| Exam Board | CAIE |
|---|---|
| Module | P1 (Pure Mathematics 1) |
| Year | 2023 |
| Session | June |
| Marks | 5 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Curve Sketching |
| Type | Trigonometric curve intersections |
| Difficulty | Standard +0.3 This is a straightforward curve sketching question requiring basic knowledge of sine transformations (amplitude, vertical shift, period adjustment) and graphical intersection counting. Part (a) is simple arithmetic with sine range, part (b) is standard sketching, and part (c) requires only visual inspection of intersections—no algebraic solving needed. Slightly easier than average due to routine nature. |
| Spec | 1.02m Graphs of functions: difference between plotting and sketching1.02q Use intersection points: of graphs to solve equations1.05f Trigonometric function graphs: symmetries and periodicities |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| Greatest \(= 5\) | B1 | No inequality required |
| Least \(= -1\) | B1 | No inequality required. Condone \((-1, 5)\) or equivalent |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| One complete cycle starting and finishing at \(y=2\), maximum and minimum in correct quadrants, shape and curvature approximately correct | B1 | One complete cycle starting and finishing at \(y=2\). Maximum and minimum in correct quadrants. Shape and curvature approximately correct |
| Maximum and minimum indicated on \(y\)-axis with numbers or lines, or labelled on graph | B1 FT | FT *their* greatest and least values. Award B1 for 5 and \(-1\) even if *their* values were incorrect in (a) |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \(1\) | B1 | WWW |
## Question 7(a):
| Answer | Marks | Guidance |
|--------|-------|----------|
| Greatest $= 5$ | B1 | No inequality required |
| Least $= -1$ | B1 | No inequality required. Condone $(-1, 5)$ or equivalent |
---
## Question 7(b):
| Answer | Marks | Guidance |
|--------|-------|----------|
| One complete cycle starting and finishing at $y=2$, maximum and minimum in correct quadrants, shape and curvature approximately correct | B1 | One complete cycle starting and finishing at $y=2$. Maximum and minimum in correct quadrants. Shape and curvature approximately correct |
| Maximum and minimum indicated on $y$-axis with numbers or lines, or labelled on graph | B1 FT | FT *their* greatest and least values. Award B1 for 5 and $-1$ even if *their* values were incorrect in (a) |
---
## Question 7(c):
| Answer | Marks | Guidance |
|--------|-------|----------|
| $1$ | B1 | WWW |
---7 A curve has equation $y = 2 + 3 \sin \frac { 1 } { 2 } x$ for $0 \leqslant x \leqslant 4 \pi$.
\begin{enumerate}[label=(\alph*)]
\item State greatest and least values of $y$.
\item Sketch the curve.\\
\includegraphics[max width=\textwidth, alt={}, center]{77f27b11-b931-481f-b4ef-5e549eff8086-09_1127_1219_904_495}
\item State the number of solutions of the equation
$$2 + 3 \sin \frac { 1 } { 2 } x = 5 - 2 x$$
for $0 \leqslant x \leqslant 4 \pi$.
\end{enumerate}
\hfill \mbox{\textit{CAIE P1 2023 Q7 [5]}}