| Exam Board | Edexcel |
|---|---|
| Module | FP2 (Further Pure Mathematics 2) |
| Year | 2009 |
| Session | June |
| Marks | 12 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Modulus function |
| Type | Sketch modulus functions involving quadratic or other non-linear |
| Difficulty | Challenging +1.2 This is a Further Maths question requiring systematic case analysis of modulus equations and inequalities with a parameter. Part (a) is routine sketching, but parts (b) and (c) require splitting into cases based on the sign of x²-a², solving resulting quadratics, and checking validity against constraints—more demanding than standard C3 modulus work but follows established FP2 techniques without requiring novel insight. |
| Spec | 1.02g Inequalities: linear and quadratic in single variable1.02l Modulus function: notation, relations, equations and inequalities1.02s Modulus graphs: sketch graph of |ax+b|1.02t Solve modulus equations: graphically with modulus function |
| Answer | Marks | Guidance |
|---|---|---|
| Working | Mark | Guidance |
| Correct shape of \(y = | x^2 - a^2 | \) |
| Correct coordinates \((\pm a, 0)\) and \((0, a^2)\) marked | B1 | Correct coordinates |
| Answer | Marks | Guidance |
|---|---|---|
| Working | Mark | Guidance |
| \(\{ | x | >a\}\): \(x^2 - a^2 = a^2 - x\) |
| \(\Rightarrow x^2 + x - 2a^2 = 0 \Rightarrow x = \frac{-1 \pm \sqrt{1+8a^2}}{2}\) | M1 | Applies quadratic formula or completes the square to find roots |
| Both correct simplified solutions \(x = \frac{-1 \pm \sqrt{1+8a^2}}{2}\) | A1 | Both correct "simplified down" solutions |
| \(\{ | x | |
| \(\Rightarrow x^2 - x = 0 \Rightarrow x(x-1)=0\) | ||
| \(x = 0\) | B1 | |
| \(x = 1\) | A1 |
| Answer | Marks | Guidance |
|---|---|---|
| Working | Mark | Guidance |
| \(x < \frac{-1-\sqrt{1+8a^2}}{2}\) | B1ft | \(x\) is less than their least value |
| \(x > \frac{-1+\sqrt{1+8a^2}}{2}\) | B1ft | \(x\) is greater than their maximum value |
| \(0 < x < 1\) | M1, A1 | For \(\{ |
## Question 7:
### Part (a):
| Working | Mark | Guidance |
|---------|------|----------|
| Correct shape of $y = |x^2 - a^2|$ | B1 | Correct shape, ignore cusps |
| Correct coordinates $(\pm a, 0)$ and $(0, a^2)$ marked | B1 | Correct coordinates |
### Part (b):
| Working | Mark | Guidance |
|---------|------|----------|
| $\{|x|>a\}$: $x^2 - a^2 = a^2 - x$ | M1 **aef** | $x^2 - a^2 = a^2 - x$ |
| $\Rightarrow x^2 + x - 2a^2 = 0 \Rightarrow x = \frac{-1 \pm \sqrt{1+8a^2}}{2}$ | M1 | Applies quadratic formula or completes the square to find roots |
| Both correct simplified solutions $x = \frac{-1 \pm \sqrt{1+8a^2}}{2}$ | A1 | Both correct "simplified down" solutions |
| $\{|x|<a\}$: $-x^2 + a^2 = a^2 - x$ | M1 **aef** | $-x^2+a^2=a^2-x$ or $x^2-a^2=x-a^2$ |
| $\Rightarrow x^2 - x = 0 \Rightarrow x(x-1)=0$ | | |
| $x = 0$ | B1 | |
| $x = 1$ | A1 | |
### Part (c):
| Working | Mark | Guidance |
|---------|------|----------|
| $x < \frac{-1-\sqrt{1+8a^2}}{2}$ | B1ft | $x$ is less than their least value |
| $x > \frac{-1+\sqrt{1+8a^2}}{2}$ | B1ft | $x$ is greater than their maximum value |
| $0 < x < 1$ | M1, A1 | For $\{|x|<a\}$: Lowest $< x <$ Highest, $0 < x < 1$ |
---\begin{enumerate}
\item (a) Sketch the graph of $y = \left| x ^ { 2 } - a ^ { 2 } \right|$, where $a > 1$, showing the coordinates of the points where the graph meets the axes.\\
(b) Solve $\left| x ^ { 2 } - a ^ { 2 } \right| = a ^ { 2 } - x , a > 1$.\\
(c) Find the set of values of $x$ for which $\left| x ^ { 2 } - a ^ { 2 } \right| > a ^ { 2 } - x , a > 1$.
\end{enumerate}
\hfill \mbox{\textit{Edexcel FP2 2009 Q7 [12]}}