| Exam Board | Edexcel |
|---|---|
| Module | FP1 (Further Pure Mathematics 1) |
| Year | 2022 |
| Session | June |
| Marks | 8 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Inequalities |
| Type | Solve absolute value inequality |
| Difficulty | Challenging +1.2 This is a multi-part Further Maths question involving absolute value functions and geometric reasoning. Part (a) requires understanding that for exactly 3 roots, the line must be tangent to one branch (discriminant = 0), which is a standard technique. Parts (b) and (c) follow straightforwardly from substitution and interpreting the graph. While it requires careful case analysis of the absolute value, the techniques are well-practiced in FP1 and the question provides significant scaffolding through its parts. |
| Spec | 1.02g Inequalities: linear and quadratic in single variable1.02i Represent inequalities: graphically on coordinate plane1.02l Modulus function: notation, relations, equations and inequalities1.02p Interpret algebraic solutions: graphically |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| Considers \(x^2-8=-(mx+c) \Rightarrow x^2+mx-8+c=0\) and sets discriminant \(=0\): \(m^2-4(-8+c)=0\) | M1 | Must see correct equation without modulus initially |
| \(m^2-4c+32=0\) | A1* | Correct result with no incorrect working seen |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| \(c=3m \Rightarrow m^2-4(3m)+32=0 \Rightarrow m=\ldots\) \((4,8)\) or \(\left(\frac{c}{3}\right)^2-4c+32=0 \Rightarrow c=\ldots\) \((12,24)\) | M1 | Substitutes \(c=3m\) into equation from (a) and solves |
| \(m="4" \Rightarrow c=\ldots\) or \(c="12" \Rightarrow m=\ldots\) | M1 | Finds corresponding value of \(c\) (or \(m\)) |
| Deduces \(m=4\) and \(c=12\) and no other values | A1 | If two sets stated, mark not achieved until \(m=8\), \(c=24\) rejected |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| Solves \(x^2-8=4x+12\) and \(x^2-8=-(4x+12)\) | M1 | Correct method to find all critical values |
| \(x=2\pm\sqrt{24}\) and \(x=-2\) | A1ft | Follow through on \(m=8\), \(c=24\) only; three critical values |
| \(x\leqslant 2-2\sqrt{6},\ x\geqslant 2+2\sqrt{6},\ x=-2\) | A1 | Accept "and"/"or" but not \(\wedge\) |
## Question 7:
### Part (a):
| Answer/Working | Mark | Guidance |
|---|---|---|
| Considers $x^2-8=-(mx+c) \Rightarrow x^2+mx-8+c=0$ and sets discriminant $=0$: $m^2-4(-8+c)=0$ | M1 | Must see correct equation without modulus initially |
| $m^2-4c+32=0$ | A1* | Correct result with no incorrect working seen |
### Part (b):
| Answer/Working | Mark | Guidance |
|---|---|---|
| $c=3m \Rightarrow m^2-4(3m)+32=0 \Rightarrow m=\ldots$ $(4,8)$ or $\left(\frac{c}{3}\right)^2-4c+32=0 \Rightarrow c=\ldots$ $(12,24)$ | M1 | Substitutes $c=3m$ into equation from (a) and solves |
| $m="4" \Rightarrow c=\ldots$ or $c="12" \Rightarrow m=\ldots$ | M1 | Finds corresponding value of $c$ (or $m$) |
| Deduces $m=4$ and $c=12$ and no other values | A1 | If two sets stated, mark not achieved until $m=8$, $c=24$ rejected |
### Part (c):
| Answer/Working | Mark | Guidance |
|---|---|---|
| Solves $x^2-8=4x+12$ and $x^2-8=-(4x+12)$ | M1 | Correct method to find all critical values |
| $x=2\pm\sqrt{24}$ and $x=-2$ | A1ft | Follow through on $m=8$, $c=24$ only; three critical values |
| $x\leqslant 2-2\sqrt{6},\ x\geqslant 2+2\sqrt{6},\ x=-2$ | A1 | Accept "and"/"or" but not $\wedge$ |
---7.
\begin{figure}[h]
\begin{center}
\includegraphics[alt={},max width=\textwidth]{7de3f581-eff1-4671-87a9-55ca1bb97890-20_591_962_312_548}
\captionsetup{labelformat=empty}
\caption{Figure 1}
\end{center}
\end{figure}
Figure 1 shows a sketch of the curve with equation $y = \left| x ^ { 2 } - 8 \right|$ and a sketch of the straight line with equation $y = m x + c$, where $m$ and $c$ are positive constants.
The equation
$$\left| x ^ { 2 } - 8 \right| = m x + c$$
has exactly 3 roots, as shown in Figure 1.
\begin{enumerate}[label=(\alph*)]
\item Show that
$$m ^ { 2 } - 4 c + 32 = 0$$
Given that $c = 3 m$
\item determine the value of $m$ and the value of $c$
\item Hence solve
$$\left| x ^ { 2 } - 8 \right| \geqslant m x + c$$
\end{enumerate}
\hfill \mbox{\textit{Edexcel FP1 2022 Q7 [8]}}