OCR MEI Paper 3 2021 November — Question 15 4 marks

Exam BoardOCR MEI
ModulePaper 3 (Paper 3)
Year2021
SessionNovember
Marks4
PaperDownload PDF ↗
Mark schemeDownload PDF ↗
TopicAddition & Double Angle Formulae
TypeFind exact trigonometric values
DifficultyChallenging +1.8 This requires applying the addition formula for arctan (tan(A+B) = (tan A + tan B)/(1 - tan A tan B)) multiple times with careful algebraic manipulation, recognizing when angles sum to π, and handling the domain restrictions of arctan. It's significantly harder than routine formula application, requiring strategic insight about how to combine the terms and interpret the result, but is a known result that appears in Further Maths contexts.
Spec1.05i Inverse trig functions: arcsin, arccos, arctan domains and graphs1.05l Double angle formulae: and compound angle formulae
15 Prove that \(\arctan 1 + \arctan 2 + \arctan 3 = \pi\), as given in line 41 . \section*{END OF QUESTION PAPER} \section*{OCR
Oxford Cambridge and RSA}
Question 15:
AnswerMarks Guidance
Working/AnswerMark Guidance
Use of \(\arctan x + \arctan\left(\frac{1}{x}\right) = \frac{\pi}{2}\)M1 e.g. \(\arctan 2 + \arctan\left(\frac{1}{2}\right) = \frac{\pi}{2}\) — Allow valid alternative methods
\(\arctan\left(\frac{1}{2}\right) + \arctan 2 + \arctan\left(\frac{1}{3}\right) + \arctan 3 = \pi\)M1 Oe
Use of \(\arctan\left(\frac{1}{2}\right) + \arctan\left(\frac{1}{3}\right) = \arctan 1\)M1 Oe
\(\Rightarrow \arctan 1 + \arctan 2 + \arctan 3 = \pi\)E1 Convincing completion possibly using the three results above (AG)
Alternative method:
AnswerMarks Guidance
Working/AnswerMark Guidance
\(\arctan 2 + \arctan 3 = \arctan\left(\frac{2+3}{1-6}\right) + \pi\)M1 Use of \(\arctan x + \arctan y = \arctan\left(\frac{x+y}{1-xy}\right) + \pi\)
\(\arctan 2 + \arctan 3 = \arctan(-1) + \pi\)M1
\(\arctan 1 = -\arctan(-1)\)M1
\(\arctan 2 + \arctan 3 = -\arctan 1 + \pi\) \(\Rightarrow \arctan 1 + \arctan 2 + \arctan 3 = \pi\)E1
Total: [4]
## Question 15:

| Working/Answer | Mark | Guidance |
|---|---|---|
| Use of $\arctan x + \arctan\left(\frac{1}{x}\right) = \frac{\pi}{2}$ | M1 | e.g. $\arctan 2 + \arctan\left(\frac{1}{2}\right) = \frac{\pi}{2}$ — Allow valid alternative methods |
| $\arctan\left(\frac{1}{2}\right) + \arctan 2 + \arctan\left(\frac{1}{3}\right) + \arctan 3 = \pi$ | M1 | Oe |
| Use of $\arctan\left(\frac{1}{2}\right) + \arctan\left(\frac{1}{3}\right) = \arctan 1$ | M1 | Oe |
| $\Rightarrow \arctan 1 + \arctan 2 + \arctan 3 = \pi$ | E1 | Convincing completion possibly using the three results above **(AG)** |

**Alternative method:**

| Working/Answer | Mark | Guidance |
|---|---|---|
| $\arctan 2 + \arctan 3 = \arctan\left(\frac{2+3}{1-6}\right) + \pi$ | M1 | Use of $\arctan x + \arctan y = \arctan\left(\frac{x+y}{1-xy}\right) + \pi$ |
| $\arctan 2 + \arctan 3 = \arctan(-1) + \pi$ | M1 | |
| $\arctan 1 = -\arctan(-1)$ | M1 | |
| $\arctan 2 + \arctan 3 = -\arctan 1 + \pi$ $\Rightarrow \arctan 1 + \arctan 2 + \arctan 3 = \pi$ | E1 | |

**Total: [4]**
15 Prove that $\arctan 1 + \arctan 2 + \arctan 3 = \pi$, as given in line 41 .

\section*{END OF QUESTION PAPER}
\section*{OCR \\
 Oxford Cambridge and RSA}

\hfill \mbox{\textit{OCR MEI Paper 3 2021 Q15 [4]}}
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Q1 5 Q2 2 Q3 7 Q4 3 Q5 8 Q6 4 Q7 3 Q8 3 Q9 11 Q10 9 Q11 5 Q12 3 Q13 3 Q14 5 Q15 4