Standard +0.3 This question requires applying Maclaurin series for sin x and cos x (standard A-level further maths content), then solving a quadratic equation. The steps are straightforward: expand cos x ≈ 1 - x²/2 and sin x ≈ x, substitute, collect terms, then solve the resulting quadratic. While it involves multiple steps and further maths content, it's a direct application of standard techniques without requiring novel insight or complex problem-solving.
3.
a) Show that when \(x\) is small, \(2 \cos x - 3 \sin x\) can be written as \(a + b x + c x ^ { 2 }\), where \(a , b\) and \(c\) are integers to be found.
b) Hence find a small positive value of \(x\) that is an approximate solution to \(2 \cos x - 3 \sin x = 7 x\)
3.\\
a) Show that when $x$ is small, $2 \cos x - 3 \sin x$ can be written as $a + b x + c x ^ { 2 }$, where $a , b$ and $c$ are integers to be found.\\
b) Hence find a small positive value of $x$ that is an approximate solution to $2 \cos x - 3 \sin x = 7 x$\\
\hfill \mbox{\textit{SPS SPS SM Pure 2023 Q3 [4]}}