| Exam Board | Edexcel |
|---|---|
| Module | C3 (Core Mathematics 3) |
| Marks | 12 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Modulus function |
| Type | Sketch y=|f(x)| or y=f(|x|) for non-linear f(x) and solve |
| Difficulty | Standard +0.3 This question involves standard transformations of modulus graphs with a quadratic function, finding composite functions, and solving a straightforward equation. While it requires understanding of modulus transformations and function composition, these are routine C3 techniques with no novel problem-solving required. The multi-part structure and algebraic manipulation with parameters make it slightly above average difficulty, but all steps follow standard procedures. |
| Spec | 1.02c Simultaneous equations: two variables by elimination and substitution1.02l Modulus function: notation, relations, equations and inequalities1.02u Functions: definition and vocabulary (domain, range, mapping)1.02w Graph transformations: simple transformations of f(x) |
| Answer | Marks |
|---|---|
| (i) Graph with minimum at \((0,0)\) and \((2a,0)\) | B3 |
| (ii) Graph with maxima at \((-2a,0)\) and \((2a,0)\), minimum at \((0,0)\) | B3 |
| Answer | Marks |
|---|---|
| \(= f(3a^2) = 9a^4 - 6a^3\) | M1 A1 |
| Answer | Marks | Guidance |
|---|---|---|
| \(g(x) = 3a(x^2-2ax)\) | M1 | |
| \(\therefore 3a(x^2-2ax) = 9a^3\) | ||
| \(x^2 - 2ax - 3a^2 = 0\) | A1 | |
| \((x+a)(x-3a) = 0\) | M1 | |
| \(x = -a, 3a\) | A1 | (12) |
**(a)**
(i) Graph with minimum at $(0,0)$ and $(2a,0)$ | B3 |
(ii) Graph with maxima at $(-2a,0)$ and $(2a,0)$, minimum at $(0,0)$ | B3 |
**(b)**
$= f(3a^2) = 9a^4 - 6a^3$ | M1 A1 |
**(c)**
$g(x) = 3a(x^2-2ax)$ | M1 |
$\therefore 3a(x^2-2ax) = 9a^3$ | |
$x^2 - 2ax - 3a^2 = 0$ | A1 |
$(x+a)(x-3a) = 0$ | M1 |
$x = -a, 3a$ | A1 | (12)
---7. The function f is defined by
$$\mathrm { f } ( x ) \equiv x ^ { 2 } - 2 a x , \quad x \in \mathbb { R } ,$$
where $a$ is a positive constant.
\begin{enumerate}[label=(\alph*)]
\item Showing the coordinates of any points where each graph meets the axes, sketch on separate diagrams the graphs of
\begin{enumerate}[label=(\roman*)]
\item $\quad y = | \mathrm { f } ( x ) |$,
\item $y = \mathrm { f } ( | x | )$.
The function g is defined by
$$\mathrm { g } ( x ) \equiv 3 a x , \quad x \in \mathbb { R } .$$
\end{enumerate}\item Find fg(a) in terms of $a$.
\item Solve the equation
$$\operatorname { gf } ( x ) = 9 a ^ { 3 }$$
\end{enumerate}
\hfill \mbox{\textit{Edexcel C3 Q7 [12]}}