Further Pure Mathematics WFM01: BINOMIAL THEOREM Topical Past Paper Questions

(a) Determine, in ascending powers of $ \left(x-\frac{\pi}{6}\right) $ up to and including the term in $ \left(x-\frac{\pi}{6}\right)^{3} $, the Taylor series expansion about $ \frac{\pi}{6} $ of $$ y=\tan\left(\frac{3x}{2}\right) $$ giving each coefficient in simplest form. (7)…

$$ \frac{\mathrm{d}y}{\mathrm{d}x}=y^{2}-x $$ (a) Show that $$ \frac{\mathrm{d}^{4}y}{\mathrm{d}x^{4}}=A y\frac{\mathrm{d}^{3}y}{\mathrm{d}x^{3}}+B\frac{\mathrm{d}y}{\mathrm{d}x}\frac{\mathrm{d}^{2}y}{\mathrm{d}x^{2}} $$ where A and B are integers to be determined. (4) Given that…