2022 RI P1 Q7
It is given that $y=\sqrt{2+{{\cos }^{2}}x}$.
(a)
Show that
(i) $2y\frac{\text{d}y}{\text{d}x}=-\sin 2x$,
[1]
(ii) $y\frac{{{\text{d}}^{2}}y}{\text{d}{{x}^{2}}}+{{\left( \frac{\text{d}y}{\text{d}x} \right)}^{2}}=-\cos 2x$.
[1]
(b)
Hence find the Maclaurin series of $y$, up to and including the term in ${{x}^{4}}$.
[5]
(c)
By substituting $x=\frac{\pi }{6}$, show that $\sqrt{11}\approx 2\sqrt{3}\left( 1-\frac{{{\pi }^{2}}}{216}+\frac{{{\pi }^{4}}}{31104} \right)$
[2]
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