| Exam Board | CAIE |
|---|---|
| Module | P2 (Pure Mathematics 2) |
| Year | 2020 |
| Session | March |
| Marks | 9 |
| 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 +0.8 This question requires sketching modulus functions with parameters, solving |f(x)| = |g(x)| by considering cases, and then applying this to an exponential substitution. Part (c) involves recognizing that 2^t is a substitution variable and interpreting the inequality solution geometrically. While systematic, it requires careful case analysis and connecting multiple representations, making it moderately challenging but within reach of strong A-level students. |
| Spec | 1.02l Modulus function: notation, relations, equations and inequalities1.02s Modulus graphs: sketch graph of |ax+b|1.02t Solve modulus equations: graphically with modulus function |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| Draw two V-shaped graphs with one vertex on negative \(x\)-axis and one vertex on positive \(x\)-axis | M1 | |
| Draw correct graphs related correctly to each other | A1 | |
| State correct coordinates \(-2k\), \(2k\), \(\frac{3}{2}k\), \(3k\) | A1 | Either given on axes or stated separately |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| State or imply non-modulus equation \((x+2k)^2 = (2x-3k)^2\) or pair of linear equations | B1 | |
| Attempt solution of 3-term quadratic equation or pair of linear equations | M1 | |
| Obtain \(x = \frac{1}{3}k\), \(x = 5k\) | A1 | |
| Obtain \(y = \frac{7}{3}k\), \(y = 7k\) | A1 | If A0A0, award A1 for one pair of correct coordinates |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| Relate \(2^t\) to larger value of \(x\) from part (b) | M1 | |
| Apply logarithms to obtain \(t = \frac{\ln(5k)}{\ln 2}\) | A1 | OE such as \(\frac{\log_{10}(5k)}{\log_{10}2}\) or \(\log_2(5k)\) |
## Question 5(a):
| Answer | Marks | Guidance |
|--------|-------|----------|
| Draw two V-shaped graphs with one vertex on negative $x$-axis and one vertex on positive $x$-axis | M1 | |
| Draw correct graphs related correctly to each other | A1 | |
| State correct coordinates $-2k$, $2k$, $\frac{3}{2}k$, $3k$ | A1 | Either given on axes or stated separately |
## Question 5(b):
| Answer | Marks | Guidance |
|--------|-------|----------|
| State or imply non-modulus equation $(x+2k)^2 = (2x-3k)^2$ or pair of linear equations | B1 | |
| Attempt solution of 3-term quadratic equation or pair of linear equations | M1 | |
| Obtain $x = \frac{1}{3}k$, $x = 5k$ | A1 | |
| Obtain $y = \frac{7}{3}k$, $y = 7k$ | A1 | If A0A0, award A1 for one pair of correct coordinates |
## Question 5(c):
| Answer | Marks | Guidance |
|--------|-------|----------|
| Relate $2^t$ to larger value of $x$ from part (b) | M1 | |
| Apply logarithms to obtain $t = \frac{\ln(5k)}{\ln 2}$ | A1 | OE such as $\frac{\log_{10}(5k)}{\log_{10}2}$ or $\log_2(5k)$ |
---5
\begin{enumerate}[label=(\alph*)]
\item Sketch, on the same diagram, the graphs of $y = | x + 2 k |$ and $y = | 2 x - 3 k |$, where $k$ is a positive constant.
Give, in terms of $k$, the coordinates of the points where each graph meets the axes.
\item Find, in terms of $k$, the coordinates of each of the two points where the graphs intersect.
\item Find, in terms of $k$, the largest value of $t$ satisfying the inequality
$$\left| 2 ^ { t } + 2 k \right| \geqslant \left| 2 ^ { t + 1 } - 3 k \right| .$$
\end{enumerate}
\hfill \mbox{\textit{CAIE P2 2020 Q5 [9]}}