \(a = 2\) and \(b = -4\)
A1: for a correct value of \(t\)
A1: oe \((t \gt 1)\ t \lt \dfrac{5}{3}\) and \(t \gt 2\)