Further Pure Mathematics WFM01: DIFFERENTIATTION Topical Past Paper Questions

$$ y=\operatorname{artanh}\left(\frac{x}{\sqrt{1+x^{2}}}\right) $$ (a) Show that $$ \frac{\mathrm{d}y}{\mathrm{d}x}=\frac{1}{\sqrt{1+x^{2}}} $$ (b) Hence, by integrating the result in part (a), prove that $$ \operatorname{artanh}\left(\frac{x}{\sqrt{1+x^{2}}}\right)=\operatorname…

The hyperbola $ H $ has equation $ \frac{x^{2}}{64} - \frac{y^{2}}{49} = 1 $ and eccentricity $ e $. (a) Show that $ e = \frac{\sqrt{113}}{8} $ (2) The point $ (8\sec\theta, 7\tan\theta) $, where $ 0 (1) The normal to H at P • has equation $8x\cos\theta + 7y\cot\theta = 113$ - me…

The ellipse E has equation $$ \frac{x^{2}}{25}+\frac{y^{2}}{9}=1 $$ The line $l$ is the normal to $E$ at the point $P(5\cos\theta,3\sin\theta)$ where $0 (5)