| Exam Board | OCR MEI |
|---|---|
| Module | C3 (Core Mathematics 3) |
| Marks | 4 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Reciprocal Trig & Identities |
| Type | Inverse trigonometric function equations |
| Difficulty | Moderate -0.8 Both parts are straightforward inverse trig equations requiring basic manipulation and knowledge of standard values. Part (i) is simple rearrangement and evaluation; part (ii) uses the complementary angle relationship sin⁻¹(x) + cos⁻¹(x) = π/2, leading to x = 1/√2. No problem-solving insight required, just direct application of definitions and standard results. |
| Spec | 1.05i Inverse trig functions: arcsin, arccos, arctan domains and graphs |
| Answer | Marks |
|---|---|
| (i) \(\arcsin x = \frac{\pi}{6}\) \(\Rightarrow\) \(x = \sin \frac{\pi}{6}\) | M1 |
| \(= \frac{1}{2}\) | A1 |
| [2] | allow unsupported answers |
| (ii) \(\sin \frac{\pi}{4} = \cos \frac{\pi}{4} = \frac{1}{\sqrt{2}}\) | B1 |
| \(\arcsin \left(\frac{1}{\sqrt{2}}\right) = \arccos \left(\frac{1}{\sqrt{2}}\right)\) | B1 |
| \(x = \frac{1}{\sqrt{2}}\) | B2 |
| [2] | o.e. e.g. \(\frac{\sqrt{2}}{2}\), must be exact; SCB1 for 0.707… |
| Answer | Marks |
|---|---|
| (i) \(a = 0, b = 3, c = 2\) or \(a = 0, b = -3, c = -2\) | B2 |
| \(B(2,1,0)\) | B1 |
| Answer | Marks |
|---|---|
| (ii) \(a = 1, b = -1, c = 1\) or \(a = 1, b = 1, c = -1\) | B2 |
| \(B(2,1,0)\) | B1 |
| Answer | Marks |
|---|---|
| Let \(\arcsin x = \theta\) | M1 |
| \(x = \sin \theta\) | M1 |
| \(\theta = \arccos y\) \(\Rightarrow\) \(y = \cos \theta\) | E1 |
| \(\sin^2 \theta + \cos^2 \theta = 1\) | M1 |
| \(x^2 + y^2 = 1\) | E1 |
| Answer | Marks |
|---|---|
| (i) period \(180°\) | B1 |
| [1] | condone \(0° \leq x \leq 180°\) or \(\pi\) |
| (ii) one-way stretch in \(x\)-direction, scale factor \(\frac{1}{2}\) | M1, A1 |
| translation in \(y\)-direction through \(\begin{pmatrix} 0 \\ 1 \end{pmatrix}\) | M1, A1 |
| [4] | condone 'squeeze', 'contract' for M1; stretch used and s.f. \(\frac{1}{2}\) |
| Answer | Marks |
|---|---|
| (iii) correct shape, touching \(x\)-axis at \(-90°, 90°\) | M1 |
| domain \(-180° \leq x \leq 180°\) | B1 |
| \((0, 2)\) marked or indicated (amplitude is 2) | A1 |
Question 1:
(i) $\arcsin x = \frac{\pi}{6}$ $\Rightarrow$ $x = \sin \frac{\pi}{6}$ | M1
$= \frac{1}{2}$ | A1
[2] | allow unsupported answers
(ii) $\sin \frac{\pi}{4} = \cos \frac{\pi}{4} = \frac{1}{\sqrt{2}}$ | B1
$\arcsin \left(\frac{1}{\sqrt{2}}\right) = \arccos \left(\frac{1}{\sqrt{2}}\right)$ | B1
$x = \frac{1}{\sqrt{2}}$ | B2
[2] | o.e. e.g. $\frac{\sqrt{2}}{2}$, must be exact; SCB1 for 0.707…
Question 2:
(i) $a = 0, b = 3, c = 2$ or $a = 0, b = -3, c = -2$ | B2
$B(2,1,0)$ | B1
[3]
(ii) $a = 1, b = -1, c = 1$ or $a = 1, b = 1, c = -1$ | B2
$B(2,1,0)$ | B1
[3]
Question 3:
Let $\arcsin x = \theta$ | M1
$x = \sin \theta$ | M1
$\theta = \arccos y$ $\Rightarrow$ $y = \cos \theta$ | E1
$\sin^2 \theta + \cos^2 \theta = 1$ | M1
$x^2 + y^2 = 1$ | E1
[3]
Question 4:
(i) period $180°$ | B1
[1] | condone $0° \leq x \leq 180°$ or $\pi$
(ii) one-way stretch in $x$-direction, scale factor $\frac{1}{2}$ | M1, A1
translation in $y$-direction through $\begin{pmatrix} 0 \\ 1 \end{pmatrix}$ | M1, A1
[4] | condone 'squeeze', 'contract' for M1; stretch used and s.f. $\frac{1}{2}$
condone 'move', 'shift', etc for M1; 'translation' used, $+1$ unit
$\begin{pmatrix} 0 \\ 1 \end{pmatrix}$ only is M1 A0
(iii) correct shape, touching $x$-axis at $-90°, 90°$ | M1
domain $-180° \leq x \leq 180°$ | B1
$(0, 2)$ marked or indicated (amplitude is 2) | A1
[3]1 Solve each of the following equations, giving your answers in exact form.\\
(i) $6 \arcsin x - \pi = 0$.\\
(ii) $\arcsin x = \arccos x$.
\hfill \mbox{\textit{OCR MEI C3 Q1 [4]}}