Moderate -0.3 This is a straightforward small-angle approximation question requiring recall of sin θ ≈ θ and tan θ ≈ θ for small θ, then substitution to show y ≈ 5 + 2x + 4x = 5 + 6x (a linear form). It's slightly easier than average as it's a standard technique with clear direction ('show that... straight line'), requiring only basic series knowledge and simple algebra, though the composite angles (x/2, x/3) add minor complexity.
Question 5:
5 | Uses small angle approximation for
sinxor tanx
Condone y =5+4x+12x for this
mark | AO1.1a | M1 | x x
y =5+4sin +12tan
2 3
sinx≈ x,tanx≈ x
x x
y ≈5+4 +12
2 3
y ≈6x+5
which is the equation of a straight
line.
Obtains correct equation
Allow unsimplified form | AO1.1b | A1
Concludes that the graph can be
approximated by a straight line.
Requires simplification of equation
(condone equals) and statement. | AO2.1 | R1
Total | 3
Q | Marking Instructions | AO | Marks | Typical Solution
Show that, for small values of $x$, the graph of $y = 5 + 4\sin\frac{x}{2} + 12\tan\frac{x}{3}$ can be approximated by a straight line.
[3 marks]
\hfill \mbox{\textit{AQA Paper 3 2018 Q5 [3]}}