October 2021 Paper 3 Q14
14
The questions in this section refer to the article on the Insert. You should read the article before attempting the questions.
The relevant parts of the article “Adding arctangents” are reproduced below; the line numbers are those printed on the Insert.
Line 7
It can be shown that \(\arctan\left(\frac{1}{2}\right) + \arctan\left(\frac{1}{3}\right) = \arctan 1\).Lines 22–23
The arctangent addition formula is a further generalization:
\(\arctan x + \arctan y = \arctan\left(\dfrac{x + y}{1 - xy}\right)\), as long as \(xy < 1\).Lines 39–40
• For \(n\) a positive integer, \(\arctan\left(\dfrac{1}{n+1}\right) + \arctan\left(\dfrac{1}{n^2+n+1}\right) = \arctan\left(\dfrac{1}{n}\right)\); this follows directly from the arctan addition formula in line 23.
\(\arctan\left(\dfrac{1}{n+1}\right) + \arctan\left(\dfrac{1}{n^2+n+1}\right) = \arctan\left(\dfrac{1}{n}\right) \Rightarrow \arctan\left(\dfrac{1}{2}\right) + \arctan\left(\dfrac{1}{3}\right) = \arctan 1.\) [1]
\(\arctan\left(\dfrac{1}{n+1}\right) + \arctan\left(\dfrac{1}{n^2+n+1}\right) = \arctan\left(\dfrac{1}{n}\right)\), as given in line 39. [4]
| Scheme | Marks | AO |
|---|---|---|
| If \(n = 1\), \(n + 1 = 2\) and \(n^2 + n + 1 = 3\) | E1 | 2.2a |
| [1] |
| Scheme | Marks | AO |
|---|---|---|
| For \(n\) a positive integer, \(\frac{1}{n}\left(\frac{1}{n^2+n+1}\right) < 1\) | E1 | 2.3 |
| \(\arctan\left(\dfrac{1}{n+1}\right) + \arctan\left(\dfrac{1}{n^2+n+1}\right)\) \(= \arctan\left(\dfrac{\left(\frac{1}{n+1}\right) + \left(\frac{1}{n^2+n+1}\right)}{1 - \left(\frac{1}{n+1}\right)\left(\frac{1}{n^2+n+1}\right)}\right)\) | M1 | 3.1a |
| \(\arctan\left(\dfrac{n^2 + n + 1 + n + 1}{(n^2 + n + 1)(n + 1) - 1}\right)\) | M1 | 1.1 |
| \(\arctan\left(\dfrac{n^2 + 2n + 2}{n^3 + 2n^2 + 2n}\right) = \arctan\left(\dfrac{n^2 + 2n + 2}{n(n^2 + 2n + 2)}\right)\) \(\qquad = \arctan\left(\dfrac{1}{n}\right)\) | A1 | 2.1 |
| [4] |
Notes
M1: Use of arctan addition formula
M1: Clearing fractions within the fraction
Condone omission of arctan for this mark and condone 1 other error
A1: Convincing completion (AG)