| Exam Board | CAIE |
|---|---|
| Module | P1 (Pure Mathematics 1) |
| Year | 2024 |
| Session | June |
| Marks | 5 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Trig Graphs & Exact Values |
| Type | Find coordinates of turning points |
| Difficulty | Moderate -0.5 Part (a) requires standard knowledge of cosine graph transformations to find where the curve crosses the x-axis and locate the minimum point—straightforward application of cos(x) = 0 and cos(x) = -1. Part (b) involves evaluating an exact inverse trig value and solving for t, which is routine manipulation. Both parts are below average difficulty as they test direct recall and basic algebraic manipulation without requiring problem-solving insight. |
| Spec | 1.05f Trigonometric function graphs: symmetries and periodicities1.05g Exact trigonometric values: for standard angles1.05i Inverse trig functions: arcsin, arccos, arctan domains and graphs |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| State \(\left(\frac{5}{3}\pi,\ 0\right)\) for point \(A\) | B1 | Or exact equivalent. Allow \(x=\frac{5}{3}\pi\) or exact equivalent. |
| \(x=\frac{19}{6}\pi\) for point \(B\) | B1 | Or exact equivalent. May be implied in coordinate or vector form. |
| \(y=-k\) for point \(B\) | B1 | May be implied in coordinate or vector form. |
| Total: 3 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| Solve at least as far as \(\sin^{-1}3t=k\pi\) with correct value for \(\cos^{-1}\!\left(\tfrac{1}{2}\sqrt{2}\right)\) | M1 | Allow use of \(\pi=3.14\ldots\) Allow \(\sin^{-1}3t=30\) |
| \(\sin^{-1}3t=\frac{1}{6}\pi\) and hence \(t=\frac{1}{6}\) | A1 | Or exact equivalent. Can use degrees if consistent. |
| Total: 2 |
## Question 2(a):
| Answer | Mark | Guidance |
|--------|------|----------|
| State $\left(\frac{5}{3}\pi,\ 0\right)$ for point $A$ | B1 | Or exact equivalent. Allow $x=\frac{5}{3}\pi$ or exact equivalent. |
| $x=\frac{19}{6}\pi$ for point $B$ | B1 | Or exact equivalent. May be implied in coordinate or vector form. |
| $y=-k$ for point $B$ | B1 | May be implied in coordinate or vector form. |
| **Total: 3** | | |
---
## Question 2(b):
| Answer | Mark | Guidance |
|--------|------|----------|
| Solve at least as far as $\sin^{-1}3t=k\pi$ with correct value for $\cos^{-1}\!\left(\tfrac{1}{2}\sqrt{2}\right)$ | M1 | Allow use of $\pi=3.14\ldots$ Allow $\sin^{-1}3t=30$ |
| $\sin^{-1}3t=\frac{1}{6}\pi$ and hence $t=\frac{1}{6}$ | A1 | Or exact equivalent. Can use degrees if consistent. |
| **Total: 2** | | |2
\begin{enumerate}[label=(\alph*)]
\item \\
\includegraphics[max width=\textwidth, alt={}, center]{d6976a4b-aecf-43f1-a3f2-bcad37d03585-04_582_922_335_575}
The diagram shows the curve $y = k \cos \left( x - \frac { 1 } { 6 } \pi \right)$ where $k$ is a positive constant and $x$ is measured in radians. The curve crosses the $x$-axis at point $A$ and $B$ is a minimum point.
Find the coordinates of $A$ and $B$.
\item Find the exact value of $t$ that satisfies the equation
$$3 \sin ^ { - 1 } ( 3 t ) + 2 \cos ^ { - 1 } \left( \frac { 1 } { 2 } \sqrt { 2 } \right) = \pi .$$
\includegraphics[max width=\textwidth, alt={}, center]{d6976a4b-aecf-43f1-a3f2-bcad37d03585-04_2718_33_141_2013}
\end{enumerate}
\hfill \mbox{\textit{CAIE P1 2024 Q2 [5]}}