June 2025 Paper 1 Q8
8 Show that, for small values of \(\boldsymbol{x}\), the graph with equation
\[y = \frac{4 + \sin 5x - 3\cos 2x}{1 + 2\tan x}\]can be approximated by the straight line with equation of the form
\[y = ax + 1\]where \(a\) is a constant to be found. [5 marks]
| Scheme | Marks | AO |
|---|---|---|
| Replaces \(\sin 5x\) with \(5x\) Or Replaces \(\tan x\) with \(x\) In \(\dfrac{4 + \sin 5x - 3\cos 2x}{1 + 2\tan x}\) | M1 | 1.1a |
| Obtains \(3\left(1 - \dfrac{(2x)^2}{2}\right)\) Condone \(3\left(1 - \dfrac{2x^2}{2}\right)\) | M1 | 1.1a |
| Obtains \(\dfrac{4 + 5x - 3\left(1 - \dfrac{(2x)^2}{2}\right)}{1 + 2x}\) OE | A1 | 1.1b |
| Uses a valid method to simplify their fraction to a linear form. May see evidence of factorising, algebraic division or comparing coefficients. | M1 | 3.1a |
| Completes a reasoned argument to obtain \(y = 3x + 1\) | R1 | 2.1 |
| (5 marks) |