| Exam Board | OCR MEI |
|---|---|
| Module | Paper 3 (Paper 3) |
| Year | 2021 |
| Session | November |
| Marks | 3 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Complex Numbers Arithmetic |
| Type | Argument relationships and tangent identities |
| Difficulty | Moderate -0.8 This question tests basic properties of arctan through geometric interpretation and function properties. Part (a) is a standard identity provable from a right triangle, part (b) is routine graph sketching, and part (c) asks for recognition that arctan is increasing (monotonic). All parts require recall and straightforward application rather than problem-solving or novel insight. |
| Spec | 1.05f Trigonometric function graphs: symmetries and periodicities1.05i Inverse trig functions: arcsin, arccos, arctan domains and graphs |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| \(\arctan\left(\frac{1}{x}\right) = \text{angle BEA} = \frac{\pi}{2} - \theta\); So \(\arctan x + \arctan\left(\frac{1}{x}\right) = \frac{\pi}{2}\) | E1 | 2.4 — Convincing explanation of given result. Must relate to triangle. Do not need to mention angle BEA |
| [1] |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| Sketch of \(y = \arctan x + \arctan\left(\frac{1}{x}\right)\): horizontal line (for \(x > 0\)), with a step/discontinuity, and horizontal line for \(x < 0\) | B1 | 1.2 — Ignore any values on axes |
| [1] |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| Increasing [function] | E1 | 2.4 |
| [1] |
## Question 13(a):
| Answer/Working | Mark | Guidance |
|---|---|---|
| $\arctan\left(\frac{1}{x}\right) = \text{angle BEA} = \frac{\pi}{2} - \theta$; So $\arctan x + \arctan\left(\frac{1}{x}\right) = \frac{\pi}{2}$ | E1 | 2.4 — Convincing explanation of given result. Must relate to triangle. Do not need to mention angle BEA |
| | **[1]** | |
---
## Question 13(b):
| Answer/Working | Mark | Guidance |
|---|---|---|
| Sketch of $y = \arctan x + \arctan\left(\frac{1}{x}\right)$: horizontal line (for $x > 0$), with a step/discontinuity, and horizontal line for $x < 0$ | B1 | 1.2 — Ignore any values on axes |
| | **[1]** | |
---
## Question 13(c):
| Answer/Working | Mark | Guidance |
|---|---|---|
| Increasing [function] | E1 | 2.4 |
| | **[1]** | |
---13
\begin{enumerate}[label=(\alph*)]
\item Use triangle ABE in Fig. C 2 to show that $\arctan x + \arctan \left( \frac { 1 } { x } \right) = \frac { \pi } { 2 }$, as given in line 29 .
\item Sketch the graph of $\mathrm { y } = \arctan \mathrm { x }$.
\item What property of the arctan function ensures that $\mathrm { y } > \frac { 1 } { \mathrm { x } } \Rightarrow \arctan y > \arctan \left( \frac { 1 } { \mathrm { x } } \right)$, as given in line 30 ?
\end{enumerate}
\hfill \mbox{\textit{OCR MEI Paper 3 2021 Q13 [3]}}