| Exam Board | CAIE |
|---|---|
| Module | P1 (Pure Mathematics 1) |
| Year | 2021 |
| Session | November |
| Marks | 5 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Curve Sketching |
| Type | Solutions from graphical analysis |
| Difficulty | Moderate -0.5 This question requires reading amplitude, vertical shift, and period from a graph (part a), then counting intersections between the given curve and linear functions (part b). These are standard P1 skills involving trigonometric graph transformations and graphical solution methods. The positive integer constraint makes part (a) straightforward, and part (b) is visual counting rather than algebraic manipulation. Slightly easier than average due to the direct reading from graph and simple intersection counting. |
| Spec | 1.02q Use intersection points: of graphs to solve equations1.05f Trigonometric function graphs: symmetries and periodicities1.05g Exact trigonometric values: for standard angles1.05o Trigonometric equations: solve in given intervals |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \(a = 5\) | B1 | |
| \(b = 2\) | B1 | |
| \(c = 3\) | B1 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \(3\) | B1 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \(2\) | B1 |
## Question 5(a):
| Answer | Marks | Guidance |
|--------|-------|----------|
| $a = 5$ | B1 | |
| $b = 2$ | B1 | |
| $c = 3$ | B1 | |
## Question 5(b)(i):
| Answer | Marks | Guidance |
|--------|-------|----------|
| $3$ | B1 | |
## Question 5(b)(ii):
| Answer | Marks | Guidance |
|--------|-------|----------|
| $2$ | B1 | |
---5\\
\includegraphics[max width=\textwidth, alt={}, center]{af7aeda9-2ded-4db4-9ff3-ed6adc67859f-07_778_878_255_630}
The diagram shows part of the graph of $y = a \cos ( b x ) + c$.
\begin{enumerate}[label=(\alph*)]
\item Find the values of the positive integers $a , b$ and $c$.
\item For these values of $a$, $b$ and $c$, use the given diagram to determine the number of solutions in the interval $0 \leqslant x \leqslant 2 \pi$ for each of the following equations.
\begin{enumerate}[label=(\roman*)]
\item $a \cos ( b x ) + c = \frac { 6 } { \pi } x$
\item $a \cos ( b x ) + c = 6 - \frac { 6 } { \pi } x$\\
The diagram shows a metal plate $A B C$ in which the sides are the straight line $A B$ and the arcs $A C$ and $B C$. The line $A B$ has length 6 cm . The arc $A C$ is part of a circle with centre $B$ and radius 6 cm , and the arc $B C$ is part of a circle with centre $A$ and radius 6 cm .
\end{enumerate}\end{enumerate}
\hfill \mbox{\textit{CAIE P1 2021 Q5 [5]}}