Given that $y-2={{\left( x+1 \right)}^{\ln \left( x+1 \right)}}$ , where $x-1$, sh

Question

Given that $y-2={{\left( x+1 \right)}^{\ln \left( x+1 \right)}}$ , where $x>-1$, show that $\left( x+1 \right)\frac{\mathrm{d}y}{\mathrm{d}x}=2\left( y-2 \right)\ln \left( x+1 \right)$.[2] Find the Maclaurin series of $y$ up to and including the term in ${{x}^{2}}$.[3] 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]

Accepted Answer

(a) $y \approx 3 + x^2$ (b) $p = 4$, $q = 4$, $r = 6$

2022 JPJC J2 MYE P2 Q2

2022 JPJC J2 MYE P2 Q2

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