2022 NYJC Promo Q7
(a)
A curve has equation $y=\sqrt{1+{{\tan }^{-1}}x}$. Show that $y\frac{{{\text{d}}^{2}}y}{\text{d}{{x}^{2}}}+{{\left( \frac{\text{d}y}{\text{d}x} \right)}^{2}}=-\frac{x}{{{\left( 1+{{x}^{2}} \right)}^{2}}}$.
By further differentiation, obtain the Maclaurin series for $y$, up to and including the term in ${{x}^{3}}$.
State the equation of the tangent to the curve at $x=0$.
[6]
(b)
Given that $x$ is a sufficiently small angle, show that $\sqrt{1+2\cos x}\approx \sqrt{3}\left( 1-\frac{{{x}^{2}}}{6} \right)$.
It is also given that $1-10\sin x=\sqrt{1+2\cos x}$, form a quadratic equation in $x$ and solve it.
[4]
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