| Exam Board | OCR MEI |
|---|---|
| Module | C3 (Core Mathematics 3) |
| Marks | 7 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Modulus function |
| Type | Solve |exponential| < constant |
| Difficulty | Standard +0.3 This question combines logarithm properties with modulus inequalities. Part (i) is straightforward substitution using given points to find constants. Part (ii) requires solving |ln expression| < 2, which splits into a compound inequality and involves exponentiating—standard technique but requires careful handling of the domain. Overall slightly easier than average due to clear structure and routine methods. |
| Spec | 1.02l Modulus function: notation, relations, equations and inequalities1.06d Natural logarithm: ln(x) function and properties |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| Substitute: \(0 = a\ln 2b \Rightarrow \ln 2b = 0 \Rightarrow 2b = 1 \Rightarrow b = \frac{1}{2}\) | M1, A1 | |
| \(1 = a\ln 2 \Rightarrow a = \frac{1}{\ln 2}\) | A1 | |
| Total: 3 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| \(\ | a\ln(bx)\ | < 2 \Rightarrow \left\ |
| \(\Rightarrow -2\ln 2 < \ln\frac{1}{2}x < 2\ln 2\) | M1 | Modulus |
| \(\Rightarrow \ln\frac{1}{4} < \ln\frac{1}{2}x < \ln 4\) | M1 | Powers of logs |
| \(\Rightarrow \frac{1}{4} < \frac{1}{2}x < 4 \Rightarrow \frac{1}{2} < x < 8\) | A1, A1 | |
| Total: 4 |
## Question 3:
### Part (i):
| Answer | Mark | Guidance |
|--------|------|----------|
| Substitute: $0 = a\ln 2b \Rightarrow \ln 2b = 0 \Rightarrow 2b = 1 \Rightarrow b = \frac{1}{2}$ | M1, A1 | |
| $1 = a\ln 2 \Rightarrow a = \frac{1}{\ln 2}$ | A1 | |
| **Total: 3** | | |
### Part (ii):
| Answer | Mark | Guidance |
|--------|------|----------|
| $\|a\ln(bx)\| < 2 \Rightarrow \left\|\frac{\ln\frac{1}{2}x}{\ln 2}\right\| < 2 \Rightarrow \left\|\ln\frac{1}{2}x\right\| < 2\ln 2$ | | |
| $\Rightarrow -2\ln 2 < \ln\frac{1}{2}x < 2\ln 2$ | M1 | Modulus |
| $\Rightarrow \ln\frac{1}{4} < \ln\frac{1}{2}x < \ln 4$ | M1 | Powers of logs |
| $\Rightarrow \frac{1}{4} < \frac{1}{2}x < 4 \Rightarrow \frac{1}{2} < x < 8$ | A1, A1 | |
| **Total: 4** | | |
---3 The graph shows part of the function $y = a \ln ( b x )$.\\
\includegraphics[max width=\textwidth, alt={}, center]{2f403099-2813-40d8-a9ae-1f7e64d41f80-2_377_762_900_685}
The graph passes through the points $( 2,0 )$ and $( 4,1 )$.\\
(i) Show that $b = \frac { 1 } { 2 }$ and find the exact value of $a$.\\
(ii) Solve the inequality $| a \ln ( b x ) | < 2$.
\hfill \mbox{\textit{OCR MEI C3 Q3 [7]}}