| Exam Board | Edexcel |
|---|---|
| Module | FP1 (Further Pure Mathematics 1) |
| Year | 2020 |
| Session | June |
| Marks | 10 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Inequalities |
| Type | Solve absolute value inequality |
| Difficulty | Challenging +1.2 This is a multi-step absolute value inequality requiring case analysis and solving quadratic inequalities, which is moderately challenging. However, it follows a standard FP1 technique (removing absolute values by considering cases) with clear structure. Part (b) is straightforward interpretation. The context adds no mathematical difficulty. Harder than routine C1/C2 work but standard for Further Maths. |
| Spec | 1.02l Modulus function: notation, relations, equations and inequalities1.02t Solve modulus equations: graphically with modulus function1.02z Models in context: use functions in modelling |
| Answer | Marks | Guidance |
|---|---|---|
| Working/Answer | Marks | Guidance |
| Establishes need for \( | 5t-31 | > |
| \(5t-31 = 3t^2-25t+8 \Rightarrow 3t^2-30t+39=0 \Rightarrow t=...\) | M1 | Attempts to find C.V.'s for the "positives"; must see attempt to find a 3TQ |
| \(t = 5 \pm 2\sqrt{3}\) | A1 | Correct C.V.'s, both required |
| \(-(5t-31) = 3t^2-25t+8 \Rightarrow 3t^2-20t-23=0 \Rightarrow t=...\) | M1 | Attempts to find the other C.V.'s |
| \(t = (-1), \frac{23}{3}\) | A1 | Correct C.V.'s; need not see negative value stated as \(t>0\) required |
| Selects "insides" \((-1<)\alpha < t < \beta,\ \gamma < t < \delta\) where \(\alpha < \beta < \gamma < \delta\) | M1 | Selects correct critical regions, shows idea that "insides" are needed |
| \((-1<)0 \leq t < 5-2\sqrt{3}\) or \(\frac{23}{3} < t < 5+2\sqrt{3}\) | A1 | One correct interval; allow loose or strict inequalities; allow any variable |
| Both regions: \(0 \leq t < 5-2\sqrt{3}\) and \(\frac{23}{3} < t < 5+2\sqrt{3}\) | A1 | Fully correct solution; must start at zero for leftmost interval; must be using \(t\) |
| Answer | Marks | Guidance |
|---|---|---|
| Working/Answer | Marks | Guidance |
| \((5t-31)^2 = (3t^2-25t+8)^2 \Rightarrow ...\) | M1 | Attempts to find C.V.'s by squaring both sides |
| \(9t^4 - 150t^3 + 648t^2 - 90t - 897 = 0\) | A1 | Correct quartic equation |
| Solves \(9t^4-150t^3+648t^2-90t-897=0\) | M1 | Attempts to solve quartic equation |
| \(t = 5\pm2\sqrt{3},\ \frac{23}{3},\ \{-1\}\) | A1 | All 4 correct exact C.V.'s; need not see negative value |
| Selects "insides" \((-1<)\alpha < t < \beta,\ \gamma < t < \delta\) | M1 | Selects correct critical regions |
| \((-1<)0 \leq t < 5-2\sqrt{3}\) or \(\frac{23}{3} < t < 5+2\sqrt{3}\) | A1 | One correct interval |
| Both regions: \(0 \leq t < 5-2\sqrt{3}\) and \(\frac{23}{3} < t < 5+2\sqrt{3}\) | A1 | Must be using \(t\) |
| Answer | Marks | Guidance |
|---|---|---|
| Working/Answer | Marks | Guidance |
| Time \(B\) closer to \(O\) than \(A\): \(5+2\sqrt{3} - \frac{23}{3} + 5 - 2\sqrt{3} = \frac{7}{3}\) seconds | M1 | Uses result from (a) to determine how long particle \(B\) is closer to \(O\) than \(A\) |
| \(\frac{7}{3}\) is considerably less than 4 seconds so model does not seem appropriate | A1ft | Draws suitable conclusion; if correct in (a) must conclude model is not suitable; must not have a negative time |
