2022 JPJC J2 MYE P2 Q2
(a)
Given that $y-2={{\left( x+1 \right)}^{\ln \left( x+1 \right)}}$ , where $x>-1$, show that
$\left( x+1 \right)\frac{\text{d}y}{\text{d}x}=2\left( y-2 \right)\ln \left( x+1 \right)$.
[2]
Find the Maclaurin series of $y$ up to and including the term in ${{x}^{2}}$.
[3]
(b)
The diagram (not drawn to scale) shows a triangle $OAB$ with $OA=6$cm, $OB=4$ cm and $\angle AOB=\theta $ radians. Given that $OB$ and $OC$ are radii of a circle with centre $O$ and $\theta$ is sufficiently small angle, show that the perimeter of $ABC$ can be approximated by $p+q\theta +r{{\theta }^{2}}$, where $p$, $q$ and $r$ are constants to be determined.
[5]
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