| Exam Board | OCR MEI |
|---|---|
| Module | Paper 1 (Paper 1) |
| Year | 2023 |
| Session | June |
| Marks | 6 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Reciprocal Trig & Identities |
| Type | Prove algebraic trigonometric identity |
| Difficulty | Moderate -0.3 This is a straightforward multi-part question testing standard double angle identities. Part (a) requires the routine identity cos(2x) = 1 - 2sin²(x), part (b) is direct observation, and part (c) involves solving a simple trigonometric equation. All techniques are standard A-level fare with no novel insight required, making it slightly easier than average. |
| Spec | 1.05f Trigonometric function graphs: symmetries and periodicities1.05l Double angle formulae: and compound angle formulae1.05o Trigonometric equations: solve in given intervals |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| Use \(\sin^2 x = \frac{1}{2}(1 - \cos 2x)\); So \(1.5 + \sin^2 x = 1.5 + \frac{1}{2}(1 - \cos 2x)\) | M1 | Attempt to write \(\sin^2 x\) in terms of \(\cos 2x\) |
| So \(y = 2 - 0.5\cos 2x\) | A1 | Allow for \(a = 2\), \(b = 0.5\) or fully correct expression |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| Period \(= \pi\) | B1 | Cao. Do not accept \(180°\) |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| Intersect when \(2 - 0.5\cos 2x = 1 + \cos 2x\); \(\cos 2x = \frac{2}{3}\) | M1 | Equate expressions in \(\cos 2x\) and attempt to rearrange |
| \(x = 0.421,\ 2.72,\ 3.56,\ 5.86\) radians (correct to 3sf) | A1, A1 | At least 1 correct value; Three other correct values and no others in the interval \(0 \leq x \leq 2\pi\). FT their first root |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \(1.5 + \sin^2 x = 2 - 2\sin^2 x\) or \(1.5 + (1 - \cos^2 x) = 1 + 2\cos^2 x - 1\); \(3\sin^2 x = 0.5\) or \(3\cos^2 x = 2.5\) | M1 | Uses correct trig identities to attempt to find a value for \(\sin^2 x\) or \(\cos^2 x\) |
| \(\sin x = \pm\sqrt{\frac{1}{6}}\left[= \pm\frac{\sqrt{6}}{6}\right]\) or \(\cos x = \pm\sqrt{\frac{5}{6}}\) | ||
| \(x = 0.421,\ 2.72,\ 3.56,\ 5.86\) radians (correct to 3sf) | A1, A1 | At least 1 correct value; Four correct values and no others in the interval \(0 \leq x \leq 2\pi\). FT their first root |
## Question 10(a):
| Answer | Marks | Guidance |
|--------|-------|----------|
| Use $\sin^2 x = \frac{1}{2}(1 - \cos 2x)$; So $1.5 + \sin^2 x = 1.5 + \frac{1}{2}(1 - \cos 2x)$ | M1 | Attempt to write $\sin^2 x$ in terms of $\cos 2x$ |
| So $y = 2 - 0.5\cos 2x$ | A1 | Allow for $a = 2$, $b = 0.5$ or fully correct expression |
---
## Question 10(b):
| Answer | Marks | Guidance |
|--------|-------|----------|
| Period $= \pi$ | B1 | Cao. Do not accept $180°$ |
---
## Question 10(c):
| Answer | Marks | Guidance |
|--------|-------|----------|
| Intersect when $2 - 0.5\cos 2x = 1 + \cos 2x$; $\cos 2x = \frac{2}{3}$ | M1 | Equate expressions in $\cos 2x$ and attempt to rearrange |
| $x = 0.421,\ 2.72,\ 3.56,\ 5.86$ radians (correct to 3sf) | A1, A1 | At least 1 correct value; Three other correct values and no others in the interval $0 \leq x \leq 2\pi$. FT their first root |
**Alternative method:**
| Answer | Marks | Guidance |
|--------|-------|----------|
| $1.5 + \sin^2 x = 2 - 2\sin^2 x$ or $1.5 + (1 - \cos^2 x) = 1 + 2\cos^2 x - 1$; $3\sin^2 x = 0.5$ or $3\cos^2 x = 2.5$ | M1 | Uses correct trig identities to attempt to find a value for $\sin^2 x$ or $\cos^2 x$ |
| $\sin x = \pm\sqrt{\frac{1}{6}}\left[= \pm\frac{\sqrt{6}}{6}\right]$ or $\cos x = \pm\sqrt{\frac{5}{6}}$ | | |
| $x = 0.421,\ 2.72,\ 3.56,\ 5.86$ radians (correct to 3sf) | A1, A1 | At least 1 correct value; Four correct values and no others in the interval $0 \leq x \leq 2\pi$. FT their first root |
---10 The diagram shows the graph of $\mathrm { y } = 1.5 + \sin ^ { 2 } \mathrm { x }$ for $0 \leqslant x \leqslant 2 \pi$.\\
\includegraphics[max width=\textwidth, alt={}, center]{8eeff88d-8b05-43c6-86a5-bd82221c0bea-07_512_1278_322_242}
\begin{enumerate}[label=(\alph*)]
\item Show that the equation of the graph can be written in the form $\mathrm { y } = \mathrm { a } - \mathrm { b } \cos 2 \mathrm { x }$ where $a$ and $b$ are constants to be determined.
\item Write down the period of the function $1.5 + \sin ^ { 2 } x$.
\item Determine the $x$-coordinates of the points of intersection of the graph of $y = 1.5 + \sin ^ { 2 } x$ with the graph of $\mathrm { y } = 1 + \cos 2 \mathrm { x }$ in the interval $0 \leqslant x \leqslant 2 \pi$.
\end{enumerate}
\hfill \mbox{\textit{OCR MEI Paper 1 2023 Q10 [6]}}