# Question 6:
## Part (a):
| Working/Answer | Marks | Guidance |
|---|---|---|
| Establishes need for $|5t-31| > |3t^2 - 25t + 8|$ and attempts to find all C.V.'s to form critical region | **M1** | Sets problem up as inequalities problem, forms complete strategy; must see attempt at all critical values and some attempt to form at least one range |
| $5t-31 = 3t^2-25t+8 \Rightarrow 3t^2-30t+39=0 \Rightarrow t=...$ | **M1** | Attempts to find C.V.'s for the "positives"; must see attempt to find a 3TQ |
| $t = 5 \pm 2\sqrt{3}$ | **A1** | Correct C.V.'s, both required |
| $-(5t-31) = 3t^2-25t+8 \Rightarrow 3t^2-20t-23=0 \Rightarrow t=...$ | **M1** | Attempts to find the other C.V.'s |
| $t = (-1), \frac{23}{3}$ | **A1** | Correct C.V.'s; need not see negative value stated as $t>0$ required |
| Selects "insides" $(-1<)\alpha < t < \beta,\ \gamma < t < \delta$ where $\alpha < \beta < \gamma < \delta$ | **M1** | Selects correct critical regions, shows idea that "insides" are needed |
| $(-1<)0 \leq t < 5-2\sqrt{3}$ or $\frac{23}{3} < t < 5+2\sqrt{3}$ | **A1** | One correct interval; allow loose or strict inequalities; allow any variable |
| Both regions: $0 \leq t < 5-2\sqrt{3}$ and $\frac{23}{3} < t < 5+2\sqrt{3}$ | **A1** | Fully correct solution; must start at zero for leftmost interval; must be using $t$ |
### Alternative Method:
| Working/Answer | Marks | Guidance |
|---|---|---|
| $(5t-31)^2 = (3t^2-25t+8)^2 \Rightarrow ...$ | **M1** | Attempts to find C.V.'s by squaring both sides |
| $9t^4 - 150t^3 + 648t^2 - 90t - 897 = 0$ | **A1** | Correct quartic equation |
| Solves $9t^4-150t^3+648t^2-90t-897=0$ | **M1** | Attempts to solve quartic equation |
| $t = 5\pm2\sqrt{3},\ \frac{23}{3},\ \{-1\}$ | **A1** | All 4 correct exact C.V.'s; need not see negative value |
| Selects "insides" $(-1<)\alpha < t < \beta,\ \gamma < t < \delta$ | **M1** | Selects correct critical regions |
| $(-1<)0 \leq t < 5-2\sqrt{3}$ or $\frac{23}{3} < t < 5+2\sqrt{3}$ | **A1** | One correct interval |
| Both regions: $0 \leq t < 5-2\sqrt{3}$ and $\frac{23}{3} < t < 5+2\sqrt{3}$ | **A1** | Must be using $t$ |
**(8 marks)**
## Part (b):
| Working/Answer | Marks | Guidance |
|---|---|---|
| Time $B$ closer to $O$ than $A$: $5+2\sqrt{3} - \frac{23}{3} + 5 - 2\sqrt{3} = \frac{7}{3}$ seconds | **M1** | Uses result from (a) to determine how long particle $B$ is closer to $O$ than $A$ |
| $\frac{7}{3}$ is considerably less than 4 seconds so model does not seem appropriate | **A1ft** | Draws suitable conclusion; if correct in (a) must conclude model is not suitable; must not have a negative time |
**(2 marks)**
---\begin{enumerate}
\item 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
\end{enumerate}
$$\begin{aligned}
& d _ { A } = | 5 t - 31 | \\
& d _ { B } = \left| 3 t ^ { 2 } - 25 t + 8 \right|
\end{aligned}$$
respectively, where $t$ is the time in seconds after motion begins.\\
(a) Use algebra to find the range of time for which particle $A$ is further away from $O$ than particle $B$ is from $O$.
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.\\
(b) Use this information to assess the validity of the model.
\hfill \mbox{\textit{Edexcel FP1 2020 Q6 [10]}}