| Exam Board | OCR |
|---|---|
| Module | C3 (Core Mathematics 3) |
| Marks | 9 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Modulus function |
| Type | Sketch y=|f(x)| for non-linear f(x) |
| Difficulty | Moderate -0.3 This is a straightforward modulus function question requiring standard techniques: sketching y=|f(x)| by reflecting negative portions of a parabola, finding a composite function value by substitution, and solving a composite function equation. All steps are routine C3 material with no novel problem-solving required, making it slightly easier than average. |
| Spec | 1.02l Modulus function: notation, relations, equations and inequalities1.02u Functions: definition and vocabulary (domain, range, mapping)1.02v Inverse and composite functions: graphs and conditions for existence |
4. 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.\\
(i) Showing the coordinates of any points where the graph meets the axes, sketch the graph of $y = | \mathrm { f } ( x ) |$.
The function $g$ is defined by
$$\mathrm { g } ( x ) \equiv 3 a x , \quad x \in \mathbb { R } .$$
(ii) Find $\mathrm { fg } ( \mathrm { a } )$ in terms of $a$.\\
(iii) Solve the equation
$$\operatorname { gf } ( x ) = 9 a ^ { 3 }$$
\hfill \mbox{\textit{OCR C3 Q4 [9]}}