| Exam Board | CAIE |
|---|---|
| Module | P2 (Pure Mathematics 2) |
| Year | 2023 |
| Session | November |
| Marks | 9 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Modulus function |
| Type | Solve |linear| < constant with sketch or follow-up application |
| Difficulty | Moderate -0.3 This is a multi-part question combining modulus inequalities with exponential inequalities. Part (a) is routine sketching, part (b) requires standard case-by-case analysis of |3-x|, part (c) is straightforward logarithm application, and part (d) combines results. While multi-step, each component uses standard techniques without requiring novel insight, making it slightly easier than average. |
| Spec | 1.02g Inequalities: linear and quadratic in single variable1.02l Modulus function: notation, relations, equations and inequalities1.06f Laws of logarithms: addition, subtraction, power rules1.06g Equations with exponentials: solve a^x = b |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| Draw V-shaped graph with vertex on positive \(x\)-axis | B1 | |
| Draw (more or less) correct graph of \(y=9-2x\) with steeper negative gradient | B1 | Dependent on first B mark, appropriately positioned with respect to first graph |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| Solve linear equation or inequality with signs of \(x\) and \(2x\) different | M1 | |
| Obtain critical value 4 | A1 | |
| Conclude \(x>4\) only | A1 | 6 must be discounted |
| Alternative Method: | ||
| State or imply non-modulus equation \((3-x)^2=(9-2x)^2\) | B1 | |
| Attempt solution of three-term quadratic equation (or inequality) | M1 | Dependent on previous B1 |
| Conclude \(x>4\) only | A1 | 6 must be discounted |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| State or imply \((3x-10)\ln 2 < \ln 500\) | B1 | Or equivalent perhaps involving different logarithm base |
| Obtain critical value 6.32 | B1 | |
| Obtain \(x<6.32\) | B1 | Or greater accuracy |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| State 5 and 6 only | B1 |
## Question 4(a):
| Answer | Marks | Guidance |
|--------|-------|----------|
| Draw V-shaped graph with vertex on positive $x$-axis | B1 | |
| Draw (more or less) correct graph of $y=9-2x$ with steeper negative gradient | B1 | Dependent on first B mark, appropriately positioned with respect to first graph |
## Question 4(b):
| Answer | Marks | Guidance |
|--------|-------|----------|
| Solve linear equation or inequality with signs of $x$ and $2x$ different | M1 | |
| Obtain critical value 4 | A1 | |
| Conclude $x>4$ only | A1 | 6 must be discounted |
| **Alternative Method:** | | |
| State or imply non-modulus equation $(3-x)^2=(9-2x)^2$ | B1 | |
| Attempt solution of three-term quadratic equation (or inequality) | M1 | Dependent on previous B1 |
| Conclude $x>4$ only | A1 | 6 must be discounted |
## Question 4(c):
| Answer | Marks | Guidance |
|--------|-------|----------|
| State or imply $(3x-10)\ln 2 < \ln 500$ | B1 | Or equivalent perhaps involving different logarithm base |
| Obtain critical value 6.32 | B1 | |
| Obtain $x<6.32$ | B1 | Or greater accuracy |
## Question 4(d):
| Answer | Marks | Guidance |
|--------|-------|----------|
| State 5 and 6 only | B1 | |4
\begin{enumerate}[label=(\alph*)]
\item Sketch, on the same diagram, the graphs of $y = | 3 - x |$ and $y = 9 - 2 x$.
\item Solve the inequality $| 3 - x | > 9 - 2 x$.
\item Use logarithms to solve the inequality $2 ^ { 3 x - 10 } < 500$. Give your answer in the form $x < a$, where the value of $a$ is given correct to 3 significant figures.
\item List the integers that satisfy both of the inequalities $| 3 - x | > 9 - 2 x$ and $2 ^ { 3 x - 10 } < 500$.
\end{enumerate}
\hfill \mbox{\textit{CAIE P2 2023 Q4 [9]}}