| Exam Board | OCR |
|---|---|
| Module | C3 (Core Mathematics 3) |
| Marks | 10 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Modulus function |
| Type | Solve |f(x)| compared to |g(x)| with parameters: sketch then solve |
| Difficulty | Standard +0.3 This is a standard modulus function question requiring evaluation of a composite function, sketching two V-shaped graphs with clearly defined vertices and intercepts, and solving a modulus equation by considering cases. While it involves multiple parts and algebraic manipulation, the techniques are routine for C3 level with no novel problem-solving required, making it slightly easier than average. |
| Spec | 1.02l Modulus function: notation, relations, equations and inequalities1.02m Graphs of functions: difference between plotting and sketching |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| \(= f(3a) = 0\) | M1 A1 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| Graph showing \(y = g(x)\) and \(y = f(x)\) with intercepts \((0, 3a)\), \((0, a)\), \((-\frac{1}{2}a, 0)\), \((3a, 0)\) | B4 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| \((x - 3a)^2 = (2x + a)^2\) | M1 | |
| \(3x^2 + 10ax - 8a^2 = 0\) | A1 | |
| \((3x - 2a)(x + 4a) = 0\) | M1 | |
| \(x = -4a\), \(\frac{2}{3}a\) | A1 | (10) |
# Question 8:
## Part (i)
| Answer/Working | Mark | Guidance |
|---|---|---|
| $= f(3a) = 0$ | M1 A1 | |
## Part (ii)
| Answer/Working | Mark | Guidance |
|---|---|---|
| Graph showing $y = g(x)$ and $y = f(x)$ with intercepts $(0, 3a)$, $(0, a)$, $(-\frac{1}{2}a, 0)$, $(3a, 0)$ | B4 | |
## Part (iii)
| Answer/Working | Mark | Guidance |
|---|---|---|
| $(x - 3a)^2 = (2x + a)^2$ | M1 | |
| $3x^2 + 10ax - 8a^2 = 0$ | A1 | |
| $(3x - 2a)(x + 4a) = 0$ | M1 | |
| $x = -4a$, $\frac{2}{3}a$ | A1 | **(10)** |
---8. The functions f and g are defined for all real values of $x$ by
$$\begin{aligned}
& \mathrm { f } : x \rightarrow | x - 3 a | \\
& \mathrm { g } : x \rightarrow | 2 x + a |
\end{aligned}$$
where $a$ is a positive constant.\\
(i) Evaluate fg(-2a).\\
(ii) Sketch on the same diagram the graphs of $y = \mathrm { f } ( x )$ and $y = \mathrm { g } ( x )$, showing the coordinates of any points where each graph meets the coordinate axes.\\
(iii) Solve the equation
$$\mathrm { f } ( x ) = \mathrm { g } ( x )$$
\hfill \mbox{\textit{OCR C3 Q8 [10]}}