| Exam Board | OCR MEI |
|---|---|
| Module | Paper 1 (Paper 1) |
| Year | 2022 |
| Session | June |
| Marks | 8 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Trig Graphs & Exact Values |
| Type | Calculate intersection coordinates algebraically |
| Difficulty | Challenging +1.2 Part (a) is routine recall of a standard inverse trig graph. Part (b) requires recognizing the double angle identity (3sin x cos x = 1.5sin 2x), setting up cos²x = 1.5sin 2x, and solving the resulting trigonometric equation over a specified interval. While it involves multiple steps and careful algebraic manipulation with trig identities, it follows standard A-level techniques without requiring novel insight—slightly above average due to the algebraic complexity and need to find all solutions systematically. |
| Spec | 1.02q Use intersection points: of graphs to solve equations1.05i Inverse trig functions: arcsin, arccos, arctan domains and graphs1.05l Double angle formulae: and compound angle formulae1.05o Trigonometric equations: solve in given intervals |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| [Graph with correct general shape] | B1 | General shape with horizontal asymptotes. Allow if asymptote not drawn provided the intention is clear. Must be a one-to-one function |
| B1 | \(y\)-values \(\pm\dfrac{\pi}{2}\) seen | |
| [2] |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| DR | ||
| Graphs intersect when \(3\sin x\cos x = \cos^2 x\) | M1 | soi |
| Either \(\cos x = 0\) | M1 | Attempt to solve \(\cos x = 0\) |
| giving \(x = -\dfrac{\pi}{2}, \dfrac{\pi}{2}\) | A1 | Both values in radians needed |
| or \(3\sin x = \cos x\) giving \(\tan x = \dfrac{1}{3}\) | M1 | Allow for \(x = \tan^{-1}\dfrac{1}{3}\) |
| \(x = 0.322,\ x = -2.82\) to 3 s.f. | A1 | Both values in radians to at least 2 s.f. needed. Do not award if additional values inside the interval \([-\pi, \pi]\). Ignore additional values outside the interval \([-\pi, \pi]\). SC1 award for 18.4° and −161.6° if 90° already seen |
| When \(x = 0.322\) or \(x = -2.82\), \(y = 0.9\) | A1 | Allow awrt 0.90. Notice 0.9 is exact. |
| Points of intersection: \((0.322, 0.9),(-2.82, 0.9)\left(-\dfrac{\pi}{2}, 0\right),\left(\dfrac{\pi}{2}, 0\right)\) | ||
| [6] |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| Graphs intersect when \(3\sin x\cos x = \cos^2 x\) | M1 | soi |
| Either \(\cos x = 0\), giving \(x = -\dfrac{\pi}{2}, \dfrac{\pi}{2}\) | M1, A1 | Attempt to solve \(\cos x = 0\); both values in radians needed |
| Or \(3\sin x = \cos x\), squaring gives \(9\sin^2 x = \cos^2 x = 1 - \sin^2 x\) | ||
| \(10\sin^2 x = 1\), \(\sin x = \pm\sqrt{0.1}\) | M1 | Complete method for finding at least one value for \(\sin x\) |
| \(x = -2.820, -0.322, 0.322, 2.820\) | ||
| Select genuine roots \(0.322, -2.820\) | A1 | Both correct roots and no others in the range |
| When \(x = 0.322\) or \(x = -2.82\), \(y = 0.9\) | A1 | Allow awrt 0.90. Notice 0.9 is exact. |
| [6] |
# Question 3:
## Part (a)
| Answer | Marks | Guidance |
|--------|-------|----------|
| [Graph with correct general shape] | B1 | General shape with horizontal asymptotes. Allow if asymptote not drawn provided the intention is clear. Must be a one-to-one function |
| | B1 | $y$-values $\pm\dfrac{\pi}{2}$ seen |
| **[2]** | | |
## Part (b)
| Answer | Marks | Guidance |
|--------|-------|----------|
| **DR** | | |
| Graphs intersect when $3\sin x\cos x = \cos^2 x$ | M1 | soi |
| Either $\cos x = 0$ | M1 | Attempt to solve $\cos x = 0$ |
| giving $x = -\dfrac{\pi}{2}, \dfrac{\pi}{2}$ | A1 | Both values in radians needed |
| or $3\sin x = \cos x$ giving $\tan x = \dfrac{1}{3}$ | M1 | Allow for $x = \tan^{-1}\dfrac{1}{3}$ |
| $x = 0.322,\ x = -2.82$ to 3 s.f. | A1 | Both values in radians to at least 2 s.f. needed. Do not award if additional values inside the interval $[-\pi, \pi]$. Ignore additional values outside the interval $[-\pi, \pi]$. SC1 award for 18.4° and −161.6° if 90° already seen |
| When $x = 0.322$ or $x = -2.82$, $y = 0.9$ | A1 | Allow awrt 0.90. Notice 0.9 is exact. |
| Points of intersection: $(0.322, 0.9),(-2.82, 0.9)\left(-\dfrac{\pi}{2}, 0\right),\left(\dfrac{\pi}{2}, 0\right)$ | | |
| **[6]** | | |
**Alternative method:**
| Answer | Marks | Guidance |
|--------|-------|----------|
| Graphs intersect when $3\sin x\cos x = \cos^2 x$ | M1 | soi |
| Either $\cos x = 0$, giving $x = -\dfrac{\pi}{2}, \dfrac{\pi}{2}$ | M1, A1 | Attempt to solve $\cos x = 0$; both values in radians needed |
| Or $3\sin x = \cos x$, squaring gives $9\sin^2 x = \cos^2 x = 1 - \sin^2 x$ | | |
| $10\sin^2 x = 1$, $\sin x = \pm\sqrt{0.1}$ | M1 | Complete method for finding at least one value for $\sin x$ |
| $x = -2.820, -0.322, 0.322, 2.820$ | | |
| Select genuine roots $0.322, -2.820$ | A1 | Both correct roots and no others in the range |
| When $x = 0.322$ or $x = -2.82$, $y = 0.9$ | A1 | Allow awrt 0.90. Notice 0.9 is exact. |
| **[6]** | | |
---3
\begin{enumerate}[label=(\alph*)]
\item Sketch the graph of $\mathrm { y } = \arctan \mathrm { x }$ where $x$ is in radians.
\item In this question you must show detailed reasoning.
Find all points of intersection of the curves $\mathrm { y } = 3 \sin \mathrm { xcos } \mathrm { x }$ and $\mathrm { y } = \cos ^ { 2 } \mathrm { x }$ for $- \pi \leqslant x \leqslant \pi$.
\end{enumerate}
\hfill \mbox{\textit{OCR MEI Paper 1 2022 Q3 [8]}}