| Exam Board | OCR MEI |
|---|---|
| Module | Paper 2 (Paper 2) |
| Year | 2024 |
| Session | June |
| Marks | 5 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Trigonometric equations in context |
| Type | Solve double/multiple angle equation |
| Difficulty | Easy -1.2 This is a straightforward two-part question on standard A-level content. Part (a) requires sketching a basic transformed sine curve (double angle, standard shape). Part (b) involves solving a simple trigonometric equation using the double angle and reference angles—a routine textbook exercise requiring only recall of standard methods with no problem-solving insight needed. |
| Spec | 1.05f Trigonometric function graphs: symmetries and periodicities1.05o Trigonometric equations: solve in given intervals |
| \cline { 2 - 4 } \multicolumn{1}{c|}{} | \(M\) | \(M ^ { \prime }\) | Total |
| \(C\) | 0.24 | 0.36 | |
| \(C ^ { \prime }\) | |||
| Total | 0.42 | 1 |
| Answer | Marks | Guidance |
|---|---|---|
| [sine wave with 2 complete cycles between \(0\) and \(2\pi\)] | M1 | sine wave with 2 complete cycles between \(0\) and \(2\pi\); condone waves of unequal amplitude or different length; allow labelling in degrees for M1 only |
| [all correct with amplitude 1] | A1 | all correct with amplitude 1; ignore graph outside \([0, 2\pi]\) |
| Answer | Marks | Guidance |
|---|---|---|
| either \(-\dfrac{\pi}{12}\) or \(-\dfrac{5\pi}{12}\) oe seen or either \(\dfrac{7\pi}{6}\) or \(\dfrac{11\pi}{6}\) oe seen | M1 | NB from use of \(\sin^{-1}\left(-\dfrac{1}{2}\right)\); decimals \(-0.262\) or \(-1.309\) correct to 2 dp or better; or 3.665 or 5.760 correct to 2 dp or better; M1 may be implied by two correct answers in radians |
| \(\dfrac{7\pi}{12}, \dfrac{11\pi}{12}, \dfrac{19\pi}{12}, \dfrac{23\pi}{12}\) | A1 | two correct values; NB awrt 1.833, 2.880, 4.974, 6.021 or correct to 2 dp |
| [all four values correct, no extras] | A1 | all four values correct with no extras for \(0 \leq \theta \leq 2\pi\); if M0A0A0 allow SCB1 for two correct answers in degrees or SCB2 for all four correct in degrees: 105, 165, 285, 345 |
## Question 4(a):
[sine wave with 2 complete cycles between $0$ and $2\pi$] | M1 | sine wave with 2 complete cycles between $0$ and $2\pi$; condone waves of unequal amplitude or different length; allow labelling in degrees for M1 only
[all correct with amplitude 1] | A1 | all correct with amplitude 1; ignore graph outside $[0, 2\pi]$
---
## Question 4(b):
either $-\dfrac{\pi}{12}$ or $-\dfrac{5\pi}{12}$ oe seen **or** either $\dfrac{7\pi}{6}$ or $\dfrac{11\pi}{6}$ oe seen | M1 | **NB** from use of $\sin^{-1}\left(-\dfrac{1}{2}\right)$; decimals $-0.262$ or $-1.309$ correct to 2 dp or better; **or** 3.665 or 5.760 correct to 2 dp or better; M1 may be implied by two correct answers in radians
$\dfrac{7\pi}{12}, \dfrac{11\pi}{12}, \dfrac{19\pi}{12}, \dfrac{23\pi}{12}$ | A1 | two correct values; **NB** awrt 1.833, 2.880, 4.974, 6.021 or correct to 2 dp
[all four values correct, no extras] | A1 | all four values correct with no extras for $0 \leq \theta \leq 2\pi$; if **M0A0A0** allow **SCB1** for two correct answers in degrees **or SCB2** for all four correct in degrees: 105, 165, 285, 3454
\begin{enumerate}[label=(\alph*)]
\item On the axes in the Printed Answer Booklet, sketch the graph of $y = \sin 2 \theta$ for $0 \leqslant \theta \leqslant 2 \pi$.
\item Solve the equation $\sin 2 \theta = - \frac { 1 } { 2 }$ for $0 \leqslant \theta \leqslant 2 \pi$.\\
$5 M$ is the event that an A-level student selected at random studies mathematics.\\
$C$ is the event that an A-level student selected at random studies chemistry.\\
You are given that $\mathrm { P } ( M ) = 0.42 , \mathrm { P } ( C ) = 0.36$ and $\mathrm { P } ( \mathrm { M }$ and $\mathrm { C } ) = 0.24$. These probabilities are shown in the two-way table below.
\begin{center}
\begin{tabular}{ | c | c | c | c | }
\cline { 2 - 4 }
\multicolumn{1}{c|}{} & $M$ & $M ^ { \prime }$ & Total \\
\hline
$C$ & 0.24 & & 0.36 \\
\hline
$C ^ { \prime }$ & & & \\
\hline
Total & 0.42 & & 1 \\
\hline
\end{tabular}
\end{center}
\end{enumerate}
\hfill \mbox{\textit{OCR MEI Paper 2 2024 Q4 [5]}}