A2 October 2020 Q6
6. A physics student is studying the movement of particles in an electric field. In one experiment, the distances in micrometres of two moving particles, \(A\) and \(B\), from a fixed point \(O\) are modelled by
\[\begin{aligned} d_A &= \left|5t - 31\right| \\ d_B &= \left|3t^2 - 25t + 8\right| \end{aligned}\]respectively, where \(t\) is the time in seconds after motion begins.
It was recorded that the distance of particle \(B\) from \(O\) was less than the distance of particle \(A\) from \(O\) for approximately 4 seconds.
| Scheme | Marks | AO |
|---|---|---|
| Establishes need for \(\left|5t - 31\right| \gt \left|3t^2 - 25t + 8\right|\) and attempts to find all C.V.’s to form the critical region, e.g. via a sketch. | M1 | 3.1a |
| \(5t - 31 = 3t^2 - 25t + 8 \Rightarrow 3t^2 - 30t + 39 = 0 \Rightarrow t = \ldots\) \(t = 5 \pm 2\sqrt{3}\) | M1 A1 | 1.1b 3.4 |
| \(-(5t - 31) = 3t^2 - 25t + 8 \Rightarrow 3t^2 - 20t - 23 = 0 \Rightarrow t = \ldots\) \(t = (-1),\ \dfrac{23}{3}\) | M1 A1 | 2.1 3.4 |
| Selects “insides” \((-1 \lt)\,\alpha \lt t \lt \beta,\ \gamma \lt t \lt \delta\) where \(\alpha \lt \beta \lt \gamma \lt \delta\) | M1 | 2.2a |
| \((-1 \lt)\,0 \leqslant t \lt 5 - 2\sqrt{3}\) or \(\dfrac{23}{3} \lt t \lt 5 + 2\sqrt{3}\) | A1 | 1.1b |
| Both regions, \(0 \leqslant t \lt 5 - 2\sqrt{3}\) and \(\dfrac{23}{3} \lt t \lt 5 + 2\sqrt{3}\) | A1 | 2.3 |
| (8) |
Notes
M1: Sets problem up as an inequalities problem, and forms complete strategy to solve – must see attempt at all critical values and some attempt to form at least one range from them. May be scored if algebra not used. This mark is for showing an overall awareness of the problem.
M1: Attempts to find C.V.’s for the “positives”. Any valid method using algebra. Must see an attempt to find a 3TQ (oe), but allow answers from calculator once a 3TQ \(= 0\) is seen.
A1: Correct C.V.’s, both required.
M1: Attempts to find the other C.V.’s (same conditions as above)
A1: Correct C.V.’s. Need not see the negative value stated as \(t \gt 0\) is required. (If both given, they must be correct)
M1: Selects correct critical regions, shows the idea the “insides” are needed. \((-1 \lt)\,\alpha \lt t \lt \beta,\ \gamma \lt t \lt \delta\) where \(\alpha \lt \beta \lt \gamma \lt \delta\) are their four critical values, possibly truncated at 0 as long as no more than 1 is negative.
A1: One correct interval. Allow with loose or strict inequalities. Allow this mark if \(-1 \lt t \lt 5 - 2\sqrt{3}\) is given. Allow any variable for this mark.
A1: Fully correct solution. Must start at zero for the leftmost interval but accept \(\lt\) or \(\leqslant\) here. Must be using \(t\)
Alternative
| Scheme | Marks | AO |
|---|---|---|
| Establishes need for \(\left|5t - 31\right| \gt \left|3t^2 - 25t + 8\right|\) and attempts to find all C.V.’s to form the critical region, e.g. via a sketch. | M1 | 3.1a |
| \(\left(5t - 31\right)^2 = \left(3t^2 - 25t + 8\right)^2 \Rightarrow \ldots\) \(9t^4 - 150t^3 + 648t^2 - 90t - 897 = 0\) | M1 A1 | 1.1b 3.4 |
| Solves \(9t^4 - 150t^3 + 648t^2 - 90t - 897 = 0 \Rightarrow t = \ldots\) \(t = 5 \pm 2\sqrt{3},\ \dfrac{23}{3},\ \{-1\}\) | M1 A1 | 2.1 3.4 |
| Selects “insides” \((-1 \lt)\,\alpha \lt t \lt \beta,\ \gamma \lt t \lt \delta\) where \(\alpha \lt \beta \lt \gamma \lt \delta\) | M1 | 2.2a |
| \((-1 \lt)\,0 \leqslant t \lt 5 - 2\sqrt{3}\) or \(\dfrac{23}{3} \lt t \lt 5 + 2\sqrt{3}\) condone any variable | A1 | 1.1b |
| Both regions, \(0 \leqslant t \lt 5 - 2\sqrt{3}\) and \(\dfrac{23}{3} \lt t \lt 5 + 2\sqrt{3}\) must be using \(t\) | A1 | 2.3 |
| (8) |
M1: Sets problem up as an inequalities problem, and forms complete strategy to solve – must see attempt at all critical values and some attempt to form at least one ranges from them. May be scored if algebra not used. This mark is for showing an overall awareness of the problem.
M1: Attempts to find C.V.’s by squaring both sides and forming a quartic equation.
A1: Correct quartic equation
M1: Attempts to solve their quartic equation
A1: All 4 correct exact C.V.’s. Need not see the negative value stated as \(t \gt 0\) is required.
M1: Selects correct critical regions, shows the idea the “insides” are needed. \((-1 \lt)\,\alpha \lt t \lt \beta,\ \gamma \lt t \lt \delta\) where \(\alpha \lt \beta \lt \gamma \lt \delta\) are their four critical values, possibly truncated at 0 as long as no more than 1 is negative.
A1: One correct interval. Allow with loose or strict inequalities. Allow this mark if \(-1 \lt t \lt 5 - 2\sqrt{3}\) is given. Allow any variable for this mark.
A1: Fully correct solution. Must start at zero for the leftmost interval but accept \(\lt\) or \(\leqslant\) here. Must be using \(t\)
| Scheme | Marks | AO |
|---|---|---|
| Time that \(B\) is closer to \(O\) than particle \(A\) is \(5 + 2\sqrt{3} - \frac{23}{3} + 5 - 2\sqrt{3} = \frac{7}{3}\) seconds. | M1 | 3.4 |
| This is considerably less than 4 seconds so the model does not seem appropriate. | A1ft | 3.5a |
| (2) | ||
| (10 marks) |
Notes
M1: Uses their result from (a) to determine how long particle \(B\) is closer to \(O\) than particle \(A\) is.
A1: Draws a suitable conclusion for their answer to (a) – if correct in (a) it is that the model is not very suitable. Must not have a negative time.